Lecture notes 11 Nonlinear Dynamics YFX1520
Lecture 11 Feigenbaumrsquos analysis of period doublingsuperstability of fixed points and period-p points renor-malisation universal limiting function discrete time dy-namics analysis methods Poincareacute section Poincareacute mapLorenz section attractor reconstruction
Contents
1 Feigenbaumrsquos analysis of period doubling 211 Feigenbaum constants 212 Superstable fixed points and period-p points 213 Renormalisation and Feigenbaum constants 514 Numerical determination of Feigenbaum α constant 7
2 Discrete time analysis methods 821 Poincareacute section Poincareacute map return map 9
211 Example Periodically driven damped Duffing oscillator 9212 Example Roumlssler attractor 11
3 Lorenz section 13
4 Attractor reconstruction 14
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Lecture notes 11 Nonlinear Dynamics YFX1520
1 Feigenbaumrsquos analysis of period doubling
In this lecture we are continuing our study of one-dimensional maps as simplified models of chaos and astools for analysing higher order differential equations We have already encountered maps in this role TheLorenz map provided strong evidence that the Lorenz attractor is truly strange and is not just a long-period limit-cycle see Lectures 9 and 10
11 Feigenbaum constants
The Feigenbaum constants were presented during Lecture 10
Slide 3
1-D unimodal maps and Feigenbaum constants
= limn1
n1
n= lim
n1
rn1 rn2
rn rn1 4669201609 (1)
crarr = limn1
dn1
dn 2502907875 (2)
DKartofelev YFX1520 3 30
In the above orbit diagram xm = max f(x) is the maximum of the unimodal map graph The Feigen-baum constants are valid up to the onset of chaos at the accumulation point r = rinfin and inside eachperiodic window for r gt rinfin The Feigenbaum constants are universal the same convergencerate appears no matter what unimodal map is iterated They are new mathematical constants asbasic to period doubling as π is to circles
Showing the values of Feigenbaum constants to 30 decimal places
δ = 4669 201 609 102 990 671 853 203 821 578 (1)
α = minus2502 907 875 095 892 822 283 902 873 218 (2)
In this lecture wersquoll take a closer look at Feigenbaum α constant
12 Superstable fixed points and period-p points
Figure 1 Superstable fixed point where f prime(xlowast) = 0 and xlowast = xm
The fixed point defined byf(xlowast) = xlowast (3)
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Lecture notes 11 Nonlinear Dynamics YFX1520
is said to be superstable if its slopef prime(xlowast) = 0 (4)
This hold true when the maximum (or minimum) of map function f is the fixed point
xlowast = xm (5)
Figures 1 2 and Slide 4 show the superstable fixed point in the case of unimodal Logistic map
Slide 4Superstable fixed point (Logistic map)
f(x r) = x
x
= xm and f0(x
r) = 0 ) r = 2 (3)
Convergence of xn about the non-trivial fixed point x
n+1 =|f 00(x
r)|2
2n +O(3n) (4)
is quadratic The iterates xn converge quadraticallyDKartofelev YFX1520 4 30
In Lecture 9 we showed that the stability of fixed point xlowast depends on the absolute slope of f at xlowastie |f prime(xlowast)| If |f prime(xlowast)| lt 1 then the fixed point is stable and for |f prime(xlowast)| gt 1 it is unstable
mdashmdashmdashmdash Skip if needed start mdashmdashmdashmdashThe linearisation of a map is derived by studying the slightly perturbed iterate
xn = xlowast + ηn (6)
where |η| ≪ 1 We study the next iterate dynamics through linearisation of map f
xn+1 = f(xn) = f(xlowast + ηn) =
983063Taylor series
about xn = xlowast
983064= f(xlowast) +
f prime(xlowast)
1(xlowast + ηn minus xlowast) +O(η2n) asymp
asymp xlowast + f prime(xlowast)ηn
(7)
xn+1 = xlowast + f prime(xlowast)ηn +O(η2n) (8)
xn+1 minus xlowast983167 983166983165 983168=ηn+1
= f prime(xlowast)ηn +O(η2n) (9)
The left-hand side of the result follows from definition (6) The linearised form of map f is obtainedby neglecting the higher order terms O(η2n)
ηn+1 = |f prime(xlowast)|ηn (10)
mdashmdashmdashmdash Skip if needed stop mdashmdashmdashmdashWe know that iterates xn converge towards stable fixed point xlowast where f prime(xlowast) lt 1 or more precisely
the slightly perturbed iterates xn will But what happens for superstable fixed point xlowast How doesthe solution converge for f prime(xlowast) = 0 Obviously the linearisation canrsquot be used We need to includean additional term in the Taylor series expansion
ηn+1 = |f prime(xlowast)|983167 983166983165 983168=0
ηn +|f primeprime(xlowast)|
2η2n +O(η3n) =
|f primeprime(xlowast)|2
η2n +O(η3n) (11)
D Kartofelev 314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
ηn+1 =|f primeprime(xlowast)|
2η2n +O(η3n) (12)
This result means that iterates converge faster than exponentially when they approach a superstablefixed point A very beneficial property to keep in mind when analysing maps numerically (less calcu-lations required to reach results)
The following interactive numerical file demonstrates the superstable convergence of iterate xn in the caseof Logistic map
Numerics cdf1 nb1Comparison of superstable and stable fixed points in Logistic mapComparison of map iterate convergence to superstable fixed point and a stable fixed point
Out[]=
|f (x)| = 0
00 02 04 06 08 1000
02
04
06
08
10
x n+1
0 5 10 15 2000
02
04
06
08
10
x n
00 02 04 06 08 1000
02
04
06
08
10
xn
x n+1
0 5 10 15 2000
02
04
06
08
10
n
x n
Figure 2 (Top) Cobweb diagram and map iterates of Logistic map featuring superstable fixed pointInitial condition x0 = 01 and parameter r = 2 (Bottom) Cobweb diagram and map iterates of Logisticmap featuring stable fixed point Initial condition x0 = 01 and parameter r = 285
Can there be superstable period-p points
Slides 5ndash7Superstable period-p point (Logistic map)
Superstable period-p orbit xm = argmax f is also a local min ormax of f p map (p-th iterate of map f)
For example in the case of period-2 point
f2(x
r) = x (x = xm) ) r = 1 +
p5 (5)
f2(x
r)0=
d
dx [f(f(x r))] = f
0(f(x r)) middot f 0(x
r) = 0 (6)
DKartofelev YFX1520 5 30
D Kartofelev 414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Superstable period-p point fixed point xlowast that is also the local minimum or maximum of map fpAlgebraically we write
fp(xlowast) = xlowast (13)
where xlowast = xm The superstable point of map f where f(xlowast) = xlowast is also one of the points in thesuperstable period-p orbit
These superstable points are shown below on the orbit diagram where rn is the onset of the period-2n
orbit (bifurcation point) and r = Rn is where the period-2n orbit is superstable The Rn occurs alwaybetween rn and rn+1 Unsurprisingly the ratios of succeeding Rn also converges to δ for n rarr infin
Orbit diagram superstable period-2n points
rn ndash stable period-2n orbit is born (bifurcation point)Rn ndash superstable period-2n point
limn1
Rn1
Rn= (7)
DKartofelev YFX1520 6 30
Period doubling bifurcation points
Values of bifurcation points and superstable points of few first perioddoublings in the Logistic map
r0 = 10 non-trivial R0 = 20 fpr1 = 30 R1 = 1 +
p5 323607 period-2
r2 = 1 +p6 344949 R2 349856 period-4
r3 354409 R3 355464 period-8r4 356441 R4 356667 period-16r5 356875 R5 356924 period-32
r1 3569945672 R1 = r1 356994567 period-21
r1 ndash onset of chaos (accumulation point)
limn1
rn1 rn2
rn rn1= lim
n1
Rn1
Rn= (8)
DKartofelev YFX1520 7 30
13 Renormalisation and Feigenbaum constants
Letrsquos try to tap into the observation presented in Lecture 10 that the orbit diagram of unimodal maps isself-similar under magnification How do the scaling constant α and δ govern the period doubling bifur-cations in the Logistic and other unimodal maps
Figure 3 (Left) Regions of interest the proximity to r = R0 and r = R1 (Right) Orbit diagram showingthe bifurcation points rn or the onsets of the period-2n orbits and superstable Rn-s where the period-2n
orbits are superstable cf Slide 6
Next we consider the local dynamics of the orbit diagram near r = R0 and then compare it to thesituation near r = R1 see Fig 3 We renormalise one region into another
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Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 8 9
Feigenbaumrsquos analysis renormalisation
f(xR0) f2(xR1)
(a) (b) (c)
DKartofelev YFX1520 8 30
Feigenbaumrsquos analysis renormalisation
f(xR0) f2(xR1) crarrf
2x
crarr R1
(a) (b) (c)
Read Mitchell J Feigenbaum ldquoThe universal metric properties ofnonlinear transformationsrdquo Journal of Statistical Physics 21(6) pp669ndash706 1979 Also relevant to the following three slides +
D Kartofelev YFX1520 9 30
The visual comparison shows that renormalised map is very similar to the original one The map max-imum xm shown on Slide 9 has beed shifted to the origin in order to simplify the math below
Letrsquos express the renormalisation procedure algebraically
f(xR0) asymp αf2983059xα R1
983060 (14)
notice that we have scaled the vertical and horizontal directions simultaneously In summary map fhas been renormalised by taking its second iterate rescaling x rarr xα shifting r to the next superstablevalue and multiplying everything by α
The higher order renormalisations of f(xR0) are given by
f(xR0) asymp α2f4983059 x
α2 R2
983060
asymp α3f8983059 x
α3 R3
983060
asymp α4f16983059 x
α4 R4
983060
asymp αnf (2n)983059 x
αn Rn
983060
(15)
What happens when we reach the accumulation point r = rinfin mdash reach the chaotic regime
limnrarrinfin
αnf (2n)983059 x
αn Rn
983060= g0(x) (16)
Feigenbaum found numerically that this limit converges only for α = minus2502907875 by producing the re-sulting limiting function g0 The function g0 is called the universal limiting function This function issimilar in shape to map f and it retains the superstability of its fixed point
The descriptor ldquouniversalrdquo in this context means that function g0 is almost independent of the unimodalfunction f How can g0 be independent of f Function g0 depends on f only through its behaviour nearx = 0 since thatrsquos all that survives in the argument xαn as n rarr infin With each renormalisation we areblowing up smaller and smaller neighbourhoods of the maximum of f until practically all of the information(global shape further away from x = 0) of f becomes meaningless The only remaining information is thequadratic nature of the maximum In the case of unimodal maps the Taylor expansion at xlowast = xm has tohave the second (quadratic) x2 term (superstability) The fixed point has to be what is called the seconddegree maximum point or the quadratic maximum
You can find the universal limiting functions related to any superstable period-2i orbit To obtain otheruniversal limiting functions gi(x) start with f(xRi) instead of f(xR0)
gi(x) = limnrarrinfin
αnf (2n)983059 x
αn Rn+i
983060 (17)
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Lecture notes 11 Nonlinear Dynamics YFX1520
What happens at the onset of chaos i = infin Letrsquos find the universal limiting function at the onset of chaoswhere Ri = Rinfin We need to renormalise f(xRinfin) one time
ginfin(x) asymp αf2983059xα Rinfin+1
983060=
983063Remainderinfin+ 1 = infin
983064= αf2
983059xα Rinfin
983060= g(x) (18)
For once we donrsquot have to shift r when we renormalise (we became independent of i) The limiting func-tion ginfin(x) usually called simply g(x) satisfies
g(x) = αg2983059xα
983060 (19)
This is a functional equation for g(x) and the universal scale factor α It is self-referential primaryunknown g(x) is defined in terms of itself
Letrsquos try to solve Eq (19) The functional equation is not complete until we specify boundary conditionson g(x) After the shift of origin all our unimodal f rsquos have a maximum at x = 0 so we require
gprime(0) = 0 (20)
Also we can set
g(0) =
983063From Def (18)we know that
983064= g(0 Rinfin) = const (21)
tog(0) = 1 (22)
without loss of generality (This just defines the normalised scale for x if g(x) is a solution of Eq (19) sois microg(xmicro) where micro isin R with the same α) Naturally this selection agrees with boundary condition (20)
Now we can solve Eq (19) for function g(x) and α at the boundary x = 0
g(0) = αg29830610
α
983062= αg2(0) = αg(g(0)) (23)
since g(0) = 1 we have1 = αg(1) (24)
α =1
g(1) (25)
The universal scaling constant α depends on the limiting function g As of now no closed form solutionexist for function g We can resort to numerical method to find the shape of the limiting function g andthe value of α Next we solve Eq (19) with boundary condition (25) using the power series solution
14 Numerical determination of Feigenbaum α constant
Slides 10 11
Limiting function g(x) and Feigenbaum constant crarr
Letrsquos consider a functional equation in the following form
g(x) = crarr g(gx
crarr
) = crarr g
2x
crarr
(9)
where crarr acts as scaling coecient and
crarr =1
g(1) (10)
The power series solution is obtained by assuming power expansion
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 + ex10 + (11)
which assumes that the map maximum is quadratic
DKartofelev YFX1520 10 30
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution1 the one term approximation where
g(x) = 1 + ax2 +O(x4) (12)
results in
a = 1
2(1 +
p3) crarr =
1
g(1) 273205 (92 error) (13)
-04 -02 00 02 04000204060810
xn
x x+1 g(x) (one term)
Logistic map
Logistic map shifted
1See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 11 30
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Lecture notes 11 Nonlinear Dynamics YFX1520
The power series expansion uses Maclaurin expansion and assumes a quadratic maximum Thequadratic maximum follows from the superstability property The above process can generate lots ofextraneous root you will need to shift through them to find the correct solution
The solution shown above is taken from the following numerical file The file is commented for the benefitof students who are interested in numerical methods in general
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
In order to find a better match more terms in the power series expansion are needed
Slide 12
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution2 the four term approximation where
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 +O(x10) (14)
Coecients a = 1528 b = 01053 c = 002631 d = 0003344and
crarr =1
g(1) 250316 (001 error) (15)
-04 -02 00 02 04000204060810
xn
x x+1 g(x)
Logistic map
Logistic map shifted
2See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 12 30
The solution shown above is taken from the following numerical file
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
The renormalisation theory also explains the value of δ Unfortunately the derivations and analysis ofthe Feigenbaum δ constant requires advanced functional analysis know-how (operators in function spaceFrechet derivatives etc) and for this reason is omitted from this courseConclusion Something qualitative (unimodality) gave us something quantitative (Feigenbaumrsquos constantsα and δ) In science it is usually the other way around
Reading suggestion
Link File name CitationPaper1 paper5pdf Mitchell J Feigenbaum ldquoThe universal metric properties of nonlinear transfor-
mationsrdquo Journal of Statistical Physics 21(6) pp 669ndash706 (1979)doi101007BF01107909
2 Discrete time analysis methods
We have already used mapping of a continuous three-dimensional flow into a discrete one-dimensional mapmdash Lorenz attractor rarr Lorenz map Our goal in this section is to generalise this approach
D Kartofelev 814 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
21 Poincareacute section Poincareacute map return map
Poincareacute section is a set of intersection points between a hyperspace hyper-surface or hyperplane anda trajectory of n-dimensional continuous time flow Poincareacute map or return map is the (algebraic)relationship between these preceding and succeeding intersection points ie point xn and xn+1 found onPoincareacute section
Slides 13ndash16
Poincare section
Intersection of an attractor trajectory with the hypersurface s
DKartofelev YFX1520 13 30
Discrete time dynamics analysis
Construction of a Poincare map (return map recurrence map)
DKartofelev YFX1520 14 30
Poincare section periodic forcing3
3Credit J Thompson H Stewart 1986 Nonlinear dynamics and chaos
geometrical methods for engineers and scientists Chichester UK Wiley
DKartofelev YFX1520 15 30
Poincare section periodic forcing
DKartofelev YFX1520 16 30
Letrsquos demonstrate these concepts using two examples
211 Example Periodically driven damped Duffing oscillator
Duffing equation (or Duffing oscillator) named after Georg Duffing (1861ndash1944) is a nonlinear second-orderdifferential equation used to model certain damped and driven oscillators
Slides 17 18The first term on the left-hand side of Eq (16 slide numbering) is the inertial term the third term isthe nonlinear stiffness and the fourth term is the damping The non-homogeneous right-hand sideof the equation describes the external 2π-periodic forcing with the amplitude F and frequency ω = 1The solution for F gt 0 may be a strange attractor The external forcing of two-dimensional systemsis often the source of chaos But wait how can two-dimensional system feature chaos Isnrsquot chaossupposed to be reserved only for three or higher-dimensional systems The external forcing term allowsfor this second order system to be represented as a three-dimensional system
D Kartofelev 914 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Periodically driven damped Dung oscillator
Figure Mechanical system featuring Dung oscillations
The non-autonomous equation of motion has the following form
x x+ x3 + x = F cost (16)
where is the damping coecient F and are the forcing strengthand frequency respectively
DKartofelev YFX1520 17 30
Periodically driven damped Dung oscillator
We introduce the variable exchange y = x and rewrite (16)
(x = y
y = x x3 y + F cost
(17)
The equivalent 3-D system is obtained by applying variable exchangez = t 8
gtlt
gt
x = y
y = x x3 y + F cos z
z =
(18)
The above folds since
z =
Zz dt =
Z dt =
Zdt = t+ C (19)
DKartofelev YFX1520 18 30
If you remove the damping and the external forcing terms the system becomes equivalent to theproblem of the particle in the double-well potential given by potential function
V (x) =1
4x4 minus 1
2x2 (26)
that was presentedstudied in Lecture 5 (go and check it out)
Figure 4 shows three ways you can think graphically about the solution of Duffing oscillator and itsPoincareacute section cf cdf3 nb3
Figure 4 (Left) Snapshots of a trajectory every 2π time units cf Slides 15 16 The 2π-periodicity isa natural choice since the external force is 2π-periodic for ω = 1 (Middle) A trajectory in the three-dimensional Cartesian xyz-space corresponding to Sys (18 slide numbering) (Right) A trajectory in thethree-dimensional space where the z-axis is 2π-periodic and is folded into a closed circle cf Slides 15 16The blue outline shows the Poincareacute section
The following numerical file shows the solution to Duffing oscillator and the quantitatively accuraterenderings of the three different ways to plot the solution of Duffing oscillator shows in Fig 4
Numerics cdf3 nb3Periodically forced damped Duffing oscillator Numerical solution and phase portrait 2π-periodicPoincareacute section of the Duffing oscillator (static)
The following slides show how the Poincareacute section of Duffing oscillator for t ≫ 1 changes as we continu-ously move it along the 2π-periodic time or z-axis as shown in Fig 5
D Kartofelev 1014 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Figure 5 Poincareacute section dynamics and periodic z-axis A moving Poincareacute section featuring a singleintersection point shown with the blue dot
Slides 19 20
Poincare section4 Dung oscillator
-15 -10 -05 00 05 10 15
-05
00
05
10
x(t)
y(t)
Poincareacute section
4See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 19 30
Poincare section dynamics Dung oscillator
No embedded video files in this pdf
DKartofelev YFX1520 20 30
Notice that the Poincareacute section is folding onto itself continuously stretching and the intersectionpoints are being progressively mixed (via re-injection process) within the region defined by theattractor This type of Poincareacute section dynamics is common for strange attractors
The dynamics of the Poincareacute section of Duffing oscillator is explored in the following numerical file
Numerics cdf4 nb4Periodically forced damped Duffing oscillator and dynamic Poincareacute section animation
212 Example Roumlssler attractor
Originally studied by Otto Roumlssler (1940ndash) The system is a product of pure imagination and it exhibitschaotic dynamics by design
D Kartofelev 1114 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 21ndash24
Rossler attractor
Rossler attractor5 is given by (as mentioned in Lecture 9)
8gtlt
gt
x = y x
y = x+ ay
z = b+ z(x c)
(20)
Chaotic solution exists for a = 01 b = 01 c = 14
5See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 21 30
Rossler attractor Poincare sections6
-10 -5 0 50
5
10
15
y(t)
z(t)
Poincareacute section yz-plane
6See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 22 30
The system has only one nonlinearity the zx term in the third equation
Rossler attractor Poincare sections7
-10 -5 0 5 100
5
10
15
20
25
30
x(t)
z(t)
Poincareacute section xz-plane
7See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 23 30
Rossler attractor return map and Poincare sections
Credit Y Maistrenko and R PaskauskasDKartofelev YFX1520 24 30
Slide 24 features the shape of Roumlssler map mdash the Poincareacute map of Roumlssler attractor
The following numerical file was used to calculate the attractor and the Poincareacute sections shown above
Numerics cdf5 nb5Numerical solution of Roumlssler attractor Poincareacute section of Roumlssler attractor
Slides 25 26
Rossler attractor orbit diagram
DKartofelev YFX1520 25 30
Rossler attractor period doubling
No embedded video files in this pdf
DKartofelev YFX1520 26 30
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Lecture notes 11 Nonlinear Dynamics YFX1520
The orbit diagram shown on Slide 25 is calculated using the Roumlssler map shown on Slide 24
The following interactive numerical file shows the period doubling bifurcation happening in the xy-projection of Roumlssler attractor cf Slide 25
Numerics cdf6 nb6Periodic orbits in Roumlssler attractor for varied c value
The following slide shows the 2π-periodic dynamics of the Poincareacute section of Roumlssler attractor
Slide 27Poincare section dynamics Rossler attractor
No embedded video files in this pdf
DKartofelev YFX1520 27 30
Once again notice the folding stretching and mixing of the Poincareacute section cf Slides 19 and20 This type of Poincareacute section dynamics will be explored further in the next weekrsquos lecture
3 Lorenz section
Slide 28
Lorenz section
Section of a sectionDKartofelev YFX1520 28 30
The slide above shows a Poincareacute section of a strange attractor where we sliced the attractor with aplane thereby exposing its cross section If we take a further one-dimensional slice or Lorenz sectionthrough the Poincareacute section we find an infinite set of points separated by gaps of various sizes Inthe next weekrsquos lecture we will look into this discovery and into the notion of Lorenz section in greaterdetail
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Lecture notes 11 Nonlinear Dynamics YFX1520
4 Attractor reconstruction
In applications it is often the case that you are able to measure or are restricted to only measuring onetime-series related to a higher order system mdash one aspect of a problem Letrsquos say you need to analyse theattractor that governs your measured one-dimensional time-series signal s(t) eg show that it is strangeAt first it might seem that there is not enough information But there exist a surprising data analysistechnique known as the attractor reconstruction
For systems governed by an attractor the dynamics in the full higher dimensional phase space canbe reconstructed from measurements of just a single time series s(t) Somehow a single variable carriessufficient information about all the others The method is based on time delays τ For instance define atwo-dimensional vector
983187x(t) = (s(t) s(t+ τ)) (27)
for some time delay τ gt 0 and signal s(t) where 983187x = (x y) Signal s(t) can therefore represent either thevariable x(t) or y(t) The time-series s(t) generates a trajectory 983187x(t) in a two-dimensional phase space thatrepresents its attractorrsquos trajectory You can also considered the attractor in three dimensions by definingthe three-dimensional vector
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)
for time delay τ gt 0 For four-dimensional attractor use
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)
etcThe following interactive numerical file demonstrates attractor reconstruction using three examples First
a sine wave is used to reconstruct its two-dimensional attractor mdash circle Second a quasi-periodic signalis used to reconstruct its three-dimensional attractor mdash torus Third time-series y(t) of Lorenz system isused to reconstruct the three-dimensional Lorenz attractor
Numerics cdf7 nb7Higher-dimensional attractor reconstruction from 1-D time-domain signals Examples sine wavequasi-periodic signal y(t) of Lorenz attractor
Revision questions
1 What are the values of Feigenbaum constants2 What are Feigenbaum constants3 Define superstable fixed point of a map4 Define superstable period-p point (or period-p orbit) of a map5 What are universals of unimodal maps6 What is the universal route to chaos7 Idea behind renormalisation8 What are universal limiting functions in the context of maps9 Name discrete time dynamics analysis methods
10 What is Poincareacute section11 What is Poincareacute map (return map)12 What is Lorenz section
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Lecture notes 11 Nonlinear Dynamics YFX1520
1 Feigenbaumrsquos analysis of period doubling
In this lecture we are continuing our study of one-dimensional maps as simplified models of chaos and astools for analysing higher order differential equations We have already encountered maps in this role TheLorenz map provided strong evidence that the Lorenz attractor is truly strange and is not just a long-period limit-cycle see Lectures 9 and 10
11 Feigenbaum constants
The Feigenbaum constants were presented during Lecture 10
Slide 3
1-D unimodal maps and Feigenbaum constants
= limn1
n1
n= lim
n1
rn1 rn2
rn rn1 4669201609 (1)
crarr = limn1
dn1
dn 2502907875 (2)
DKartofelev YFX1520 3 30
In the above orbit diagram xm = max f(x) is the maximum of the unimodal map graph The Feigen-baum constants are valid up to the onset of chaos at the accumulation point r = rinfin and inside eachperiodic window for r gt rinfin The Feigenbaum constants are universal the same convergencerate appears no matter what unimodal map is iterated They are new mathematical constants asbasic to period doubling as π is to circles
Showing the values of Feigenbaum constants to 30 decimal places
δ = 4669 201 609 102 990 671 853 203 821 578 (1)
α = minus2502 907 875 095 892 822 283 902 873 218 (2)
In this lecture wersquoll take a closer look at Feigenbaum α constant
12 Superstable fixed points and period-p points
Figure 1 Superstable fixed point where f prime(xlowast) = 0 and xlowast = xm
The fixed point defined byf(xlowast) = xlowast (3)
D Kartofelev 214 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
is said to be superstable if its slopef prime(xlowast) = 0 (4)
This hold true when the maximum (or minimum) of map function f is the fixed point
xlowast = xm (5)
Figures 1 2 and Slide 4 show the superstable fixed point in the case of unimodal Logistic map
Slide 4Superstable fixed point (Logistic map)
f(x r) = x
x
= xm and f0(x
r) = 0 ) r = 2 (3)
Convergence of xn about the non-trivial fixed point x
n+1 =|f 00(x
r)|2
2n +O(3n) (4)
is quadratic The iterates xn converge quadraticallyDKartofelev YFX1520 4 30
In Lecture 9 we showed that the stability of fixed point xlowast depends on the absolute slope of f at xlowastie |f prime(xlowast)| If |f prime(xlowast)| lt 1 then the fixed point is stable and for |f prime(xlowast)| gt 1 it is unstable
mdashmdashmdashmdash Skip if needed start mdashmdashmdashmdashThe linearisation of a map is derived by studying the slightly perturbed iterate
xn = xlowast + ηn (6)
where |η| ≪ 1 We study the next iterate dynamics through linearisation of map f
xn+1 = f(xn) = f(xlowast + ηn) =
983063Taylor series
about xn = xlowast
983064= f(xlowast) +
f prime(xlowast)
1(xlowast + ηn minus xlowast) +O(η2n) asymp
asymp xlowast + f prime(xlowast)ηn
(7)
xn+1 = xlowast + f prime(xlowast)ηn +O(η2n) (8)
xn+1 minus xlowast983167 983166983165 983168=ηn+1
= f prime(xlowast)ηn +O(η2n) (9)
The left-hand side of the result follows from definition (6) The linearised form of map f is obtainedby neglecting the higher order terms O(η2n)
ηn+1 = |f prime(xlowast)|ηn (10)
mdashmdashmdashmdash Skip if needed stop mdashmdashmdashmdashWe know that iterates xn converge towards stable fixed point xlowast where f prime(xlowast) lt 1 or more precisely
the slightly perturbed iterates xn will But what happens for superstable fixed point xlowast How doesthe solution converge for f prime(xlowast) = 0 Obviously the linearisation canrsquot be used We need to includean additional term in the Taylor series expansion
ηn+1 = |f prime(xlowast)|983167 983166983165 983168=0
ηn +|f primeprime(xlowast)|
2η2n +O(η3n) =
|f primeprime(xlowast)|2
η2n +O(η3n) (11)
D Kartofelev 314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
ηn+1 =|f primeprime(xlowast)|
2η2n +O(η3n) (12)
This result means that iterates converge faster than exponentially when they approach a superstablefixed point A very beneficial property to keep in mind when analysing maps numerically (less calcu-lations required to reach results)
The following interactive numerical file demonstrates the superstable convergence of iterate xn in the caseof Logistic map
Numerics cdf1 nb1Comparison of superstable and stable fixed points in Logistic mapComparison of map iterate convergence to superstable fixed point and a stable fixed point
Out[]=
|f (x)| = 0
00 02 04 06 08 1000
02
04
06
08
10
x n+1
0 5 10 15 2000
02
04
06
08
10
x n
00 02 04 06 08 1000
02
04
06
08
10
xn
x n+1
0 5 10 15 2000
02
04
06
08
10
n
x n
Figure 2 (Top) Cobweb diagram and map iterates of Logistic map featuring superstable fixed pointInitial condition x0 = 01 and parameter r = 2 (Bottom) Cobweb diagram and map iterates of Logisticmap featuring stable fixed point Initial condition x0 = 01 and parameter r = 285
Can there be superstable period-p points
Slides 5ndash7Superstable period-p point (Logistic map)
Superstable period-p orbit xm = argmax f is also a local min ormax of f p map (p-th iterate of map f)
For example in the case of period-2 point
f2(x
r) = x (x = xm) ) r = 1 +
p5 (5)
f2(x
r)0=
d
dx [f(f(x r))] = f
0(f(x r)) middot f 0(x
r) = 0 (6)
DKartofelev YFX1520 5 30
D Kartofelev 414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Superstable period-p point fixed point xlowast that is also the local minimum or maximum of map fpAlgebraically we write
fp(xlowast) = xlowast (13)
where xlowast = xm The superstable point of map f where f(xlowast) = xlowast is also one of the points in thesuperstable period-p orbit
These superstable points are shown below on the orbit diagram where rn is the onset of the period-2n
orbit (bifurcation point) and r = Rn is where the period-2n orbit is superstable The Rn occurs alwaybetween rn and rn+1 Unsurprisingly the ratios of succeeding Rn also converges to δ for n rarr infin
Orbit diagram superstable period-2n points
rn ndash stable period-2n orbit is born (bifurcation point)Rn ndash superstable period-2n point
limn1
Rn1
Rn= (7)
DKartofelev YFX1520 6 30
Period doubling bifurcation points
Values of bifurcation points and superstable points of few first perioddoublings in the Logistic map
r0 = 10 non-trivial R0 = 20 fpr1 = 30 R1 = 1 +
p5 323607 period-2
r2 = 1 +p6 344949 R2 349856 period-4
r3 354409 R3 355464 period-8r4 356441 R4 356667 period-16r5 356875 R5 356924 period-32
r1 3569945672 R1 = r1 356994567 period-21
r1 ndash onset of chaos (accumulation point)
limn1
rn1 rn2
rn rn1= lim
n1
Rn1
Rn= (8)
DKartofelev YFX1520 7 30
13 Renormalisation and Feigenbaum constants
Letrsquos try to tap into the observation presented in Lecture 10 that the orbit diagram of unimodal maps isself-similar under magnification How do the scaling constant α and δ govern the period doubling bifur-cations in the Logistic and other unimodal maps
Figure 3 (Left) Regions of interest the proximity to r = R0 and r = R1 (Right) Orbit diagram showingthe bifurcation points rn or the onsets of the period-2n orbits and superstable Rn-s where the period-2n
orbits are superstable cf Slide 6
Next we consider the local dynamics of the orbit diagram near r = R0 and then compare it to thesituation near r = R1 see Fig 3 We renormalise one region into another
D Kartofelev 514 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 8 9
Feigenbaumrsquos analysis renormalisation
f(xR0) f2(xR1)
(a) (b) (c)
DKartofelev YFX1520 8 30
Feigenbaumrsquos analysis renormalisation
f(xR0) f2(xR1) crarrf
2x
crarr R1
(a) (b) (c)
Read Mitchell J Feigenbaum ldquoThe universal metric properties ofnonlinear transformationsrdquo Journal of Statistical Physics 21(6) pp669ndash706 1979 Also relevant to the following three slides +
D Kartofelev YFX1520 9 30
The visual comparison shows that renormalised map is very similar to the original one The map max-imum xm shown on Slide 9 has beed shifted to the origin in order to simplify the math below
Letrsquos express the renormalisation procedure algebraically
f(xR0) asymp αf2983059xα R1
983060 (14)
notice that we have scaled the vertical and horizontal directions simultaneously In summary map fhas been renormalised by taking its second iterate rescaling x rarr xα shifting r to the next superstablevalue and multiplying everything by α
The higher order renormalisations of f(xR0) are given by
f(xR0) asymp α2f4983059 x
α2 R2
983060
asymp α3f8983059 x
α3 R3
983060
asymp α4f16983059 x
α4 R4
983060
asymp αnf (2n)983059 x
αn Rn
983060
(15)
What happens when we reach the accumulation point r = rinfin mdash reach the chaotic regime
limnrarrinfin
αnf (2n)983059 x
αn Rn
983060= g0(x) (16)
Feigenbaum found numerically that this limit converges only for α = minus2502907875 by producing the re-sulting limiting function g0 The function g0 is called the universal limiting function This function issimilar in shape to map f and it retains the superstability of its fixed point
The descriptor ldquouniversalrdquo in this context means that function g0 is almost independent of the unimodalfunction f How can g0 be independent of f Function g0 depends on f only through its behaviour nearx = 0 since thatrsquos all that survives in the argument xαn as n rarr infin With each renormalisation we areblowing up smaller and smaller neighbourhoods of the maximum of f until practically all of the information(global shape further away from x = 0) of f becomes meaningless The only remaining information is thequadratic nature of the maximum In the case of unimodal maps the Taylor expansion at xlowast = xm has tohave the second (quadratic) x2 term (superstability) The fixed point has to be what is called the seconddegree maximum point or the quadratic maximum
You can find the universal limiting functions related to any superstable period-2i orbit To obtain otheruniversal limiting functions gi(x) start with f(xRi) instead of f(xR0)
gi(x) = limnrarrinfin
αnf (2n)983059 x
αn Rn+i
983060 (17)
D Kartofelev 614 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
What happens at the onset of chaos i = infin Letrsquos find the universal limiting function at the onset of chaoswhere Ri = Rinfin We need to renormalise f(xRinfin) one time
ginfin(x) asymp αf2983059xα Rinfin+1
983060=
983063Remainderinfin+ 1 = infin
983064= αf2
983059xα Rinfin
983060= g(x) (18)
For once we donrsquot have to shift r when we renormalise (we became independent of i) The limiting func-tion ginfin(x) usually called simply g(x) satisfies
g(x) = αg2983059xα
983060 (19)
This is a functional equation for g(x) and the universal scale factor α It is self-referential primaryunknown g(x) is defined in terms of itself
Letrsquos try to solve Eq (19) The functional equation is not complete until we specify boundary conditionson g(x) After the shift of origin all our unimodal f rsquos have a maximum at x = 0 so we require
gprime(0) = 0 (20)
Also we can set
g(0) =
983063From Def (18)we know that
983064= g(0 Rinfin) = const (21)
tog(0) = 1 (22)
without loss of generality (This just defines the normalised scale for x if g(x) is a solution of Eq (19) sois microg(xmicro) where micro isin R with the same α) Naturally this selection agrees with boundary condition (20)
Now we can solve Eq (19) for function g(x) and α at the boundary x = 0
g(0) = αg29830610
α
983062= αg2(0) = αg(g(0)) (23)
since g(0) = 1 we have1 = αg(1) (24)
α =1
g(1) (25)
The universal scaling constant α depends on the limiting function g As of now no closed form solutionexist for function g We can resort to numerical method to find the shape of the limiting function g andthe value of α Next we solve Eq (19) with boundary condition (25) using the power series solution
14 Numerical determination of Feigenbaum α constant
Slides 10 11
Limiting function g(x) and Feigenbaum constant crarr
Letrsquos consider a functional equation in the following form
g(x) = crarr g(gx
crarr
) = crarr g
2x
crarr
(9)
where crarr acts as scaling coecient and
crarr =1
g(1) (10)
The power series solution is obtained by assuming power expansion
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 + ex10 + (11)
which assumes that the map maximum is quadratic
DKartofelev YFX1520 10 30
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution1 the one term approximation where
g(x) = 1 + ax2 +O(x4) (12)
results in
a = 1
2(1 +
p3) crarr =
1
g(1) 273205 (92 error) (13)
-04 -02 00 02 04000204060810
xn
x x+1 g(x) (one term)
Logistic map
Logistic map shifted
1See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 11 30
D Kartofelev 714 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The power series expansion uses Maclaurin expansion and assumes a quadratic maximum Thequadratic maximum follows from the superstability property The above process can generate lots ofextraneous root you will need to shift through them to find the correct solution
The solution shown above is taken from the following numerical file The file is commented for the benefitof students who are interested in numerical methods in general
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
In order to find a better match more terms in the power series expansion are needed
Slide 12
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution2 the four term approximation where
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 +O(x10) (14)
Coecients a = 1528 b = 01053 c = 002631 d = 0003344and
crarr =1
g(1) 250316 (001 error) (15)
-04 -02 00 02 04000204060810
xn
x x+1 g(x)
Logistic map
Logistic map shifted
2See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 12 30
The solution shown above is taken from the following numerical file
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
The renormalisation theory also explains the value of δ Unfortunately the derivations and analysis ofthe Feigenbaum δ constant requires advanced functional analysis know-how (operators in function spaceFrechet derivatives etc) and for this reason is omitted from this courseConclusion Something qualitative (unimodality) gave us something quantitative (Feigenbaumrsquos constantsα and δ) In science it is usually the other way around
Reading suggestion
Link File name CitationPaper1 paper5pdf Mitchell J Feigenbaum ldquoThe universal metric properties of nonlinear transfor-
mationsrdquo Journal of Statistical Physics 21(6) pp 669ndash706 (1979)doi101007BF01107909
2 Discrete time analysis methods
We have already used mapping of a continuous three-dimensional flow into a discrete one-dimensional mapmdash Lorenz attractor rarr Lorenz map Our goal in this section is to generalise this approach
D Kartofelev 814 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
21 Poincareacute section Poincareacute map return map
Poincareacute section is a set of intersection points between a hyperspace hyper-surface or hyperplane anda trajectory of n-dimensional continuous time flow Poincareacute map or return map is the (algebraic)relationship between these preceding and succeeding intersection points ie point xn and xn+1 found onPoincareacute section
Slides 13ndash16
Poincare section
Intersection of an attractor trajectory with the hypersurface s
DKartofelev YFX1520 13 30
Discrete time dynamics analysis
Construction of a Poincare map (return map recurrence map)
DKartofelev YFX1520 14 30
Poincare section periodic forcing3
3Credit J Thompson H Stewart 1986 Nonlinear dynamics and chaos
geometrical methods for engineers and scientists Chichester UK Wiley
DKartofelev YFX1520 15 30
Poincare section periodic forcing
DKartofelev YFX1520 16 30
Letrsquos demonstrate these concepts using two examples
211 Example Periodically driven damped Duffing oscillator
Duffing equation (or Duffing oscillator) named after Georg Duffing (1861ndash1944) is a nonlinear second-orderdifferential equation used to model certain damped and driven oscillators
Slides 17 18The first term on the left-hand side of Eq (16 slide numbering) is the inertial term the third term isthe nonlinear stiffness and the fourth term is the damping The non-homogeneous right-hand sideof the equation describes the external 2π-periodic forcing with the amplitude F and frequency ω = 1The solution for F gt 0 may be a strange attractor The external forcing of two-dimensional systemsis often the source of chaos But wait how can two-dimensional system feature chaos Isnrsquot chaossupposed to be reserved only for three or higher-dimensional systems The external forcing term allowsfor this second order system to be represented as a three-dimensional system
D Kartofelev 914 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Periodically driven damped Dung oscillator
Figure Mechanical system featuring Dung oscillations
The non-autonomous equation of motion has the following form
x x+ x3 + x = F cost (16)
where is the damping coecient F and are the forcing strengthand frequency respectively
DKartofelev YFX1520 17 30
Periodically driven damped Dung oscillator
We introduce the variable exchange y = x and rewrite (16)
(x = y
y = x x3 y + F cost
(17)
The equivalent 3-D system is obtained by applying variable exchangez = t 8
gtlt
gt
x = y
y = x x3 y + F cos z
z =
(18)
The above folds since
z =
Zz dt =
Z dt =
Zdt = t+ C (19)
DKartofelev YFX1520 18 30
If you remove the damping and the external forcing terms the system becomes equivalent to theproblem of the particle in the double-well potential given by potential function
V (x) =1
4x4 minus 1
2x2 (26)
that was presentedstudied in Lecture 5 (go and check it out)
Figure 4 shows three ways you can think graphically about the solution of Duffing oscillator and itsPoincareacute section cf cdf3 nb3
Figure 4 (Left) Snapshots of a trajectory every 2π time units cf Slides 15 16 The 2π-periodicity isa natural choice since the external force is 2π-periodic for ω = 1 (Middle) A trajectory in the three-dimensional Cartesian xyz-space corresponding to Sys (18 slide numbering) (Right) A trajectory in thethree-dimensional space where the z-axis is 2π-periodic and is folded into a closed circle cf Slides 15 16The blue outline shows the Poincareacute section
The following numerical file shows the solution to Duffing oscillator and the quantitatively accuraterenderings of the three different ways to plot the solution of Duffing oscillator shows in Fig 4
Numerics cdf3 nb3Periodically forced damped Duffing oscillator Numerical solution and phase portrait 2π-periodicPoincareacute section of the Duffing oscillator (static)
The following slides show how the Poincareacute section of Duffing oscillator for t ≫ 1 changes as we continu-ously move it along the 2π-periodic time or z-axis as shown in Fig 5
D Kartofelev 1014 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Figure 5 Poincareacute section dynamics and periodic z-axis A moving Poincareacute section featuring a singleintersection point shown with the blue dot
Slides 19 20
Poincare section4 Dung oscillator
-15 -10 -05 00 05 10 15
-05
00
05
10
x(t)
y(t)
Poincareacute section
4See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 19 30
Poincare section dynamics Dung oscillator
No embedded video files in this pdf
DKartofelev YFX1520 20 30
Notice that the Poincareacute section is folding onto itself continuously stretching and the intersectionpoints are being progressively mixed (via re-injection process) within the region defined by theattractor This type of Poincareacute section dynamics is common for strange attractors
The dynamics of the Poincareacute section of Duffing oscillator is explored in the following numerical file
Numerics cdf4 nb4Periodically forced damped Duffing oscillator and dynamic Poincareacute section animation
212 Example Roumlssler attractor
Originally studied by Otto Roumlssler (1940ndash) The system is a product of pure imagination and it exhibitschaotic dynamics by design
D Kartofelev 1114 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 21ndash24
Rossler attractor
Rossler attractor5 is given by (as mentioned in Lecture 9)
8gtlt
gt
x = y x
y = x+ ay
z = b+ z(x c)
(20)
Chaotic solution exists for a = 01 b = 01 c = 14
5See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 21 30
Rossler attractor Poincare sections6
-10 -5 0 50
5
10
15
y(t)
z(t)
Poincareacute section yz-plane
6See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 22 30
The system has only one nonlinearity the zx term in the third equation
Rossler attractor Poincare sections7
-10 -5 0 5 100
5
10
15
20
25
30
x(t)
z(t)
Poincareacute section xz-plane
7See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 23 30
Rossler attractor return map and Poincare sections
Credit Y Maistrenko and R PaskauskasDKartofelev YFX1520 24 30
Slide 24 features the shape of Roumlssler map mdash the Poincareacute map of Roumlssler attractor
The following numerical file was used to calculate the attractor and the Poincareacute sections shown above
Numerics cdf5 nb5Numerical solution of Roumlssler attractor Poincareacute section of Roumlssler attractor
Slides 25 26
Rossler attractor orbit diagram
DKartofelev YFX1520 25 30
Rossler attractor period doubling
No embedded video files in this pdf
DKartofelev YFX1520 26 30
D Kartofelev 1214 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The orbit diagram shown on Slide 25 is calculated using the Roumlssler map shown on Slide 24
The following interactive numerical file shows the period doubling bifurcation happening in the xy-projection of Roumlssler attractor cf Slide 25
Numerics cdf6 nb6Periodic orbits in Roumlssler attractor for varied c value
The following slide shows the 2π-periodic dynamics of the Poincareacute section of Roumlssler attractor
Slide 27Poincare section dynamics Rossler attractor
No embedded video files in this pdf
DKartofelev YFX1520 27 30
Once again notice the folding stretching and mixing of the Poincareacute section cf Slides 19 and20 This type of Poincareacute section dynamics will be explored further in the next weekrsquos lecture
3 Lorenz section
Slide 28
Lorenz section
Section of a sectionDKartofelev YFX1520 28 30
The slide above shows a Poincareacute section of a strange attractor where we sliced the attractor with aplane thereby exposing its cross section If we take a further one-dimensional slice or Lorenz sectionthrough the Poincareacute section we find an infinite set of points separated by gaps of various sizes Inthe next weekrsquos lecture we will look into this discovery and into the notion of Lorenz section in greaterdetail
D Kartofelev 1314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
4 Attractor reconstruction
In applications it is often the case that you are able to measure or are restricted to only measuring onetime-series related to a higher order system mdash one aspect of a problem Letrsquos say you need to analyse theattractor that governs your measured one-dimensional time-series signal s(t) eg show that it is strangeAt first it might seem that there is not enough information But there exist a surprising data analysistechnique known as the attractor reconstruction
For systems governed by an attractor the dynamics in the full higher dimensional phase space canbe reconstructed from measurements of just a single time series s(t) Somehow a single variable carriessufficient information about all the others The method is based on time delays τ For instance define atwo-dimensional vector
983187x(t) = (s(t) s(t+ τ)) (27)
for some time delay τ gt 0 and signal s(t) where 983187x = (x y) Signal s(t) can therefore represent either thevariable x(t) or y(t) The time-series s(t) generates a trajectory 983187x(t) in a two-dimensional phase space thatrepresents its attractorrsquos trajectory You can also considered the attractor in three dimensions by definingthe three-dimensional vector
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)
for time delay τ gt 0 For four-dimensional attractor use
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)
etcThe following interactive numerical file demonstrates attractor reconstruction using three examples First
a sine wave is used to reconstruct its two-dimensional attractor mdash circle Second a quasi-periodic signalis used to reconstruct its three-dimensional attractor mdash torus Third time-series y(t) of Lorenz system isused to reconstruct the three-dimensional Lorenz attractor
Numerics cdf7 nb7Higher-dimensional attractor reconstruction from 1-D time-domain signals Examples sine wavequasi-periodic signal y(t) of Lorenz attractor
Revision questions
1 What are the values of Feigenbaum constants2 What are Feigenbaum constants3 Define superstable fixed point of a map4 Define superstable period-p point (or period-p orbit) of a map5 What are universals of unimodal maps6 What is the universal route to chaos7 Idea behind renormalisation8 What are universal limiting functions in the context of maps9 Name discrete time dynamics analysis methods
10 What is Poincareacute section11 What is Poincareacute map (return map)12 What is Lorenz section
D Kartofelev 1414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
is said to be superstable if its slopef prime(xlowast) = 0 (4)
This hold true when the maximum (or minimum) of map function f is the fixed point
xlowast = xm (5)
Figures 1 2 and Slide 4 show the superstable fixed point in the case of unimodal Logistic map
Slide 4Superstable fixed point (Logistic map)
f(x r) = x
x
= xm and f0(x
r) = 0 ) r = 2 (3)
Convergence of xn about the non-trivial fixed point x
n+1 =|f 00(x
r)|2
2n +O(3n) (4)
is quadratic The iterates xn converge quadraticallyDKartofelev YFX1520 4 30
In Lecture 9 we showed that the stability of fixed point xlowast depends on the absolute slope of f at xlowastie |f prime(xlowast)| If |f prime(xlowast)| lt 1 then the fixed point is stable and for |f prime(xlowast)| gt 1 it is unstable
mdashmdashmdashmdash Skip if needed start mdashmdashmdashmdashThe linearisation of a map is derived by studying the slightly perturbed iterate
xn = xlowast + ηn (6)
where |η| ≪ 1 We study the next iterate dynamics through linearisation of map f
xn+1 = f(xn) = f(xlowast + ηn) =
983063Taylor series
about xn = xlowast
983064= f(xlowast) +
f prime(xlowast)
1(xlowast + ηn minus xlowast) +O(η2n) asymp
asymp xlowast + f prime(xlowast)ηn
(7)
xn+1 = xlowast + f prime(xlowast)ηn +O(η2n) (8)
xn+1 minus xlowast983167 983166983165 983168=ηn+1
= f prime(xlowast)ηn +O(η2n) (9)
The left-hand side of the result follows from definition (6) The linearised form of map f is obtainedby neglecting the higher order terms O(η2n)
ηn+1 = |f prime(xlowast)|ηn (10)
mdashmdashmdashmdash Skip if needed stop mdashmdashmdashmdashWe know that iterates xn converge towards stable fixed point xlowast where f prime(xlowast) lt 1 or more precisely
the slightly perturbed iterates xn will But what happens for superstable fixed point xlowast How doesthe solution converge for f prime(xlowast) = 0 Obviously the linearisation canrsquot be used We need to includean additional term in the Taylor series expansion
ηn+1 = |f prime(xlowast)|983167 983166983165 983168=0
ηn +|f primeprime(xlowast)|
2η2n +O(η3n) =
|f primeprime(xlowast)|2
η2n +O(η3n) (11)
D Kartofelev 314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
ηn+1 =|f primeprime(xlowast)|
2η2n +O(η3n) (12)
This result means that iterates converge faster than exponentially when they approach a superstablefixed point A very beneficial property to keep in mind when analysing maps numerically (less calcu-lations required to reach results)
The following interactive numerical file demonstrates the superstable convergence of iterate xn in the caseof Logistic map
Numerics cdf1 nb1Comparison of superstable and stable fixed points in Logistic mapComparison of map iterate convergence to superstable fixed point and a stable fixed point
Out[]=
|f (x)| = 0
00 02 04 06 08 1000
02
04
06
08
10
x n+1
0 5 10 15 2000
02
04
06
08
10
x n
00 02 04 06 08 1000
02
04
06
08
10
xn
x n+1
0 5 10 15 2000
02
04
06
08
10
n
x n
Figure 2 (Top) Cobweb diagram and map iterates of Logistic map featuring superstable fixed pointInitial condition x0 = 01 and parameter r = 2 (Bottom) Cobweb diagram and map iterates of Logisticmap featuring stable fixed point Initial condition x0 = 01 and parameter r = 285
Can there be superstable period-p points
Slides 5ndash7Superstable period-p point (Logistic map)
Superstable period-p orbit xm = argmax f is also a local min ormax of f p map (p-th iterate of map f)
For example in the case of period-2 point
f2(x
r) = x (x = xm) ) r = 1 +
p5 (5)
f2(x
r)0=
d
dx [f(f(x r))] = f
0(f(x r)) middot f 0(x
r) = 0 (6)
DKartofelev YFX1520 5 30
D Kartofelev 414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Superstable period-p point fixed point xlowast that is also the local minimum or maximum of map fpAlgebraically we write
fp(xlowast) = xlowast (13)
where xlowast = xm The superstable point of map f where f(xlowast) = xlowast is also one of the points in thesuperstable period-p orbit
These superstable points are shown below on the orbit diagram where rn is the onset of the period-2n
orbit (bifurcation point) and r = Rn is where the period-2n orbit is superstable The Rn occurs alwaybetween rn and rn+1 Unsurprisingly the ratios of succeeding Rn also converges to δ for n rarr infin
Orbit diagram superstable period-2n points
rn ndash stable period-2n orbit is born (bifurcation point)Rn ndash superstable period-2n point
limn1
Rn1
Rn= (7)
DKartofelev YFX1520 6 30
Period doubling bifurcation points
Values of bifurcation points and superstable points of few first perioddoublings in the Logistic map
r0 = 10 non-trivial R0 = 20 fpr1 = 30 R1 = 1 +
p5 323607 period-2
r2 = 1 +p6 344949 R2 349856 period-4
r3 354409 R3 355464 period-8r4 356441 R4 356667 period-16r5 356875 R5 356924 period-32
r1 3569945672 R1 = r1 356994567 period-21
r1 ndash onset of chaos (accumulation point)
limn1
rn1 rn2
rn rn1= lim
n1
Rn1
Rn= (8)
DKartofelev YFX1520 7 30
13 Renormalisation and Feigenbaum constants
Letrsquos try to tap into the observation presented in Lecture 10 that the orbit diagram of unimodal maps isself-similar under magnification How do the scaling constant α and δ govern the period doubling bifur-cations in the Logistic and other unimodal maps
Figure 3 (Left) Regions of interest the proximity to r = R0 and r = R1 (Right) Orbit diagram showingthe bifurcation points rn or the onsets of the period-2n orbits and superstable Rn-s where the period-2n
orbits are superstable cf Slide 6
Next we consider the local dynamics of the orbit diagram near r = R0 and then compare it to thesituation near r = R1 see Fig 3 We renormalise one region into another
D Kartofelev 514 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 8 9
Feigenbaumrsquos analysis renormalisation
f(xR0) f2(xR1)
(a) (b) (c)
DKartofelev YFX1520 8 30
Feigenbaumrsquos analysis renormalisation
f(xR0) f2(xR1) crarrf
2x
crarr R1
(a) (b) (c)
Read Mitchell J Feigenbaum ldquoThe universal metric properties ofnonlinear transformationsrdquo Journal of Statistical Physics 21(6) pp669ndash706 1979 Also relevant to the following three slides +
D Kartofelev YFX1520 9 30
The visual comparison shows that renormalised map is very similar to the original one The map max-imum xm shown on Slide 9 has beed shifted to the origin in order to simplify the math below
Letrsquos express the renormalisation procedure algebraically
f(xR0) asymp αf2983059xα R1
983060 (14)
notice that we have scaled the vertical and horizontal directions simultaneously In summary map fhas been renormalised by taking its second iterate rescaling x rarr xα shifting r to the next superstablevalue and multiplying everything by α
The higher order renormalisations of f(xR0) are given by
f(xR0) asymp α2f4983059 x
α2 R2
983060
asymp α3f8983059 x
α3 R3
983060
asymp α4f16983059 x
α4 R4
983060
asymp αnf (2n)983059 x
αn Rn
983060
(15)
What happens when we reach the accumulation point r = rinfin mdash reach the chaotic regime
limnrarrinfin
αnf (2n)983059 x
αn Rn
983060= g0(x) (16)
Feigenbaum found numerically that this limit converges only for α = minus2502907875 by producing the re-sulting limiting function g0 The function g0 is called the universal limiting function This function issimilar in shape to map f and it retains the superstability of its fixed point
The descriptor ldquouniversalrdquo in this context means that function g0 is almost independent of the unimodalfunction f How can g0 be independent of f Function g0 depends on f only through its behaviour nearx = 0 since thatrsquos all that survives in the argument xαn as n rarr infin With each renormalisation we areblowing up smaller and smaller neighbourhoods of the maximum of f until practically all of the information(global shape further away from x = 0) of f becomes meaningless The only remaining information is thequadratic nature of the maximum In the case of unimodal maps the Taylor expansion at xlowast = xm has tohave the second (quadratic) x2 term (superstability) The fixed point has to be what is called the seconddegree maximum point or the quadratic maximum
You can find the universal limiting functions related to any superstable period-2i orbit To obtain otheruniversal limiting functions gi(x) start with f(xRi) instead of f(xR0)
gi(x) = limnrarrinfin
αnf (2n)983059 x
αn Rn+i
983060 (17)
D Kartofelev 614 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
What happens at the onset of chaos i = infin Letrsquos find the universal limiting function at the onset of chaoswhere Ri = Rinfin We need to renormalise f(xRinfin) one time
ginfin(x) asymp αf2983059xα Rinfin+1
983060=
983063Remainderinfin+ 1 = infin
983064= αf2
983059xα Rinfin
983060= g(x) (18)
For once we donrsquot have to shift r when we renormalise (we became independent of i) The limiting func-tion ginfin(x) usually called simply g(x) satisfies
g(x) = αg2983059xα
983060 (19)
This is a functional equation for g(x) and the universal scale factor α It is self-referential primaryunknown g(x) is defined in terms of itself
Letrsquos try to solve Eq (19) The functional equation is not complete until we specify boundary conditionson g(x) After the shift of origin all our unimodal f rsquos have a maximum at x = 0 so we require
gprime(0) = 0 (20)
Also we can set
g(0) =
983063From Def (18)we know that
983064= g(0 Rinfin) = const (21)
tog(0) = 1 (22)
without loss of generality (This just defines the normalised scale for x if g(x) is a solution of Eq (19) sois microg(xmicro) where micro isin R with the same α) Naturally this selection agrees with boundary condition (20)
Now we can solve Eq (19) for function g(x) and α at the boundary x = 0
g(0) = αg29830610
α
983062= αg2(0) = αg(g(0)) (23)
since g(0) = 1 we have1 = αg(1) (24)
α =1
g(1) (25)
The universal scaling constant α depends on the limiting function g As of now no closed form solutionexist for function g We can resort to numerical method to find the shape of the limiting function g andthe value of α Next we solve Eq (19) with boundary condition (25) using the power series solution
14 Numerical determination of Feigenbaum α constant
Slides 10 11
Limiting function g(x) and Feigenbaum constant crarr
Letrsquos consider a functional equation in the following form
g(x) = crarr g(gx
crarr
) = crarr g
2x
crarr
(9)
where crarr acts as scaling coecient and
crarr =1
g(1) (10)
The power series solution is obtained by assuming power expansion
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 + ex10 + (11)
which assumes that the map maximum is quadratic
DKartofelev YFX1520 10 30
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution1 the one term approximation where
g(x) = 1 + ax2 +O(x4) (12)
results in
a = 1
2(1 +
p3) crarr =
1
g(1) 273205 (92 error) (13)
-04 -02 00 02 04000204060810
xn
x x+1 g(x) (one term)
Logistic map
Logistic map shifted
1See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 11 30
D Kartofelev 714 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The power series expansion uses Maclaurin expansion and assumes a quadratic maximum Thequadratic maximum follows from the superstability property The above process can generate lots ofextraneous root you will need to shift through them to find the correct solution
The solution shown above is taken from the following numerical file The file is commented for the benefitof students who are interested in numerical methods in general
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
In order to find a better match more terms in the power series expansion are needed
Slide 12
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution2 the four term approximation where
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 +O(x10) (14)
Coecients a = 1528 b = 01053 c = 002631 d = 0003344and
crarr =1
g(1) 250316 (001 error) (15)
-04 -02 00 02 04000204060810
xn
x x+1 g(x)
Logistic map
Logistic map shifted
2See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 12 30
The solution shown above is taken from the following numerical file
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
The renormalisation theory also explains the value of δ Unfortunately the derivations and analysis ofthe Feigenbaum δ constant requires advanced functional analysis know-how (operators in function spaceFrechet derivatives etc) and for this reason is omitted from this courseConclusion Something qualitative (unimodality) gave us something quantitative (Feigenbaumrsquos constantsα and δ) In science it is usually the other way around
Reading suggestion
Link File name CitationPaper1 paper5pdf Mitchell J Feigenbaum ldquoThe universal metric properties of nonlinear transfor-
mationsrdquo Journal of Statistical Physics 21(6) pp 669ndash706 (1979)doi101007BF01107909
2 Discrete time analysis methods
We have already used mapping of a continuous three-dimensional flow into a discrete one-dimensional mapmdash Lorenz attractor rarr Lorenz map Our goal in this section is to generalise this approach
D Kartofelev 814 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
21 Poincareacute section Poincareacute map return map
Poincareacute section is a set of intersection points between a hyperspace hyper-surface or hyperplane anda trajectory of n-dimensional continuous time flow Poincareacute map or return map is the (algebraic)relationship between these preceding and succeeding intersection points ie point xn and xn+1 found onPoincareacute section
Slides 13ndash16
Poincare section
Intersection of an attractor trajectory with the hypersurface s
DKartofelev YFX1520 13 30
Discrete time dynamics analysis
Construction of a Poincare map (return map recurrence map)
DKartofelev YFX1520 14 30
Poincare section periodic forcing3
3Credit J Thompson H Stewart 1986 Nonlinear dynamics and chaos
geometrical methods for engineers and scientists Chichester UK Wiley
DKartofelev YFX1520 15 30
Poincare section periodic forcing
DKartofelev YFX1520 16 30
Letrsquos demonstrate these concepts using two examples
211 Example Periodically driven damped Duffing oscillator
Duffing equation (or Duffing oscillator) named after Georg Duffing (1861ndash1944) is a nonlinear second-orderdifferential equation used to model certain damped and driven oscillators
Slides 17 18The first term on the left-hand side of Eq (16 slide numbering) is the inertial term the third term isthe nonlinear stiffness and the fourth term is the damping The non-homogeneous right-hand sideof the equation describes the external 2π-periodic forcing with the amplitude F and frequency ω = 1The solution for F gt 0 may be a strange attractor The external forcing of two-dimensional systemsis often the source of chaos But wait how can two-dimensional system feature chaos Isnrsquot chaossupposed to be reserved only for three or higher-dimensional systems The external forcing term allowsfor this second order system to be represented as a three-dimensional system
D Kartofelev 914 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Periodically driven damped Dung oscillator
Figure Mechanical system featuring Dung oscillations
The non-autonomous equation of motion has the following form
x x+ x3 + x = F cost (16)
where is the damping coecient F and are the forcing strengthand frequency respectively
DKartofelev YFX1520 17 30
Periodically driven damped Dung oscillator
We introduce the variable exchange y = x and rewrite (16)
(x = y
y = x x3 y + F cost
(17)
The equivalent 3-D system is obtained by applying variable exchangez = t 8
gtlt
gt
x = y
y = x x3 y + F cos z
z =
(18)
The above folds since
z =
Zz dt =
Z dt =
Zdt = t+ C (19)
DKartofelev YFX1520 18 30
If you remove the damping and the external forcing terms the system becomes equivalent to theproblem of the particle in the double-well potential given by potential function
V (x) =1
4x4 minus 1
2x2 (26)
that was presentedstudied in Lecture 5 (go and check it out)
Figure 4 shows three ways you can think graphically about the solution of Duffing oscillator and itsPoincareacute section cf cdf3 nb3
Figure 4 (Left) Snapshots of a trajectory every 2π time units cf Slides 15 16 The 2π-periodicity isa natural choice since the external force is 2π-periodic for ω = 1 (Middle) A trajectory in the three-dimensional Cartesian xyz-space corresponding to Sys (18 slide numbering) (Right) A trajectory in thethree-dimensional space where the z-axis is 2π-periodic and is folded into a closed circle cf Slides 15 16The blue outline shows the Poincareacute section
The following numerical file shows the solution to Duffing oscillator and the quantitatively accuraterenderings of the three different ways to plot the solution of Duffing oscillator shows in Fig 4
Numerics cdf3 nb3Periodically forced damped Duffing oscillator Numerical solution and phase portrait 2π-periodicPoincareacute section of the Duffing oscillator (static)
The following slides show how the Poincareacute section of Duffing oscillator for t ≫ 1 changes as we continu-ously move it along the 2π-periodic time or z-axis as shown in Fig 5
D Kartofelev 1014 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Figure 5 Poincareacute section dynamics and periodic z-axis A moving Poincareacute section featuring a singleintersection point shown with the blue dot
Slides 19 20
Poincare section4 Dung oscillator
-15 -10 -05 00 05 10 15
-05
00
05
10
x(t)
y(t)
Poincareacute section
4See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 19 30
Poincare section dynamics Dung oscillator
No embedded video files in this pdf
DKartofelev YFX1520 20 30
Notice that the Poincareacute section is folding onto itself continuously stretching and the intersectionpoints are being progressively mixed (via re-injection process) within the region defined by theattractor This type of Poincareacute section dynamics is common for strange attractors
The dynamics of the Poincareacute section of Duffing oscillator is explored in the following numerical file
Numerics cdf4 nb4Periodically forced damped Duffing oscillator and dynamic Poincareacute section animation
212 Example Roumlssler attractor
Originally studied by Otto Roumlssler (1940ndash) The system is a product of pure imagination and it exhibitschaotic dynamics by design
D Kartofelev 1114 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 21ndash24
Rossler attractor
Rossler attractor5 is given by (as mentioned in Lecture 9)
8gtlt
gt
x = y x
y = x+ ay
z = b+ z(x c)
(20)
Chaotic solution exists for a = 01 b = 01 c = 14
5See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 21 30
Rossler attractor Poincare sections6
-10 -5 0 50
5
10
15
y(t)
z(t)
Poincareacute section yz-plane
6See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 22 30
The system has only one nonlinearity the zx term in the third equation
Rossler attractor Poincare sections7
-10 -5 0 5 100
5
10
15
20
25
30
x(t)
z(t)
Poincareacute section xz-plane
7See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 23 30
Rossler attractor return map and Poincare sections
Credit Y Maistrenko and R PaskauskasDKartofelev YFX1520 24 30
Slide 24 features the shape of Roumlssler map mdash the Poincareacute map of Roumlssler attractor
The following numerical file was used to calculate the attractor and the Poincareacute sections shown above
Numerics cdf5 nb5Numerical solution of Roumlssler attractor Poincareacute section of Roumlssler attractor
Slides 25 26
Rossler attractor orbit diagram
DKartofelev YFX1520 25 30
Rossler attractor period doubling
No embedded video files in this pdf
DKartofelev YFX1520 26 30
D Kartofelev 1214 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The orbit diagram shown on Slide 25 is calculated using the Roumlssler map shown on Slide 24
The following interactive numerical file shows the period doubling bifurcation happening in the xy-projection of Roumlssler attractor cf Slide 25
Numerics cdf6 nb6Periodic orbits in Roumlssler attractor for varied c value
The following slide shows the 2π-periodic dynamics of the Poincareacute section of Roumlssler attractor
Slide 27Poincare section dynamics Rossler attractor
No embedded video files in this pdf
DKartofelev YFX1520 27 30
Once again notice the folding stretching and mixing of the Poincareacute section cf Slides 19 and20 This type of Poincareacute section dynamics will be explored further in the next weekrsquos lecture
3 Lorenz section
Slide 28
Lorenz section
Section of a sectionDKartofelev YFX1520 28 30
The slide above shows a Poincareacute section of a strange attractor where we sliced the attractor with aplane thereby exposing its cross section If we take a further one-dimensional slice or Lorenz sectionthrough the Poincareacute section we find an infinite set of points separated by gaps of various sizes Inthe next weekrsquos lecture we will look into this discovery and into the notion of Lorenz section in greaterdetail
D Kartofelev 1314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
4 Attractor reconstruction
In applications it is often the case that you are able to measure or are restricted to only measuring onetime-series related to a higher order system mdash one aspect of a problem Letrsquos say you need to analyse theattractor that governs your measured one-dimensional time-series signal s(t) eg show that it is strangeAt first it might seem that there is not enough information But there exist a surprising data analysistechnique known as the attractor reconstruction
For systems governed by an attractor the dynamics in the full higher dimensional phase space canbe reconstructed from measurements of just a single time series s(t) Somehow a single variable carriessufficient information about all the others The method is based on time delays τ For instance define atwo-dimensional vector
983187x(t) = (s(t) s(t+ τ)) (27)
for some time delay τ gt 0 and signal s(t) where 983187x = (x y) Signal s(t) can therefore represent either thevariable x(t) or y(t) The time-series s(t) generates a trajectory 983187x(t) in a two-dimensional phase space thatrepresents its attractorrsquos trajectory You can also considered the attractor in three dimensions by definingthe three-dimensional vector
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)
for time delay τ gt 0 For four-dimensional attractor use
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)
etcThe following interactive numerical file demonstrates attractor reconstruction using three examples First
a sine wave is used to reconstruct its two-dimensional attractor mdash circle Second a quasi-periodic signalis used to reconstruct its three-dimensional attractor mdash torus Third time-series y(t) of Lorenz system isused to reconstruct the three-dimensional Lorenz attractor
Numerics cdf7 nb7Higher-dimensional attractor reconstruction from 1-D time-domain signals Examples sine wavequasi-periodic signal y(t) of Lorenz attractor
Revision questions
1 What are the values of Feigenbaum constants2 What are Feigenbaum constants3 Define superstable fixed point of a map4 Define superstable period-p point (or period-p orbit) of a map5 What are universals of unimodal maps6 What is the universal route to chaos7 Idea behind renormalisation8 What are universal limiting functions in the context of maps9 Name discrete time dynamics analysis methods
10 What is Poincareacute section11 What is Poincareacute map (return map)12 What is Lorenz section
D Kartofelev 1414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
ηn+1 =|f primeprime(xlowast)|
2η2n +O(η3n) (12)
This result means that iterates converge faster than exponentially when they approach a superstablefixed point A very beneficial property to keep in mind when analysing maps numerically (less calcu-lations required to reach results)
The following interactive numerical file demonstrates the superstable convergence of iterate xn in the caseof Logistic map
Numerics cdf1 nb1Comparison of superstable and stable fixed points in Logistic mapComparison of map iterate convergence to superstable fixed point and a stable fixed point
Out[]=
|f (x)| = 0
00 02 04 06 08 1000
02
04
06
08
10
x n+1
0 5 10 15 2000
02
04
06
08
10
x n
00 02 04 06 08 1000
02
04
06
08
10
xn
x n+1
0 5 10 15 2000
02
04
06
08
10
n
x n
Figure 2 (Top) Cobweb diagram and map iterates of Logistic map featuring superstable fixed pointInitial condition x0 = 01 and parameter r = 2 (Bottom) Cobweb diagram and map iterates of Logisticmap featuring stable fixed point Initial condition x0 = 01 and parameter r = 285
Can there be superstable period-p points
Slides 5ndash7Superstable period-p point (Logistic map)
Superstable period-p orbit xm = argmax f is also a local min ormax of f p map (p-th iterate of map f)
For example in the case of period-2 point
f2(x
r) = x (x = xm) ) r = 1 +
p5 (5)
f2(x
r)0=
d
dx [f(f(x r))] = f
0(f(x r)) middot f 0(x
r) = 0 (6)
DKartofelev YFX1520 5 30
D Kartofelev 414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Superstable period-p point fixed point xlowast that is also the local minimum or maximum of map fpAlgebraically we write
fp(xlowast) = xlowast (13)
where xlowast = xm The superstable point of map f where f(xlowast) = xlowast is also one of the points in thesuperstable period-p orbit
These superstable points are shown below on the orbit diagram where rn is the onset of the period-2n
orbit (bifurcation point) and r = Rn is where the period-2n orbit is superstable The Rn occurs alwaybetween rn and rn+1 Unsurprisingly the ratios of succeeding Rn also converges to δ for n rarr infin
Orbit diagram superstable period-2n points
rn ndash stable period-2n orbit is born (bifurcation point)Rn ndash superstable period-2n point
limn1
Rn1
Rn= (7)
DKartofelev YFX1520 6 30
Period doubling bifurcation points
Values of bifurcation points and superstable points of few first perioddoublings in the Logistic map
r0 = 10 non-trivial R0 = 20 fpr1 = 30 R1 = 1 +
p5 323607 period-2
r2 = 1 +p6 344949 R2 349856 period-4
r3 354409 R3 355464 period-8r4 356441 R4 356667 period-16r5 356875 R5 356924 period-32
r1 3569945672 R1 = r1 356994567 period-21
r1 ndash onset of chaos (accumulation point)
limn1
rn1 rn2
rn rn1= lim
n1
Rn1
Rn= (8)
DKartofelev YFX1520 7 30
13 Renormalisation and Feigenbaum constants
Letrsquos try to tap into the observation presented in Lecture 10 that the orbit diagram of unimodal maps isself-similar under magnification How do the scaling constant α and δ govern the period doubling bifur-cations in the Logistic and other unimodal maps
Figure 3 (Left) Regions of interest the proximity to r = R0 and r = R1 (Right) Orbit diagram showingthe bifurcation points rn or the onsets of the period-2n orbits and superstable Rn-s where the period-2n
orbits are superstable cf Slide 6
Next we consider the local dynamics of the orbit diagram near r = R0 and then compare it to thesituation near r = R1 see Fig 3 We renormalise one region into another
D Kartofelev 514 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 8 9
Feigenbaumrsquos analysis renormalisation
f(xR0) f2(xR1)
(a) (b) (c)
DKartofelev YFX1520 8 30
Feigenbaumrsquos analysis renormalisation
f(xR0) f2(xR1) crarrf
2x
crarr R1
(a) (b) (c)
Read Mitchell J Feigenbaum ldquoThe universal metric properties ofnonlinear transformationsrdquo Journal of Statistical Physics 21(6) pp669ndash706 1979 Also relevant to the following three slides +
D Kartofelev YFX1520 9 30
The visual comparison shows that renormalised map is very similar to the original one The map max-imum xm shown on Slide 9 has beed shifted to the origin in order to simplify the math below
Letrsquos express the renormalisation procedure algebraically
f(xR0) asymp αf2983059xα R1
983060 (14)
notice that we have scaled the vertical and horizontal directions simultaneously In summary map fhas been renormalised by taking its second iterate rescaling x rarr xα shifting r to the next superstablevalue and multiplying everything by α
The higher order renormalisations of f(xR0) are given by
f(xR0) asymp α2f4983059 x
α2 R2
983060
asymp α3f8983059 x
α3 R3
983060
asymp α4f16983059 x
α4 R4
983060
asymp αnf (2n)983059 x
αn Rn
983060
(15)
What happens when we reach the accumulation point r = rinfin mdash reach the chaotic regime
limnrarrinfin
αnf (2n)983059 x
αn Rn
983060= g0(x) (16)
Feigenbaum found numerically that this limit converges only for α = minus2502907875 by producing the re-sulting limiting function g0 The function g0 is called the universal limiting function This function issimilar in shape to map f and it retains the superstability of its fixed point
The descriptor ldquouniversalrdquo in this context means that function g0 is almost independent of the unimodalfunction f How can g0 be independent of f Function g0 depends on f only through its behaviour nearx = 0 since thatrsquos all that survives in the argument xαn as n rarr infin With each renormalisation we areblowing up smaller and smaller neighbourhoods of the maximum of f until practically all of the information(global shape further away from x = 0) of f becomes meaningless The only remaining information is thequadratic nature of the maximum In the case of unimodal maps the Taylor expansion at xlowast = xm has tohave the second (quadratic) x2 term (superstability) The fixed point has to be what is called the seconddegree maximum point or the quadratic maximum
You can find the universal limiting functions related to any superstable period-2i orbit To obtain otheruniversal limiting functions gi(x) start with f(xRi) instead of f(xR0)
gi(x) = limnrarrinfin
αnf (2n)983059 x
αn Rn+i
983060 (17)
D Kartofelev 614 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
What happens at the onset of chaos i = infin Letrsquos find the universal limiting function at the onset of chaoswhere Ri = Rinfin We need to renormalise f(xRinfin) one time
ginfin(x) asymp αf2983059xα Rinfin+1
983060=
983063Remainderinfin+ 1 = infin
983064= αf2
983059xα Rinfin
983060= g(x) (18)
For once we donrsquot have to shift r when we renormalise (we became independent of i) The limiting func-tion ginfin(x) usually called simply g(x) satisfies
g(x) = αg2983059xα
983060 (19)
This is a functional equation for g(x) and the universal scale factor α It is self-referential primaryunknown g(x) is defined in terms of itself
Letrsquos try to solve Eq (19) The functional equation is not complete until we specify boundary conditionson g(x) After the shift of origin all our unimodal f rsquos have a maximum at x = 0 so we require
gprime(0) = 0 (20)
Also we can set
g(0) =
983063From Def (18)we know that
983064= g(0 Rinfin) = const (21)
tog(0) = 1 (22)
without loss of generality (This just defines the normalised scale for x if g(x) is a solution of Eq (19) sois microg(xmicro) where micro isin R with the same α) Naturally this selection agrees with boundary condition (20)
Now we can solve Eq (19) for function g(x) and α at the boundary x = 0
g(0) = αg29830610
α
983062= αg2(0) = αg(g(0)) (23)
since g(0) = 1 we have1 = αg(1) (24)
α =1
g(1) (25)
The universal scaling constant α depends on the limiting function g As of now no closed form solutionexist for function g We can resort to numerical method to find the shape of the limiting function g andthe value of α Next we solve Eq (19) with boundary condition (25) using the power series solution
14 Numerical determination of Feigenbaum α constant
Slides 10 11
Limiting function g(x) and Feigenbaum constant crarr
Letrsquos consider a functional equation in the following form
g(x) = crarr g(gx
crarr
) = crarr g
2x
crarr
(9)
where crarr acts as scaling coecient and
crarr =1
g(1) (10)
The power series solution is obtained by assuming power expansion
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 + ex10 + (11)
which assumes that the map maximum is quadratic
DKartofelev YFX1520 10 30
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution1 the one term approximation where
g(x) = 1 + ax2 +O(x4) (12)
results in
a = 1
2(1 +
p3) crarr =
1
g(1) 273205 (92 error) (13)
-04 -02 00 02 04000204060810
xn
x x+1 g(x) (one term)
Logistic map
Logistic map shifted
1See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 11 30
D Kartofelev 714 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The power series expansion uses Maclaurin expansion and assumes a quadratic maximum Thequadratic maximum follows from the superstability property The above process can generate lots ofextraneous root you will need to shift through them to find the correct solution
The solution shown above is taken from the following numerical file The file is commented for the benefitof students who are interested in numerical methods in general
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
In order to find a better match more terms in the power series expansion are needed
Slide 12
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution2 the four term approximation where
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 +O(x10) (14)
Coecients a = 1528 b = 01053 c = 002631 d = 0003344and
crarr =1
g(1) 250316 (001 error) (15)
-04 -02 00 02 04000204060810
xn
x x+1 g(x)
Logistic map
Logistic map shifted
2See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 12 30
The solution shown above is taken from the following numerical file
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
The renormalisation theory also explains the value of δ Unfortunately the derivations and analysis ofthe Feigenbaum δ constant requires advanced functional analysis know-how (operators in function spaceFrechet derivatives etc) and for this reason is omitted from this courseConclusion Something qualitative (unimodality) gave us something quantitative (Feigenbaumrsquos constantsα and δ) In science it is usually the other way around
Reading suggestion
Link File name CitationPaper1 paper5pdf Mitchell J Feigenbaum ldquoThe universal metric properties of nonlinear transfor-
mationsrdquo Journal of Statistical Physics 21(6) pp 669ndash706 (1979)doi101007BF01107909
2 Discrete time analysis methods
We have already used mapping of a continuous three-dimensional flow into a discrete one-dimensional mapmdash Lorenz attractor rarr Lorenz map Our goal in this section is to generalise this approach
D Kartofelev 814 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
21 Poincareacute section Poincareacute map return map
Poincareacute section is a set of intersection points between a hyperspace hyper-surface or hyperplane anda trajectory of n-dimensional continuous time flow Poincareacute map or return map is the (algebraic)relationship between these preceding and succeeding intersection points ie point xn and xn+1 found onPoincareacute section
Slides 13ndash16
Poincare section
Intersection of an attractor trajectory with the hypersurface s
DKartofelev YFX1520 13 30
Discrete time dynamics analysis
Construction of a Poincare map (return map recurrence map)
DKartofelev YFX1520 14 30
Poincare section periodic forcing3
3Credit J Thompson H Stewart 1986 Nonlinear dynamics and chaos
geometrical methods for engineers and scientists Chichester UK Wiley
DKartofelev YFX1520 15 30
Poincare section periodic forcing
DKartofelev YFX1520 16 30
Letrsquos demonstrate these concepts using two examples
211 Example Periodically driven damped Duffing oscillator
Duffing equation (or Duffing oscillator) named after Georg Duffing (1861ndash1944) is a nonlinear second-orderdifferential equation used to model certain damped and driven oscillators
Slides 17 18The first term on the left-hand side of Eq (16 slide numbering) is the inertial term the third term isthe nonlinear stiffness and the fourth term is the damping The non-homogeneous right-hand sideof the equation describes the external 2π-periodic forcing with the amplitude F and frequency ω = 1The solution for F gt 0 may be a strange attractor The external forcing of two-dimensional systemsis often the source of chaos But wait how can two-dimensional system feature chaos Isnrsquot chaossupposed to be reserved only for three or higher-dimensional systems The external forcing term allowsfor this second order system to be represented as a three-dimensional system
D Kartofelev 914 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Periodically driven damped Dung oscillator
Figure Mechanical system featuring Dung oscillations
The non-autonomous equation of motion has the following form
x x+ x3 + x = F cost (16)
where is the damping coecient F and are the forcing strengthand frequency respectively
DKartofelev YFX1520 17 30
Periodically driven damped Dung oscillator
We introduce the variable exchange y = x and rewrite (16)
(x = y
y = x x3 y + F cost
(17)
The equivalent 3-D system is obtained by applying variable exchangez = t 8
gtlt
gt
x = y
y = x x3 y + F cos z
z =
(18)
The above folds since
z =
Zz dt =
Z dt =
Zdt = t+ C (19)
DKartofelev YFX1520 18 30
If you remove the damping and the external forcing terms the system becomes equivalent to theproblem of the particle in the double-well potential given by potential function
V (x) =1
4x4 minus 1
2x2 (26)
that was presentedstudied in Lecture 5 (go and check it out)
Figure 4 shows three ways you can think graphically about the solution of Duffing oscillator and itsPoincareacute section cf cdf3 nb3
Figure 4 (Left) Snapshots of a trajectory every 2π time units cf Slides 15 16 The 2π-periodicity isa natural choice since the external force is 2π-periodic for ω = 1 (Middle) A trajectory in the three-dimensional Cartesian xyz-space corresponding to Sys (18 slide numbering) (Right) A trajectory in thethree-dimensional space where the z-axis is 2π-periodic and is folded into a closed circle cf Slides 15 16The blue outline shows the Poincareacute section
The following numerical file shows the solution to Duffing oscillator and the quantitatively accuraterenderings of the three different ways to plot the solution of Duffing oscillator shows in Fig 4
Numerics cdf3 nb3Periodically forced damped Duffing oscillator Numerical solution and phase portrait 2π-periodicPoincareacute section of the Duffing oscillator (static)
The following slides show how the Poincareacute section of Duffing oscillator for t ≫ 1 changes as we continu-ously move it along the 2π-periodic time or z-axis as shown in Fig 5
D Kartofelev 1014 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Figure 5 Poincareacute section dynamics and periodic z-axis A moving Poincareacute section featuring a singleintersection point shown with the blue dot
Slides 19 20
Poincare section4 Dung oscillator
-15 -10 -05 00 05 10 15
-05
00
05
10
x(t)
y(t)
Poincareacute section
4See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 19 30
Poincare section dynamics Dung oscillator
No embedded video files in this pdf
DKartofelev YFX1520 20 30
Notice that the Poincareacute section is folding onto itself continuously stretching and the intersectionpoints are being progressively mixed (via re-injection process) within the region defined by theattractor This type of Poincareacute section dynamics is common for strange attractors
The dynamics of the Poincareacute section of Duffing oscillator is explored in the following numerical file
Numerics cdf4 nb4Periodically forced damped Duffing oscillator and dynamic Poincareacute section animation
212 Example Roumlssler attractor
Originally studied by Otto Roumlssler (1940ndash) The system is a product of pure imagination and it exhibitschaotic dynamics by design
D Kartofelev 1114 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 21ndash24
Rossler attractor
Rossler attractor5 is given by (as mentioned in Lecture 9)
8gtlt
gt
x = y x
y = x+ ay
z = b+ z(x c)
(20)
Chaotic solution exists for a = 01 b = 01 c = 14
5See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 21 30
Rossler attractor Poincare sections6
-10 -5 0 50
5
10
15
y(t)
z(t)
Poincareacute section yz-plane
6See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 22 30
The system has only one nonlinearity the zx term in the third equation
Rossler attractor Poincare sections7
-10 -5 0 5 100
5
10
15
20
25
30
x(t)
z(t)
Poincareacute section xz-plane
7See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 23 30
Rossler attractor return map and Poincare sections
Credit Y Maistrenko and R PaskauskasDKartofelev YFX1520 24 30
Slide 24 features the shape of Roumlssler map mdash the Poincareacute map of Roumlssler attractor
The following numerical file was used to calculate the attractor and the Poincareacute sections shown above
Numerics cdf5 nb5Numerical solution of Roumlssler attractor Poincareacute section of Roumlssler attractor
Slides 25 26
Rossler attractor orbit diagram
DKartofelev YFX1520 25 30
Rossler attractor period doubling
No embedded video files in this pdf
DKartofelev YFX1520 26 30
D Kartofelev 1214 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The orbit diagram shown on Slide 25 is calculated using the Roumlssler map shown on Slide 24
The following interactive numerical file shows the period doubling bifurcation happening in the xy-projection of Roumlssler attractor cf Slide 25
Numerics cdf6 nb6Periodic orbits in Roumlssler attractor for varied c value
The following slide shows the 2π-periodic dynamics of the Poincareacute section of Roumlssler attractor
Slide 27Poincare section dynamics Rossler attractor
No embedded video files in this pdf
DKartofelev YFX1520 27 30
Once again notice the folding stretching and mixing of the Poincareacute section cf Slides 19 and20 This type of Poincareacute section dynamics will be explored further in the next weekrsquos lecture
3 Lorenz section
Slide 28
Lorenz section
Section of a sectionDKartofelev YFX1520 28 30
The slide above shows a Poincareacute section of a strange attractor where we sliced the attractor with aplane thereby exposing its cross section If we take a further one-dimensional slice or Lorenz sectionthrough the Poincareacute section we find an infinite set of points separated by gaps of various sizes Inthe next weekrsquos lecture we will look into this discovery and into the notion of Lorenz section in greaterdetail
D Kartofelev 1314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
4 Attractor reconstruction
In applications it is often the case that you are able to measure or are restricted to only measuring onetime-series related to a higher order system mdash one aspect of a problem Letrsquos say you need to analyse theattractor that governs your measured one-dimensional time-series signal s(t) eg show that it is strangeAt first it might seem that there is not enough information But there exist a surprising data analysistechnique known as the attractor reconstruction
For systems governed by an attractor the dynamics in the full higher dimensional phase space canbe reconstructed from measurements of just a single time series s(t) Somehow a single variable carriessufficient information about all the others The method is based on time delays τ For instance define atwo-dimensional vector
983187x(t) = (s(t) s(t+ τ)) (27)
for some time delay τ gt 0 and signal s(t) where 983187x = (x y) Signal s(t) can therefore represent either thevariable x(t) or y(t) The time-series s(t) generates a trajectory 983187x(t) in a two-dimensional phase space thatrepresents its attractorrsquos trajectory You can also considered the attractor in three dimensions by definingthe three-dimensional vector
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)
for time delay τ gt 0 For four-dimensional attractor use
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)
etcThe following interactive numerical file demonstrates attractor reconstruction using three examples First
a sine wave is used to reconstruct its two-dimensional attractor mdash circle Second a quasi-periodic signalis used to reconstruct its three-dimensional attractor mdash torus Third time-series y(t) of Lorenz system isused to reconstruct the three-dimensional Lorenz attractor
Numerics cdf7 nb7Higher-dimensional attractor reconstruction from 1-D time-domain signals Examples sine wavequasi-periodic signal y(t) of Lorenz attractor
Revision questions
1 What are the values of Feigenbaum constants2 What are Feigenbaum constants3 Define superstable fixed point of a map4 Define superstable period-p point (or period-p orbit) of a map5 What are universals of unimodal maps6 What is the universal route to chaos7 Idea behind renormalisation8 What are universal limiting functions in the context of maps9 Name discrete time dynamics analysis methods
10 What is Poincareacute section11 What is Poincareacute map (return map)12 What is Lorenz section
D Kartofelev 1414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Superstable period-p point fixed point xlowast that is also the local minimum or maximum of map fpAlgebraically we write
fp(xlowast) = xlowast (13)
where xlowast = xm The superstable point of map f where f(xlowast) = xlowast is also one of the points in thesuperstable period-p orbit
These superstable points are shown below on the orbit diagram where rn is the onset of the period-2n
orbit (bifurcation point) and r = Rn is where the period-2n orbit is superstable The Rn occurs alwaybetween rn and rn+1 Unsurprisingly the ratios of succeeding Rn also converges to δ for n rarr infin
Orbit diagram superstable period-2n points
rn ndash stable period-2n orbit is born (bifurcation point)Rn ndash superstable period-2n point
limn1
Rn1
Rn= (7)
DKartofelev YFX1520 6 30
Period doubling bifurcation points
Values of bifurcation points and superstable points of few first perioddoublings in the Logistic map
r0 = 10 non-trivial R0 = 20 fpr1 = 30 R1 = 1 +
p5 323607 period-2
r2 = 1 +p6 344949 R2 349856 period-4
r3 354409 R3 355464 period-8r4 356441 R4 356667 period-16r5 356875 R5 356924 period-32
r1 3569945672 R1 = r1 356994567 period-21
r1 ndash onset of chaos (accumulation point)
limn1
rn1 rn2
rn rn1= lim
n1
Rn1
Rn= (8)
DKartofelev YFX1520 7 30
13 Renormalisation and Feigenbaum constants
Letrsquos try to tap into the observation presented in Lecture 10 that the orbit diagram of unimodal maps isself-similar under magnification How do the scaling constant α and δ govern the period doubling bifur-cations in the Logistic and other unimodal maps
Figure 3 (Left) Regions of interest the proximity to r = R0 and r = R1 (Right) Orbit diagram showingthe bifurcation points rn or the onsets of the period-2n orbits and superstable Rn-s where the period-2n
orbits are superstable cf Slide 6
Next we consider the local dynamics of the orbit diagram near r = R0 and then compare it to thesituation near r = R1 see Fig 3 We renormalise one region into another
D Kartofelev 514 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 8 9
Feigenbaumrsquos analysis renormalisation
f(xR0) f2(xR1)
(a) (b) (c)
DKartofelev YFX1520 8 30
Feigenbaumrsquos analysis renormalisation
f(xR0) f2(xR1) crarrf
2x
crarr R1
(a) (b) (c)
Read Mitchell J Feigenbaum ldquoThe universal metric properties ofnonlinear transformationsrdquo Journal of Statistical Physics 21(6) pp669ndash706 1979 Also relevant to the following three slides +
D Kartofelev YFX1520 9 30
The visual comparison shows that renormalised map is very similar to the original one The map max-imum xm shown on Slide 9 has beed shifted to the origin in order to simplify the math below
Letrsquos express the renormalisation procedure algebraically
f(xR0) asymp αf2983059xα R1
983060 (14)
notice that we have scaled the vertical and horizontal directions simultaneously In summary map fhas been renormalised by taking its second iterate rescaling x rarr xα shifting r to the next superstablevalue and multiplying everything by α
The higher order renormalisations of f(xR0) are given by
f(xR0) asymp α2f4983059 x
α2 R2
983060
asymp α3f8983059 x
α3 R3
983060
asymp α4f16983059 x
α4 R4
983060
asymp αnf (2n)983059 x
αn Rn
983060
(15)
What happens when we reach the accumulation point r = rinfin mdash reach the chaotic regime
limnrarrinfin
αnf (2n)983059 x
αn Rn
983060= g0(x) (16)
Feigenbaum found numerically that this limit converges only for α = minus2502907875 by producing the re-sulting limiting function g0 The function g0 is called the universal limiting function This function issimilar in shape to map f and it retains the superstability of its fixed point
The descriptor ldquouniversalrdquo in this context means that function g0 is almost independent of the unimodalfunction f How can g0 be independent of f Function g0 depends on f only through its behaviour nearx = 0 since thatrsquos all that survives in the argument xαn as n rarr infin With each renormalisation we areblowing up smaller and smaller neighbourhoods of the maximum of f until practically all of the information(global shape further away from x = 0) of f becomes meaningless The only remaining information is thequadratic nature of the maximum In the case of unimodal maps the Taylor expansion at xlowast = xm has tohave the second (quadratic) x2 term (superstability) The fixed point has to be what is called the seconddegree maximum point or the quadratic maximum
You can find the universal limiting functions related to any superstable period-2i orbit To obtain otheruniversal limiting functions gi(x) start with f(xRi) instead of f(xR0)
gi(x) = limnrarrinfin
αnf (2n)983059 x
αn Rn+i
983060 (17)
D Kartofelev 614 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
What happens at the onset of chaos i = infin Letrsquos find the universal limiting function at the onset of chaoswhere Ri = Rinfin We need to renormalise f(xRinfin) one time
ginfin(x) asymp αf2983059xα Rinfin+1
983060=
983063Remainderinfin+ 1 = infin
983064= αf2
983059xα Rinfin
983060= g(x) (18)
For once we donrsquot have to shift r when we renormalise (we became independent of i) The limiting func-tion ginfin(x) usually called simply g(x) satisfies
g(x) = αg2983059xα
983060 (19)
This is a functional equation for g(x) and the universal scale factor α It is self-referential primaryunknown g(x) is defined in terms of itself
Letrsquos try to solve Eq (19) The functional equation is not complete until we specify boundary conditionson g(x) After the shift of origin all our unimodal f rsquos have a maximum at x = 0 so we require
gprime(0) = 0 (20)
Also we can set
g(0) =
983063From Def (18)we know that
983064= g(0 Rinfin) = const (21)
tog(0) = 1 (22)
without loss of generality (This just defines the normalised scale for x if g(x) is a solution of Eq (19) sois microg(xmicro) where micro isin R with the same α) Naturally this selection agrees with boundary condition (20)
Now we can solve Eq (19) for function g(x) and α at the boundary x = 0
g(0) = αg29830610
α
983062= αg2(0) = αg(g(0)) (23)
since g(0) = 1 we have1 = αg(1) (24)
α =1
g(1) (25)
The universal scaling constant α depends on the limiting function g As of now no closed form solutionexist for function g We can resort to numerical method to find the shape of the limiting function g andthe value of α Next we solve Eq (19) with boundary condition (25) using the power series solution
14 Numerical determination of Feigenbaum α constant
Slides 10 11
Limiting function g(x) and Feigenbaum constant crarr
Letrsquos consider a functional equation in the following form
g(x) = crarr g(gx
crarr
) = crarr g
2x
crarr
(9)
where crarr acts as scaling coecient and
crarr =1
g(1) (10)
The power series solution is obtained by assuming power expansion
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 + ex10 + (11)
which assumes that the map maximum is quadratic
DKartofelev YFX1520 10 30
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution1 the one term approximation where
g(x) = 1 + ax2 +O(x4) (12)
results in
a = 1
2(1 +
p3) crarr =
1
g(1) 273205 (92 error) (13)
-04 -02 00 02 04000204060810
xn
x x+1 g(x) (one term)
Logistic map
Logistic map shifted
1See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 11 30
D Kartofelev 714 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The power series expansion uses Maclaurin expansion and assumes a quadratic maximum Thequadratic maximum follows from the superstability property The above process can generate lots ofextraneous root you will need to shift through them to find the correct solution
The solution shown above is taken from the following numerical file The file is commented for the benefitof students who are interested in numerical methods in general
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
In order to find a better match more terms in the power series expansion are needed
Slide 12
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution2 the four term approximation where
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 +O(x10) (14)
Coecients a = 1528 b = 01053 c = 002631 d = 0003344and
crarr =1
g(1) 250316 (001 error) (15)
-04 -02 00 02 04000204060810
xn
x x+1 g(x)
Logistic map
Logistic map shifted
2See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 12 30
The solution shown above is taken from the following numerical file
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
The renormalisation theory also explains the value of δ Unfortunately the derivations and analysis ofthe Feigenbaum δ constant requires advanced functional analysis know-how (operators in function spaceFrechet derivatives etc) and for this reason is omitted from this courseConclusion Something qualitative (unimodality) gave us something quantitative (Feigenbaumrsquos constantsα and δ) In science it is usually the other way around
Reading suggestion
Link File name CitationPaper1 paper5pdf Mitchell J Feigenbaum ldquoThe universal metric properties of nonlinear transfor-
mationsrdquo Journal of Statistical Physics 21(6) pp 669ndash706 (1979)doi101007BF01107909
2 Discrete time analysis methods
We have already used mapping of a continuous three-dimensional flow into a discrete one-dimensional mapmdash Lorenz attractor rarr Lorenz map Our goal in this section is to generalise this approach
D Kartofelev 814 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
21 Poincareacute section Poincareacute map return map
Poincareacute section is a set of intersection points between a hyperspace hyper-surface or hyperplane anda trajectory of n-dimensional continuous time flow Poincareacute map or return map is the (algebraic)relationship between these preceding and succeeding intersection points ie point xn and xn+1 found onPoincareacute section
Slides 13ndash16
Poincare section
Intersection of an attractor trajectory with the hypersurface s
DKartofelev YFX1520 13 30
Discrete time dynamics analysis
Construction of a Poincare map (return map recurrence map)
DKartofelev YFX1520 14 30
Poincare section periodic forcing3
3Credit J Thompson H Stewart 1986 Nonlinear dynamics and chaos
geometrical methods for engineers and scientists Chichester UK Wiley
DKartofelev YFX1520 15 30
Poincare section periodic forcing
DKartofelev YFX1520 16 30
Letrsquos demonstrate these concepts using two examples
211 Example Periodically driven damped Duffing oscillator
Duffing equation (or Duffing oscillator) named after Georg Duffing (1861ndash1944) is a nonlinear second-orderdifferential equation used to model certain damped and driven oscillators
Slides 17 18The first term on the left-hand side of Eq (16 slide numbering) is the inertial term the third term isthe nonlinear stiffness and the fourth term is the damping The non-homogeneous right-hand sideof the equation describes the external 2π-periodic forcing with the amplitude F and frequency ω = 1The solution for F gt 0 may be a strange attractor The external forcing of two-dimensional systemsis often the source of chaos But wait how can two-dimensional system feature chaos Isnrsquot chaossupposed to be reserved only for three or higher-dimensional systems The external forcing term allowsfor this second order system to be represented as a three-dimensional system
D Kartofelev 914 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Periodically driven damped Dung oscillator
Figure Mechanical system featuring Dung oscillations
The non-autonomous equation of motion has the following form
x x+ x3 + x = F cost (16)
where is the damping coecient F and are the forcing strengthand frequency respectively
DKartofelev YFX1520 17 30
Periodically driven damped Dung oscillator
We introduce the variable exchange y = x and rewrite (16)
(x = y
y = x x3 y + F cost
(17)
The equivalent 3-D system is obtained by applying variable exchangez = t 8
gtlt
gt
x = y
y = x x3 y + F cos z
z =
(18)
The above folds since
z =
Zz dt =
Z dt =
Zdt = t+ C (19)
DKartofelev YFX1520 18 30
If you remove the damping and the external forcing terms the system becomes equivalent to theproblem of the particle in the double-well potential given by potential function
V (x) =1
4x4 minus 1
2x2 (26)
that was presentedstudied in Lecture 5 (go and check it out)
Figure 4 shows three ways you can think graphically about the solution of Duffing oscillator and itsPoincareacute section cf cdf3 nb3
Figure 4 (Left) Snapshots of a trajectory every 2π time units cf Slides 15 16 The 2π-periodicity isa natural choice since the external force is 2π-periodic for ω = 1 (Middle) A trajectory in the three-dimensional Cartesian xyz-space corresponding to Sys (18 slide numbering) (Right) A trajectory in thethree-dimensional space where the z-axis is 2π-periodic and is folded into a closed circle cf Slides 15 16The blue outline shows the Poincareacute section
The following numerical file shows the solution to Duffing oscillator and the quantitatively accuraterenderings of the three different ways to plot the solution of Duffing oscillator shows in Fig 4
Numerics cdf3 nb3Periodically forced damped Duffing oscillator Numerical solution and phase portrait 2π-periodicPoincareacute section of the Duffing oscillator (static)
The following slides show how the Poincareacute section of Duffing oscillator for t ≫ 1 changes as we continu-ously move it along the 2π-periodic time or z-axis as shown in Fig 5
D Kartofelev 1014 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Figure 5 Poincareacute section dynamics and periodic z-axis A moving Poincareacute section featuring a singleintersection point shown with the blue dot
Slides 19 20
Poincare section4 Dung oscillator
-15 -10 -05 00 05 10 15
-05
00
05
10
x(t)
y(t)
Poincareacute section
4See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 19 30
Poincare section dynamics Dung oscillator
No embedded video files in this pdf
DKartofelev YFX1520 20 30
Notice that the Poincareacute section is folding onto itself continuously stretching and the intersectionpoints are being progressively mixed (via re-injection process) within the region defined by theattractor This type of Poincareacute section dynamics is common for strange attractors
The dynamics of the Poincareacute section of Duffing oscillator is explored in the following numerical file
Numerics cdf4 nb4Periodically forced damped Duffing oscillator and dynamic Poincareacute section animation
212 Example Roumlssler attractor
Originally studied by Otto Roumlssler (1940ndash) The system is a product of pure imagination and it exhibitschaotic dynamics by design
D Kartofelev 1114 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 21ndash24
Rossler attractor
Rossler attractor5 is given by (as mentioned in Lecture 9)
8gtlt
gt
x = y x
y = x+ ay
z = b+ z(x c)
(20)
Chaotic solution exists for a = 01 b = 01 c = 14
5See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 21 30
Rossler attractor Poincare sections6
-10 -5 0 50
5
10
15
y(t)
z(t)
Poincareacute section yz-plane
6See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 22 30
The system has only one nonlinearity the zx term in the third equation
Rossler attractor Poincare sections7
-10 -5 0 5 100
5
10
15
20
25
30
x(t)
z(t)
Poincareacute section xz-plane
7See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 23 30
Rossler attractor return map and Poincare sections
Credit Y Maistrenko and R PaskauskasDKartofelev YFX1520 24 30
Slide 24 features the shape of Roumlssler map mdash the Poincareacute map of Roumlssler attractor
The following numerical file was used to calculate the attractor and the Poincareacute sections shown above
Numerics cdf5 nb5Numerical solution of Roumlssler attractor Poincareacute section of Roumlssler attractor
Slides 25 26
Rossler attractor orbit diagram
DKartofelev YFX1520 25 30
Rossler attractor period doubling
No embedded video files in this pdf
DKartofelev YFX1520 26 30
D Kartofelev 1214 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The orbit diagram shown on Slide 25 is calculated using the Roumlssler map shown on Slide 24
The following interactive numerical file shows the period doubling bifurcation happening in the xy-projection of Roumlssler attractor cf Slide 25
Numerics cdf6 nb6Periodic orbits in Roumlssler attractor for varied c value
The following slide shows the 2π-periodic dynamics of the Poincareacute section of Roumlssler attractor
Slide 27Poincare section dynamics Rossler attractor
No embedded video files in this pdf
DKartofelev YFX1520 27 30
Once again notice the folding stretching and mixing of the Poincareacute section cf Slides 19 and20 This type of Poincareacute section dynamics will be explored further in the next weekrsquos lecture
3 Lorenz section
Slide 28
Lorenz section
Section of a sectionDKartofelev YFX1520 28 30
The slide above shows a Poincareacute section of a strange attractor where we sliced the attractor with aplane thereby exposing its cross section If we take a further one-dimensional slice or Lorenz sectionthrough the Poincareacute section we find an infinite set of points separated by gaps of various sizes Inthe next weekrsquos lecture we will look into this discovery and into the notion of Lorenz section in greaterdetail
D Kartofelev 1314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
4 Attractor reconstruction
In applications it is often the case that you are able to measure or are restricted to only measuring onetime-series related to a higher order system mdash one aspect of a problem Letrsquos say you need to analyse theattractor that governs your measured one-dimensional time-series signal s(t) eg show that it is strangeAt first it might seem that there is not enough information But there exist a surprising data analysistechnique known as the attractor reconstruction
For systems governed by an attractor the dynamics in the full higher dimensional phase space canbe reconstructed from measurements of just a single time series s(t) Somehow a single variable carriessufficient information about all the others The method is based on time delays τ For instance define atwo-dimensional vector
983187x(t) = (s(t) s(t+ τ)) (27)
for some time delay τ gt 0 and signal s(t) where 983187x = (x y) Signal s(t) can therefore represent either thevariable x(t) or y(t) The time-series s(t) generates a trajectory 983187x(t) in a two-dimensional phase space thatrepresents its attractorrsquos trajectory You can also considered the attractor in three dimensions by definingthe three-dimensional vector
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)
for time delay τ gt 0 For four-dimensional attractor use
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)
etcThe following interactive numerical file demonstrates attractor reconstruction using three examples First
a sine wave is used to reconstruct its two-dimensional attractor mdash circle Second a quasi-periodic signalis used to reconstruct its three-dimensional attractor mdash torus Third time-series y(t) of Lorenz system isused to reconstruct the three-dimensional Lorenz attractor
Numerics cdf7 nb7Higher-dimensional attractor reconstruction from 1-D time-domain signals Examples sine wavequasi-periodic signal y(t) of Lorenz attractor
Revision questions
1 What are the values of Feigenbaum constants2 What are Feigenbaum constants3 Define superstable fixed point of a map4 Define superstable period-p point (or period-p orbit) of a map5 What are universals of unimodal maps6 What is the universal route to chaos7 Idea behind renormalisation8 What are universal limiting functions in the context of maps9 Name discrete time dynamics analysis methods
10 What is Poincareacute section11 What is Poincareacute map (return map)12 What is Lorenz section
D Kartofelev 1414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 8 9
Feigenbaumrsquos analysis renormalisation
f(xR0) f2(xR1)
(a) (b) (c)
DKartofelev YFX1520 8 30
Feigenbaumrsquos analysis renormalisation
f(xR0) f2(xR1) crarrf
2x
crarr R1
(a) (b) (c)
Read Mitchell J Feigenbaum ldquoThe universal metric properties ofnonlinear transformationsrdquo Journal of Statistical Physics 21(6) pp669ndash706 1979 Also relevant to the following three slides +
D Kartofelev YFX1520 9 30
The visual comparison shows that renormalised map is very similar to the original one The map max-imum xm shown on Slide 9 has beed shifted to the origin in order to simplify the math below
Letrsquos express the renormalisation procedure algebraically
f(xR0) asymp αf2983059xα R1
983060 (14)
notice that we have scaled the vertical and horizontal directions simultaneously In summary map fhas been renormalised by taking its second iterate rescaling x rarr xα shifting r to the next superstablevalue and multiplying everything by α
The higher order renormalisations of f(xR0) are given by
f(xR0) asymp α2f4983059 x
α2 R2
983060
asymp α3f8983059 x
α3 R3
983060
asymp α4f16983059 x
α4 R4
983060
asymp αnf (2n)983059 x
αn Rn
983060
(15)
What happens when we reach the accumulation point r = rinfin mdash reach the chaotic regime
limnrarrinfin
αnf (2n)983059 x
αn Rn
983060= g0(x) (16)
Feigenbaum found numerically that this limit converges only for α = minus2502907875 by producing the re-sulting limiting function g0 The function g0 is called the universal limiting function This function issimilar in shape to map f and it retains the superstability of its fixed point
The descriptor ldquouniversalrdquo in this context means that function g0 is almost independent of the unimodalfunction f How can g0 be independent of f Function g0 depends on f only through its behaviour nearx = 0 since thatrsquos all that survives in the argument xαn as n rarr infin With each renormalisation we areblowing up smaller and smaller neighbourhoods of the maximum of f until practically all of the information(global shape further away from x = 0) of f becomes meaningless The only remaining information is thequadratic nature of the maximum In the case of unimodal maps the Taylor expansion at xlowast = xm has tohave the second (quadratic) x2 term (superstability) The fixed point has to be what is called the seconddegree maximum point or the quadratic maximum
You can find the universal limiting functions related to any superstable period-2i orbit To obtain otheruniversal limiting functions gi(x) start with f(xRi) instead of f(xR0)
gi(x) = limnrarrinfin
αnf (2n)983059 x
αn Rn+i
983060 (17)
D Kartofelev 614 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
What happens at the onset of chaos i = infin Letrsquos find the universal limiting function at the onset of chaoswhere Ri = Rinfin We need to renormalise f(xRinfin) one time
ginfin(x) asymp αf2983059xα Rinfin+1
983060=
983063Remainderinfin+ 1 = infin
983064= αf2
983059xα Rinfin
983060= g(x) (18)
For once we donrsquot have to shift r when we renormalise (we became independent of i) The limiting func-tion ginfin(x) usually called simply g(x) satisfies
g(x) = αg2983059xα
983060 (19)
This is a functional equation for g(x) and the universal scale factor α It is self-referential primaryunknown g(x) is defined in terms of itself
Letrsquos try to solve Eq (19) The functional equation is not complete until we specify boundary conditionson g(x) After the shift of origin all our unimodal f rsquos have a maximum at x = 0 so we require
gprime(0) = 0 (20)
Also we can set
g(0) =
983063From Def (18)we know that
983064= g(0 Rinfin) = const (21)
tog(0) = 1 (22)
without loss of generality (This just defines the normalised scale for x if g(x) is a solution of Eq (19) sois microg(xmicro) where micro isin R with the same α) Naturally this selection agrees with boundary condition (20)
Now we can solve Eq (19) for function g(x) and α at the boundary x = 0
g(0) = αg29830610
α
983062= αg2(0) = αg(g(0)) (23)
since g(0) = 1 we have1 = αg(1) (24)
α =1
g(1) (25)
The universal scaling constant α depends on the limiting function g As of now no closed form solutionexist for function g We can resort to numerical method to find the shape of the limiting function g andthe value of α Next we solve Eq (19) with boundary condition (25) using the power series solution
14 Numerical determination of Feigenbaum α constant
Slides 10 11
Limiting function g(x) and Feigenbaum constant crarr
Letrsquos consider a functional equation in the following form
g(x) = crarr g(gx
crarr
) = crarr g
2x
crarr
(9)
where crarr acts as scaling coecient and
crarr =1
g(1) (10)
The power series solution is obtained by assuming power expansion
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 + ex10 + (11)
which assumes that the map maximum is quadratic
DKartofelev YFX1520 10 30
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution1 the one term approximation where
g(x) = 1 + ax2 +O(x4) (12)
results in
a = 1
2(1 +
p3) crarr =
1
g(1) 273205 (92 error) (13)
-04 -02 00 02 04000204060810
xn
x x+1 g(x) (one term)
Logistic map
Logistic map shifted
1See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 11 30
D Kartofelev 714 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The power series expansion uses Maclaurin expansion and assumes a quadratic maximum Thequadratic maximum follows from the superstability property The above process can generate lots ofextraneous root you will need to shift through them to find the correct solution
The solution shown above is taken from the following numerical file The file is commented for the benefitof students who are interested in numerical methods in general
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
In order to find a better match more terms in the power series expansion are needed
Slide 12
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution2 the four term approximation where
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 +O(x10) (14)
Coecients a = 1528 b = 01053 c = 002631 d = 0003344and
crarr =1
g(1) 250316 (001 error) (15)
-04 -02 00 02 04000204060810
xn
x x+1 g(x)
Logistic map
Logistic map shifted
2See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 12 30
The solution shown above is taken from the following numerical file
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
The renormalisation theory also explains the value of δ Unfortunately the derivations and analysis ofthe Feigenbaum δ constant requires advanced functional analysis know-how (operators in function spaceFrechet derivatives etc) and for this reason is omitted from this courseConclusion Something qualitative (unimodality) gave us something quantitative (Feigenbaumrsquos constantsα and δ) In science it is usually the other way around
Reading suggestion
Link File name CitationPaper1 paper5pdf Mitchell J Feigenbaum ldquoThe universal metric properties of nonlinear transfor-
mationsrdquo Journal of Statistical Physics 21(6) pp 669ndash706 (1979)doi101007BF01107909
2 Discrete time analysis methods
We have already used mapping of a continuous three-dimensional flow into a discrete one-dimensional mapmdash Lorenz attractor rarr Lorenz map Our goal in this section is to generalise this approach
D Kartofelev 814 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
21 Poincareacute section Poincareacute map return map
Poincareacute section is a set of intersection points between a hyperspace hyper-surface or hyperplane anda trajectory of n-dimensional continuous time flow Poincareacute map or return map is the (algebraic)relationship between these preceding and succeeding intersection points ie point xn and xn+1 found onPoincareacute section
Slides 13ndash16
Poincare section
Intersection of an attractor trajectory with the hypersurface s
DKartofelev YFX1520 13 30
Discrete time dynamics analysis
Construction of a Poincare map (return map recurrence map)
DKartofelev YFX1520 14 30
Poincare section periodic forcing3
3Credit J Thompson H Stewart 1986 Nonlinear dynamics and chaos
geometrical methods for engineers and scientists Chichester UK Wiley
DKartofelev YFX1520 15 30
Poincare section periodic forcing
DKartofelev YFX1520 16 30
Letrsquos demonstrate these concepts using two examples
211 Example Periodically driven damped Duffing oscillator
Duffing equation (or Duffing oscillator) named after Georg Duffing (1861ndash1944) is a nonlinear second-orderdifferential equation used to model certain damped and driven oscillators
Slides 17 18The first term on the left-hand side of Eq (16 slide numbering) is the inertial term the third term isthe nonlinear stiffness and the fourth term is the damping The non-homogeneous right-hand sideof the equation describes the external 2π-periodic forcing with the amplitude F and frequency ω = 1The solution for F gt 0 may be a strange attractor The external forcing of two-dimensional systemsis often the source of chaos But wait how can two-dimensional system feature chaos Isnrsquot chaossupposed to be reserved only for three or higher-dimensional systems The external forcing term allowsfor this second order system to be represented as a three-dimensional system
D Kartofelev 914 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Periodically driven damped Dung oscillator
Figure Mechanical system featuring Dung oscillations
The non-autonomous equation of motion has the following form
x x+ x3 + x = F cost (16)
where is the damping coecient F and are the forcing strengthand frequency respectively
DKartofelev YFX1520 17 30
Periodically driven damped Dung oscillator
We introduce the variable exchange y = x and rewrite (16)
(x = y
y = x x3 y + F cost
(17)
The equivalent 3-D system is obtained by applying variable exchangez = t 8
gtlt
gt
x = y
y = x x3 y + F cos z
z =
(18)
The above folds since
z =
Zz dt =
Z dt =
Zdt = t+ C (19)
DKartofelev YFX1520 18 30
If you remove the damping and the external forcing terms the system becomes equivalent to theproblem of the particle in the double-well potential given by potential function
V (x) =1
4x4 minus 1
2x2 (26)
that was presentedstudied in Lecture 5 (go and check it out)
Figure 4 shows three ways you can think graphically about the solution of Duffing oscillator and itsPoincareacute section cf cdf3 nb3
Figure 4 (Left) Snapshots of a trajectory every 2π time units cf Slides 15 16 The 2π-periodicity isa natural choice since the external force is 2π-periodic for ω = 1 (Middle) A trajectory in the three-dimensional Cartesian xyz-space corresponding to Sys (18 slide numbering) (Right) A trajectory in thethree-dimensional space where the z-axis is 2π-periodic and is folded into a closed circle cf Slides 15 16The blue outline shows the Poincareacute section
The following numerical file shows the solution to Duffing oscillator and the quantitatively accuraterenderings of the three different ways to plot the solution of Duffing oscillator shows in Fig 4
Numerics cdf3 nb3Periodically forced damped Duffing oscillator Numerical solution and phase portrait 2π-periodicPoincareacute section of the Duffing oscillator (static)
The following slides show how the Poincareacute section of Duffing oscillator for t ≫ 1 changes as we continu-ously move it along the 2π-periodic time or z-axis as shown in Fig 5
D Kartofelev 1014 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Figure 5 Poincareacute section dynamics and periodic z-axis A moving Poincareacute section featuring a singleintersection point shown with the blue dot
Slides 19 20
Poincare section4 Dung oscillator
-15 -10 -05 00 05 10 15
-05
00
05
10
x(t)
y(t)
Poincareacute section
4See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 19 30
Poincare section dynamics Dung oscillator
No embedded video files in this pdf
DKartofelev YFX1520 20 30
Notice that the Poincareacute section is folding onto itself continuously stretching and the intersectionpoints are being progressively mixed (via re-injection process) within the region defined by theattractor This type of Poincareacute section dynamics is common for strange attractors
The dynamics of the Poincareacute section of Duffing oscillator is explored in the following numerical file
Numerics cdf4 nb4Periodically forced damped Duffing oscillator and dynamic Poincareacute section animation
212 Example Roumlssler attractor
Originally studied by Otto Roumlssler (1940ndash) The system is a product of pure imagination and it exhibitschaotic dynamics by design
D Kartofelev 1114 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 21ndash24
Rossler attractor
Rossler attractor5 is given by (as mentioned in Lecture 9)
8gtlt
gt
x = y x
y = x+ ay
z = b+ z(x c)
(20)
Chaotic solution exists for a = 01 b = 01 c = 14
5See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 21 30
Rossler attractor Poincare sections6
-10 -5 0 50
5
10
15
y(t)
z(t)
Poincareacute section yz-plane
6See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 22 30
The system has only one nonlinearity the zx term in the third equation
Rossler attractor Poincare sections7
-10 -5 0 5 100
5
10
15
20
25
30
x(t)
z(t)
Poincareacute section xz-plane
7See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 23 30
Rossler attractor return map and Poincare sections
Credit Y Maistrenko and R PaskauskasDKartofelev YFX1520 24 30
Slide 24 features the shape of Roumlssler map mdash the Poincareacute map of Roumlssler attractor
The following numerical file was used to calculate the attractor and the Poincareacute sections shown above
Numerics cdf5 nb5Numerical solution of Roumlssler attractor Poincareacute section of Roumlssler attractor
Slides 25 26
Rossler attractor orbit diagram
DKartofelev YFX1520 25 30
Rossler attractor period doubling
No embedded video files in this pdf
DKartofelev YFX1520 26 30
D Kartofelev 1214 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The orbit diagram shown on Slide 25 is calculated using the Roumlssler map shown on Slide 24
The following interactive numerical file shows the period doubling bifurcation happening in the xy-projection of Roumlssler attractor cf Slide 25
Numerics cdf6 nb6Periodic orbits in Roumlssler attractor for varied c value
The following slide shows the 2π-periodic dynamics of the Poincareacute section of Roumlssler attractor
Slide 27Poincare section dynamics Rossler attractor
No embedded video files in this pdf
DKartofelev YFX1520 27 30
Once again notice the folding stretching and mixing of the Poincareacute section cf Slides 19 and20 This type of Poincareacute section dynamics will be explored further in the next weekrsquos lecture
3 Lorenz section
Slide 28
Lorenz section
Section of a sectionDKartofelev YFX1520 28 30
The slide above shows a Poincareacute section of a strange attractor where we sliced the attractor with aplane thereby exposing its cross section If we take a further one-dimensional slice or Lorenz sectionthrough the Poincareacute section we find an infinite set of points separated by gaps of various sizes Inthe next weekrsquos lecture we will look into this discovery and into the notion of Lorenz section in greaterdetail
D Kartofelev 1314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
4 Attractor reconstruction
In applications it is often the case that you are able to measure or are restricted to only measuring onetime-series related to a higher order system mdash one aspect of a problem Letrsquos say you need to analyse theattractor that governs your measured one-dimensional time-series signal s(t) eg show that it is strangeAt first it might seem that there is not enough information But there exist a surprising data analysistechnique known as the attractor reconstruction
For systems governed by an attractor the dynamics in the full higher dimensional phase space canbe reconstructed from measurements of just a single time series s(t) Somehow a single variable carriessufficient information about all the others The method is based on time delays τ For instance define atwo-dimensional vector
983187x(t) = (s(t) s(t+ τ)) (27)
for some time delay τ gt 0 and signal s(t) where 983187x = (x y) Signal s(t) can therefore represent either thevariable x(t) or y(t) The time-series s(t) generates a trajectory 983187x(t) in a two-dimensional phase space thatrepresents its attractorrsquos trajectory You can also considered the attractor in three dimensions by definingthe three-dimensional vector
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)
for time delay τ gt 0 For four-dimensional attractor use
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)
etcThe following interactive numerical file demonstrates attractor reconstruction using three examples First
a sine wave is used to reconstruct its two-dimensional attractor mdash circle Second a quasi-periodic signalis used to reconstruct its three-dimensional attractor mdash torus Third time-series y(t) of Lorenz system isused to reconstruct the three-dimensional Lorenz attractor
Numerics cdf7 nb7Higher-dimensional attractor reconstruction from 1-D time-domain signals Examples sine wavequasi-periodic signal y(t) of Lorenz attractor
Revision questions
1 What are the values of Feigenbaum constants2 What are Feigenbaum constants3 Define superstable fixed point of a map4 Define superstable period-p point (or period-p orbit) of a map5 What are universals of unimodal maps6 What is the universal route to chaos7 Idea behind renormalisation8 What are universal limiting functions in the context of maps9 Name discrete time dynamics analysis methods
10 What is Poincareacute section11 What is Poincareacute map (return map)12 What is Lorenz section
D Kartofelev 1414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
What happens at the onset of chaos i = infin Letrsquos find the universal limiting function at the onset of chaoswhere Ri = Rinfin We need to renormalise f(xRinfin) one time
ginfin(x) asymp αf2983059xα Rinfin+1
983060=
983063Remainderinfin+ 1 = infin
983064= αf2
983059xα Rinfin
983060= g(x) (18)
For once we donrsquot have to shift r when we renormalise (we became independent of i) The limiting func-tion ginfin(x) usually called simply g(x) satisfies
g(x) = αg2983059xα
983060 (19)
This is a functional equation for g(x) and the universal scale factor α It is self-referential primaryunknown g(x) is defined in terms of itself
Letrsquos try to solve Eq (19) The functional equation is not complete until we specify boundary conditionson g(x) After the shift of origin all our unimodal f rsquos have a maximum at x = 0 so we require
gprime(0) = 0 (20)
Also we can set
g(0) =
983063From Def (18)we know that
983064= g(0 Rinfin) = const (21)
tog(0) = 1 (22)
without loss of generality (This just defines the normalised scale for x if g(x) is a solution of Eq (19) sois microg(xmicro) where micro isin R with the same α) Naturally this selection agrees with boundary condition (20)
Now we can solve Eq (19) for function g(x) and α at the boundary x = 0
g(0) = αg29830610
α
983062= αg2(0) = αg(g(0)) (23)
since g(0) = 1 we have1 = αg(1) (24)
α =1
g(1) (25)
The universal scaling constant α depends on the limiting function g As of now no closed form solutionexist for function g We can resort to numerical method to find the shape of the limiting function g andthe value of α Next we solve Eq (19) with boundary condition (25) using the power series solution
14 Numerical determination of Feigenbaum α constant
Slides 10 11
Limiting function g(x) and Feigenbaum constant crarr
Letrsquos consider a functional equation in the following form
g(x) = crarr g(gx
crarr
) = crarr g
2x
crarr
(9)
where crarr acts as scaling coecient and
crarr =1
g(1) (10)
The power series solution is obtained by assuming power expansion
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 + ex10 + (11)
which assumes that the map maximum is quadratic
DKartofelev YFX1520 10 30
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution1 the one term approximation where
g(x) = 1 + ax2 +O(x4) (12)
results in
a = 1
2(1 +
p3) crarr =
1
g(1) 273205 (92 error) (13)
-04 -02 00 02 04000204060810
xn
x x+1 g(x) (one term)
Logistic map
Logistic map shifted
1See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 11 30
D Kartofelev 714 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The power series expansion uses Maclaurin expansion and assumes a quadratic maximum Thequadratic maximum follows from the superstability property The above process can generate lots ofextraneous root you will need to shift through them to find the correct solution
The solution shown above is taken from the following numerical file The file is commented for the benefitof students who are interested in numerical methods in general
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
In order to find a better match more terms in the power series expansion are needed
Slide 12
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution2 the four term approximation where
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 +O(x10) (14)
Coecients a = 1528 b = 01053 c = 002631 d = 0003344and
crarr =1
g(1) 250316 (001 error) (15)
-04 -02 00 02 04000204060810
xn
x x+1 g(x)
Logistic map
Logistic map shifted
2See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 12 30
The solution shown above is taken from the following numerical file
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
The renormalisation theory also explains the value of δ Unfortunately the derivations and analysis ofthe Feigenbaum δ constant requires advanced functional analysis know-how (operators in function spaceFrechet derivatives etc) and for this reason is omitted from this courseConclusion Something qualitative (unimodality) gave us something quantitative (Feigenbaumrsquos constantsα and δ) In science it is usually the other way around
Reading suggestion
Link File name CitationPaper1 paper5pdf Mitchell J Feigenbaum ldquoThe universal metric properties of nonlinear transfor-
mationsrdquo Journal of Statistical Physics 21(6) pp 669ndash706 (1979)doi101007BF01107909
2 Discrete time analysis methods
We have already used mapping of a continuous three-dimensional flow into a discrete one-dimensional mapmdash Lorenz attractor rarr Lorenz map Our goal in this section is to generalise this approach
D Kartofelev 814 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
21 Poincareacute section Poincareacute map return map
Poincareacute section is a set of intersection points between a hyperspace hyper-surface or hyperplane anda trajectory of n-dimensional continuous time flow Poincareacute map or return map is the (algebraic)relationship between these preceding and succeeding intersection points ie point xn and xn+1 found onPoincareacute section
Slides 13ndash16
Poincare section
Intersection of an attractor trajectory with the hypersurface s
DKartofelev YFX1520 13 30
Discrete time dynamics analysis
Construction of a Poincare map (return map recurrence map)
DKartofelev YFX1520 14 30
Poincare section periodic forcing3
3Credit J Thompson H Stewart 1986 Nonlinear dynamics and chaos
geometrical methods for engineers and scientists Chichester UK Wiley
DKartofelev YFX1520 15 30
Poincare section periodic forcing
DKartofelev YFX1520 16 30
Letrsquos demonstrate these concepts using two examples
211 Example Periodically driven damped Duffing oscillator
Duffing equation (or Duffing oscillator) named after Georg Duffing (1861ndash1944) is a nonlinear second-orderdifferential equation used to model certain damped and driven oscillators
Slides 17 18The first term on the left-hand side of Eq (16 slide numbering) is the inertial term the third term isthe nonlinear stiffness and the fourth term is the damping The non-homogeneous right-hand sideof the equation describes the external 2π-periodic forcing with the amplitude F and frequency ω = 1The solution for F gt 0 may be a strange attractor The external forcing of two-dimensional systemsis often the source of chaos But wait how can two-dimensional system feature chaos Isnrsquot chaossupposed to be reserved only for three or higher-dimensional systems The external forcing term allowsfor this second order system to be represented as a three-dimensional system
D Kartofelev 914 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Periodically driven damped Dung oscillator
Figure Mechanical system featuring Dung oscillations
The non-autonomous equation of motion has the following form
x x+ x3 + x = F cost (16)
where is the damping coecient F and are the forcing strengthand frequency respectively
DKartofelev YFX1520 17 30
Periodically driven damped Dung oscillator
We introduce the variable exchange y = x and rewrite (16)
(x = y
y = x x3 y + F cost
(17)
The equivalent 3-D system is obtained by applying variable exchangez = t 8
gtlt
gt
x = y
y = x x3 y + F cos z
z =
(18)
The above folds since
z =
Zz dt =
Z dt =
Zdt = t+ C (19)
DKartofelev YFX1520 18 30
If you remove the damping and the external forcing terms the system becomes equivalent to theproblem of the particle in the double-well potential given by potential function
V (x) =1
4x4 minus 1
2x2 (26)
that was presentedstudied in Lecture 5 (go and check it out)
Figure 4 shows three ways you can think graphically about the solution of Duffing oscillator and itsPoincareacute section cf cdf3 nb3
Figure 4 (Left) Snapshots of a trajectory every 2π time units cf Slides 15 16 The 2π-periodicity isa natural choice since the external force is 2π-periodic for ω = 1 (Middle) A trajectory in the three-dimensional Cartesian xyz-space corresponding to Sys (18 slide numbering) (Right) A trajectory in thethree-dimensional space where the z-axis is 2π-periodic and is folded into a closed circle cf Slides 15 16The blue outline shows the Poincareacute section
The following numerical file shows the solution to Duffing oscillator and the quantitatively accuraterenderings of the three different ways to plot the solution of Duffing oscillator shows in Fig 4
Numerics cdf3 nb3Periodically forced damped Duffing oscillator Numerical solution and phase portrait 2π-periodicPoincareacute section of the Duffing oscillator (static)
The following slides show how the Poincareacute section of Duffing oscillator for t ≫ 1 changes as we continu-ously move it along the 2π-periodic time or z-axis as shown in Fig 5
D Kartofelev 1014 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Figure 5 Poincareacute section dynamics and periodic z-axis A moving Poincareacute section featuring a singleintersection point shown with the blue dot
Slides 19 20
Poincare section4 Dung oscillator
-15 -10 -05 00 05 10 15
-05
00
05
10
x(t)
y(t)
Poincareacute section
4See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 19 30
Poincare section dynamics Dung oscillator
No embedded video files in this pdf
DKartofelev YFX1520 20 30
Notice that the Poincareacute section is folding onto itself continuously stretching and the intersectionpoints are being progressively mixed (via re-injection process) within the region defined by theattractor This type of Poincareacute section dynamics is common for strange attractors
The dynamics of the Poincareacute section of Duffing oscillator is explored in the following numerical file
Numerics cdf4 nb4Periodically forced damped Duffing oscillator and dynamic Poincareacute section animation
212 Example Roumlssler attractor
Originally studied by Otto Roumlssler (1940ndash) The system is a product of pure imagination and it exhibitschaotic dynamics by design
D Kartofelev 1114 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 21ndash24
Rossler attractor
Rossler attractor5 is given by (as mentioned in Lecture 9)
8gtlt
gt
x = y x
y = x+ ay
z = b+ z(x c)
(20)
Chaotic solution exists for a = 01 b = 01 c = 14
5See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 21 30
Rossler attractor Poincare sections6
-10 -5 0 50
5
10
15
y(t)
z(t)
Poincareacute section yz-plane
6See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 22 30
The system has only one nonlinearity the zx term in the third equation
Rossler attractor Poincare sections7
-10 -5 0 5 100
5
10
15
20
25
30
x(t)
z(t)
Poincareacute section xz-plane
7See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 23 30
Rossler attractor return map and Poincare sections
Credit Y Maistrenko and R PaskauskasDKartofelev YFX1520 24 30
Slide 24 features the shape of Roumlssler map mdash the Poincareacute map of Roumlssler attractor
The following numerical file was used to calculate the attractor and the Poincareacute sections shown above
Numerics cdf5 nb5Numerical solution of Roumlssler attractor Poincareacute section of Roumlssler attractor
Slides 25 26
Rossler attractor orbit diagram
DKartofelev YFX1520 25 30
Rossler attractor period doubling
No embedded video files in this pdf
DKartofelev YFX1520 26 30
D Kartofelev 1214 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The orbit diagram shown on Slide 25 is calculated using the Roumlssler map shown on Slide 24
The following interactive numerical file shows the period doubling bifurcation happening in the xy-projection of Roumlssler attractor cf Slide 25
Numerics cdf6 nb6Periodic orbits in Roumlssler attractor for varied c value
The following slide shows the 2π-periodic dynamics of the Poincareacute section of Roumlssler attractor
Slide 27Poincare section dynamics Rossler attractor
No embedded video files in this pdf
DKartofelev YFX1520 27 30
Once again notice the folding stretching and mixing of the Poincareacute section cf Slides 19 and20 This type of Poincareacute section dynamics will be explored further in the next weekrsquos lecture
3 Lorenz section
Slide 28
Lorenz section
Section of a sectionDKartofelev YFX1520 28 30
The slide above shows a Poincareacute section of a strange attractor where we sliced the attractor with aplane thereby exposing its cross section If we take a further one-dimensional slice or Lorenz sectionthrough the Poincareacute section we find an infinite set of points separated by gaps of various sizes Inthe next weekrsquos lecture we will look into this discovery and into the notion of Lorenz section in greaterdetail
D Kartofelev 1314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
4 Attractor reconstruction
In applications it is often the case that you are able to measure or are restricted to only measuring onetime-series related to a higher order system mdash one aspect of a problem Letrsquos say you need to analyse theattractor that governs your measured one-dimensional time-series signal s(t) eg show that it is strangeAt first it might seem that there is not enough information But there exist a surprising data analysistechnique known as the attractor reconstruction
For systems governed by an attractor the dynamics in the full higher dimensional phase space canbe reconstructed from measurements of just a single time series s(t) Somehow a single variable carriessufficient information about all the others The method is based on time delays τ For instance define atwo-dimensional vector
983187x(t) = (s(t) s(t+ τ)) (27)
for some time delay τ gt 0 and signal s(t) where 983187x = (x y) Signal s(t) can therefore represent either thevariable x(t) or y(t) The time-series s(t) generates a trajectory 983187x(t) in a two-dimensional phase space thatrepresents its attractorrsquos trajectory You can also considered the attractor in three dimensions by definingthe three-dimensional vector
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)
for time delay τ gt 0 For four-dimensional attractor use
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)
etcThe following interactive numerical file demonstrates attractor reconstruction using three examples First
a sine wave is used to reconstruct its two-dimensional attractor mdash circle Second a quasi-periodic signalis used to reconstruct its three-dimensional attractor mdash torus Third time-series y(t) of Lorenz system isused to reconstruct the three-dimensional Lorenz attractor
Numerics cdf7 nb7Higher-dimensional attractor reconstruction from 1-D time-domain signals Examples sine wavequasi-periodic signal y(t) of Lorenz attractor
Revision questions
1 What are the values of Feigenbaum constants2 What are Feigenbaum constants3 Define superstable fixed point of a map4 Define superstable period-p point (or period-p orbit) of a map5 What are universals of unimodal maps6 What is the universal route to chaos7 Idea behind renormalisation8 What are universal limiting functions in the context of maps9 Name discrete time dynamics analysis methods
10 What is Poincareacute section11 What is Poincareacute map (return map)12 What is Lorenz section
D Kartofelev 1414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The power series expansion uses Maclaurin expansion and assumes a quadratic maximum Thequadratic maximum follows from the superstability property The above process can generate lots ofextraneous root you will need to shift through them to find the correct solution
The solution shown above is taken from the following numerical file The file is commented for the benefitof students who are interested in numerical methods in general
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
In order to find a better match more terms in the power series expansion are needed
Slide 12
Limiting function g(x) and Feigenbaum constant crarr
Numeric solution2 the four term approximation where
g(x) = 1 + ax2 + bx
4 + cx6 + dx
8 +O(x10) (14)
Coecients a = 1528 b = 01053 c = 002631 d = 0003344and
crarr =1
g(1) 250316 (001 error) (15)
-04 -02 00 02 04000204060810
xn
x x+1 g(x)
Logistic map
Logistic map shifted
2See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 12 30
The solution shown above is taken from the following numerical file
Numerics cdf2 nb2Calculation of universal limiting function g and value of Feigenbaum constant α in the case of Logisticmap using power series expansion (one and four term approximations)
The renormalisation theory also explains the value of δ Unfortunately the derivations and analysis ofthe Feigenbaum δ constant requires advanced functional analysis know-how (operators in function spaceFrechet derivatives etc) and for this reason is omitted from this courseConclusion Something qualitative (unimodality) gave us something quantitative (Feigenbaumrsquos constantsα and δ) In science it is usually the other way around
Reading suggestion
Link File name CitationPaper1 paper5pdf Mitchell J Feigenbaum ldquoThe universal metric properties of nonlinear transfor-
mationsrdquo Journal of Statistical Physics 21(6) pp 669ndash706 (1979)doi101007BF01107909
2 Discrete time analysis methods
We have already used mapping of a continuous three-dimensional flow into a discrete one-dimensional mapmdash Lorenz attractor rarr Lorenz map Our goal in this section is to generalise this approach
D Kartofelev 814 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
21 Poincareacute section Poincareacute map return map
Poincareacute section is a set of intersection points between a hyperspace hyper-surface or hyperplane anda trajectory of n-dimensional continuous time flow Poincareacute map or return map is the (algebraic)relationship between these preceding and succeeding intersection points ie point xn and xn+1 found onPoincareacute section
Slides 13ndash16
Poincare section
Intersection of an attractor trajectory with the hypersurface s
DKartofelev YFX1520 13 30
Discrete time dynamics analysis
Construction of a Poincare map (return map recurrence map)
DKartofelev YFX1520 14 30
Poincare section periodic forcing3
3Credit J Thompson H Stewart 1986 Nonlinear dynamics and chaos
geometrical methods for engineers and scientists Chichester UK Wiley
DKartofelev YFX1520 15 30
Poincare section periodic forcing
DKartofelev YFX1520 16 30
Letrsquos demonstrate these concepts using two examples
211 Example Periodically driven damped Duffing oscillator
Duffing equation (or Duffing oscillator) named after Georg Duffing (1861ndash1944) is a nonlinear second-orderdifferential equation used to model certain damped and driven oscillators
Slides 17 18The first term on the left-hand side of Eq (16 slide numbering) is the inertial term the third term isthe nonlinear stiffness and the fourth term is the damping The non-homogeneous right-hand sideof the equation describes the external 2π-periodic forcing with the amplitude F and frequency ω = 1The solution for F gt 0 may be a strange attractor The external forcing of two-dimensional systemsis often the source of chaos But wait how can two-dimensional system feature chaos Isnrsquot chaossupposed to be reserved only for three or higher-dimensional systems The external forcing term allowsfor this second order system to be represented as a three-dimensional system
D Kartofelev 914 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Periodically driven damped Dung oscillator
Figure Mechanical system featuring Dung oscillations
The non-autonomous equation of motion has the following form
x x+ x3 + x = F cost (16)
where is the damping coecient F and are the forcing strengthand frequency respectively
DKartofelev YFX1520 17 30
Periodically driven damped Dung oscillator
We introduce the variable exchange y = x and rewrite (16)
(x = y
y = x x3 y + F cost
(17)
The equivalent 3-D system is obtained by applying variable exchangez = t 8
gtlt
gt
x = y
y = x x3 y + F cos z
z =
(18)
The above folds since
z =
Zz dt =
Z dt =
Zdt = t+ C (19)
DKartofelev YFX1520 18 30
If you remove the damping and the external forcing terms the system becomes equivalent to theproblem of the particle in the double-well potential given by potential function
V (x) =1
4x4 minus 1
2x2 (26)
that was presentedstudied in Lecture 5 (go and check it out)
Figure 4 shows three ways you can think graphically about the solution of Duffing oscillator and itsPoincareacute section cf cdf3 nb3
Figure 4 (Left) Snapshots of a trajectory every 2π time units cf Slides 15 16 The 2π-periodicity isa natural choice since the external force is 2π-periodic for ω = 1 (Middle) A trajectory in the three-dimensional Cartesian xyz-space corresponding to Sys (18 slide numbering) (Right) A trajectory in thethree-dimensional space where the z-axis is 2π-periodic and is folded into a closed circle cf Slides 15 16The blue outline shows the Poincareacute section
The following numerical file shows the solution to Duffing oscillator and the quantitatively accuraterenderings of the three different ways to plot the solution of Duffing oscillator shows in Fig 4
Numerics cdf3 nb3Periodically forced damped Duffing oscillator Numerical solution and phase portrait 2π-periodicPoincareacute section of the Duffing oscillator (static)
The following slides show how the Poincareacute section of Duffing oscillator for t ≫ 1 changes as we continu-ously move it along the 2π-periodic time or z-axis as shown in Fig 5
D Kartofelev 1014 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Figure 5 Poincareacute section dynamics and periodic z-axis A moving Poincareacute section featuring a singleintersection point shown with the blue dot
Slides 19 20
Poincare section4 Dung oscillator
-15 -10 -05 00 05 10 15
-05
00
05
10
x(t)
y(t)
Poincareacute section
4See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 19 30
Poincare section dynamics Dung oscillator
No embedded video files in this pdf
DKartofelev YFX1520 20 30
Notice that the Poincareacute section is folding onto itself continuously stretching and the intersectionpoints are being progressively mixed (via re-injection process) within the region defined by theattractor This type of Poincareacute section dynamics is common for strange attractors
The dynamics of the Poincareacute section of Duffing oscillator is explored in the following numerical file
Numerics cdf4 nb4Periodically forced damped Duffing oscillator and dynamic Poincareacute section animation
212 Example Roumlssler attractor
Originally studied by Otto Roumlssler (1940ndash) The system is a product of pure imagination and it exhibitschaotic dynamics by design
D Kartofelev 1114 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 21ndash24
Rossler attractor
Rossler attractor5 is given by (as mentioned in Lecture 9)
8gtlt
gt
x = y x
y = x+ ay
z = b+ z(x c)
(20)
Chaotic solution exists for a = 01 b = 01 c = 14
5See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 21 30
Rossler attractor Poincare sections6
-10 -5 0 50
5
10
15
y(t)
z(t)
Poincareacute section yz-plane
6See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 22 30
The system has only one nonlinearity the zx term in the third equation
Rossler attractor Poincare sections7
-10 -5 0 5 100
5
10
15
20
25
30
x(t)
z(t)
Poincareacute section xz-plane
7See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 23 30
Rossler attractor return map and Poincare sections
Credit Y Maistrenko and R PaskauskasDKartofelev YFX1520 24 30
Slide 24 features the shape of Roumlssler map mdash the Poincareacute map of Roumlssler attractor
The following numerical file was used to calculate the attractor and the Poincareacute sections shown above
Numerics cdf5 nb5Numerical solution of Roumlssler attractor Poincareacute section of Roumlssler attractor
Slides 25 26
Rossler attractor orbit diagram
DKartofelev YFX1520 25 30
Rossler attractor period doubling
No embedded video files in this pdf
DKartofelev YFX1520 26 30
D Kartofelev 1214 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The orbit diagram shown on Slide 25 is calculated using the Roumlssler map shown on Slide 24
The following interactive numerical file shows the period doubling bifurcation happening in the xy-projection of Roumlssler attractor cf Slide 25
Numerics cdf6 nb6Periodic orbits in Roumlssler attractor for varied c value
The following slide shows the 2π-periodic dynamics of the Poincareacute section of Roumlssler attractor
Slide 27Poincare section dynamics Rossler attractor
No embedded video files in this pdf
DKartofelev YFX1520 27 30
Once again notice the folding stretching and mixing of the Poincareacute section cf Slides 19 and20 This type of Poincareacute section dynamics will be explored further in the next weekrsquos lecture
3 Lorenz section
Slide 28
Lorenz section
Section of a sectionDKartofelev YFX1520 28 30
The slide above shows a Poincareacute section of a strange attractor where we sliced the attractor with aplane thereby exposing its cross section If we take a further one-dimensional slice or Lorenz sectionthrough the Poincareacute section we find an infinite set of points separated by gaps of various sizes Inthe next weekrsquos lecture we will look into this discovery and into the notion of Lorenz section in greaterdetail
D Kartofelev 1314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
4 Attractor reconstruction
In applications it is often the case that you are able to measure or are restricted to only measuring onetime-series related to a higher order system mdash one aspect of a problem Letrsquos say you need to analyse theattractor that governs your measured one-dimensional time-series signal s(t) eg show that it is strangeAt first it might seem that there is not enough information But there exist a surprising data analysistechnique known as the attractor reconstruction
For systems governed by an attractor the dynamics in the full higher dimensional phase space canbe reconstructed from measurements of just a single time series s(t) Somehow a single variable carriessufficient information about all the others The method is based on time delays τ For instance define atwo-dimensional vector
983187x(t) = (s(t) s(t+ τ)) (27)
for some time delay τ gt 0 and signal s(t) where 983187x = (x y) Signal s(t) can therefore represent either thevariable x(t) or y(t) The time-series s(t) generates a trajectory 983187x(t) in a two-dimensional phase space thatrepresents its attractorrsquos trajectory You can also considered the attractor in three dimensions by definingthe three-dimensional vector
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)
for time delay τ gt 0 For four-dimensional attractor use
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)
etcThe following interactive numerical file demonstrates attractor reconstruction using three examples First
a sine wave is used to reconstruct its two-dimensional attractor mdash circle Second a quasi-periodic signalis used to reconstruct its three-dimensional attractor mdash torus Third time-series y(t) of Lorenz system isused to reconstruct the three-dimensional Lorenz attractor
Numerics cdf7 nb7Higher-dimensional attractor reconstruction from 1-D time-domain signals Examples sine wavequasi-periodic signal y(t) of Lorenz attractor
Revision questions
1 What are the values of Feigenbaum constants2 What are Feigenbaum constants3 Define superstable fixed point of a map4 Define superstable period-p point (or period-p orbit) of a map5 What are universals of unimodal maps6 What is the universal route to chaos7 Idea behind renormalisation8 What are universal limiting functions in the context of maps9 Name discrete time dynamics analysis methods
10 What is Poincareacute section11 What is Poincareacute map (return map)12 What is Lorenz section
D Kartofelev 1414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
21 Poincareacute section Poincareacute map return map
Poincareacute section is a set of intersection points between a hyperspace hyper-surface or hyperplane anda trajectory of n-dimensional continuous time flow Poincareacute map or return map is the (algebraic)relationship between these preceding and succeeding intersection points ie point xn and xn+1 found onPoincareacute section
Slides 13ndash16
Poincare section
Intersection of an attractor trajectory with the hypersurface s
DKartofelev YFX1520 13 30
Discrete time dynamics analysis
Construction of a Poincare map (return map recurrence map)
DKartofelev YFX1520 14 30
Poincare section periodic forcing3
3Credit J Thompson H Stewart 1986 Nonlinear dynamics and chaos
geometrical methods for engineers and scientists Chichester UK Wiley
DKartofelev YFX1520 15 30
Poincare section periodic forcing
DKartofelev YFX1520 16 30
Letrsquos demonstrate these concepts using two examples
211 Example Periodically driven damped Duffing oscillator
Duffing equation (or Duffing oscillator) named after Georg Duffing (1861ndash1944) is a nonlinear second-orderdifferential equation used to model certain damped and driven oscillators
Slides 17 18The first term on the left-hand side of Eq (16 slide numbering) is the inertial term the third term isthe nonlinear stiffness and the fourth term is the damping The non-homogeneous right-hand sideof the equation describes the external 2π-periodic forcing with the amplitude F and frequency ω = 1The solution for F gt 0 may be a strange attractor The external forcing of two-dimensional systemsis often the source of chaos But wait how can two-dimensional system feature chaos Isnrsquot chaossupposed to be reserved only for three or higher-dimensional systems The external forcing term allowsfor this second order system to be represented as a three-dimensional system
D Kartofelev 914 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Periodically driven damped Dung oscillator
Figure Mechanical system featuring Dung oscillations
The non-autonomous equation of motion has the following form
x x+ x3 + x = F cost (16)
where is the damping coecient F and are the forcing strengthand frequency respectively
DKartofelev YFX1520 17 30
Periodically driven damped Dung oscillator
We introduce the variable exchange y = x and rewrite (16)
(x = y
y = x x3 y + F cost
(17)
The equivalent 3-D system is obtained by applying variable exchangez = t 8
gtlt
gt
x = y
y = x x3 y + F cos z
z =
(18)
The above folds since
z =
Zz dt =
Z dt =
Zdt = t+ C (19)
DKartofelev YFX1520 18 30
If you remove the damping and the external forcing terms the system becomes equivalent to theproblem of the particle in the double-well potential given by potential function
V (x) =1
4x4 minus 1
2x2 (26)
that was presentedstudied in Lecture 5 (go and check it out)
Figure 4 shows three ways you can think graphically about the solution of Duffing oscillator and itsPoincareacute section cf cdf3 nb3
Figure 4 (Left) Snapshots of a trajectory every 2π time units cf Slides 15 16 The 2π-periodicity isa natural choice since the external force is 2π-periodic for ω = 1 (Middle) A trajectory in the three-dimensional Cartesian xyz-space corresponding to Sys (18 slide numbering) (Right) A trajectory in thethree-dimensional space where the z-axis is 2π-periodic and is folded into a closed circle cf Slides 15 16The blue outline shows the Poincareacute section
The following numerical file shows the solution to Duffing oscillator and the quantitatively accuraterenderings of the three different ways to plot the solution of Duffing oscillator shows in Fig 4
Numerics cdf3 nb3Periodically forced damped Duffing oscillator Numerical solution and phase portrait 2π-periodicPoincareacute section of the Duffing oscillator (static)
The following slides show how the Poincareacute section of Duffing oscillator for t ≫ 1 changes as we continu-ously move it along the 2π-periodic time or z-axis as shown in Fig 5
D Kartofelev 1014 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Figure 5 Poincareacute section dynamics and periodic z-axis A moving Poincareacute section featuring a singleintersection point shown with the blue dot
Slides 19 20
Poincare section4 Dung oscillator
-15 -10 -05 00 05 10 15
-05
00
05
10
x(t)
y(t)
Poincareacute section
4See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 19 30
Poincare section dynamics Dung oscillator
No embedded video files in this pdf
DKartofelev YFX1520 20 30
Notice that the Poincareacute section is folding onto itself continuously stretching and the intersectionpoints are being progressively mixed (via re-injection process) within the region defined by theattractor This type of Poincareacute section dynamics is common for strange attractors
The dynamics of the Poincareacute section of Duffing oscillator is explored in the following numerical file
Numerics cdf4 nb4Periodically forced damped Duffing oscillator and dynamic Poincareacute section animation
212 Example Roumlssler attractor
Originally studied by Otto Roumlssler (1940ndash) The system is a product of pure imagination and it exhibitschaotic dynamics by design
D Kartofelev 1114 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 21ndash24
Rossler attractor
Rossler attractor5 is given by (as mentioned in Lecture 9)
8gtlt
gt
x = y x
y = x+ ay
z = b+ z(x c)
(20)
Chaotic solution exists for a = 01 b = 01 c = 14
5See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 21 30
Rossler attractor Poincare sections6
-10 -5 0 50
5
10
15
y(t)
z(t)
Poincareacute section yz-plane
6See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 22 30
The system has only one nonlinearity the zx term in the third equation
Rossler attractor Poincare sections7
-10 -5 0 5 100
5
10
15
20
25
30
x(t)
z(t)
Poincareacute section xz-plane
7See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 23 30
Rossler attractor return map and Poincare sections
Credit Y Maistrenko and R PaskauskasDKartofelev YFX1520 24 30
Slide 24 features the shape of Roumlssler map mdash the Poincareacute map of Roumlssler attractor
The following numerical file was used to calculate the attractor and the Poincareacute sections shown above
Numerics cdf5 nb5Numerical solution of Roumlssler attractor Poincareacute section of Roumlssler attractor
Slides 25 26
Rossler attractor orbit diagram
DKartofelev YFX1520 25 30
Rossler attractor period doubling
No embedded video files in this pdf
DKartofelev YFX1520 26 30
D Kartofelev 1214 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The orbit diagram shown on Slide 25 is calculated using the Roumlssler map shown on Slide 24
The following interactive numerical file shows the period doubling bifurcation happening in the xy-projection of Roumlssler attractor cf Slide 25
Numerics cdf6 nb6Periodic orbits in Roumlssler attractor for varied c value
The following slide shows the 2π-periodic dynamics of the Poincareacute section of Roumlssler attractor
Slide 27Poincare section dynamics Rossler attractor
No embedded video files in this pdf
DKartofelev YFX1520 27 30
Once again notice the folding stretching and mixing of the Poincareacute section cf Slides 19 and20 This type of Poincareacute section dynamics will be explored further in the next weekrsquos lecture
3 Lorenz section
Slide 28
Lorenz section
Section of a sectionDKartofelev YFX1520 28 30
The slide above shows a Poincareacute section of a strange attractor where we sliced the attractor with aplane thereby exposing its cross section If we take a further one-dimensional slice or Lorenz sectionthrough the Poincareacute section we find an infinite set of points separated by gaps of various sizes Inthe next weekrsquos lecture we will look into this discovery and into the notion of Lorenz section in greaterdetail
D Kartofelev 1314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
4 Attractor reconstruction
In applications it is often the case that you are able to measure or are restricted to only measuring onetime-series related to a higher order system mdash one aspect of a problem Letrsquos say you need to analyse theattractor that governs your measured one-dimensional time-series signal s(t) eg show that it is strangeAt first it might seem that there is not enough information But there exist a surprising data analysistechnique known as the attractor reconstruction
For systems governed by an attractor the dynamics in the full higher dimensional phase space canbe reconstructed from measurements of just a single time series s(t) Somehow a single variable carriessufficient information about all the others The method is based on time delays τ For instance define atwo-dimensional vector
983187x(t) = (s(t) s(t+ τ)) (27)
for some time delay τ gt 0 and signal s(t) where 983187x = (x y) Signal s(t) can therefore represent either thevariable x(t) or y(t) The time-series s(t) generates a trajectory 983187x(t) in a two-dimensional phase space thatrepresents its attractorrsquos trajectory You can also considered the attractor in three dimensions by definingthe three-dimensional vector
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)
for time delay τ gt 0 For four-dimensional attractor use
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)
etcThe following interactive numerical file demonstrates attractor reconstruction using three examples First
a sine wave is used to reconstruct its two-dimensional attractor mdash circle Second a quasi-periodic signalis used to reconstruct its three-dimensional attractor mdash torus Third time-series y(t) of Lorenz system isused to reconstruct the three-dimensional Lorenz attractor
Numerics cdf7 nb7Higher-dimensional attractor reconstruction from 1-D time-domain signals Examples sine wavequasi-periodic signal y(t) of Lorenz attractor
Revision questions
1 What are the values of Feigenbaum constants2 What are Feigenbaum constants3 Define superstable fixed point of a map4 Define superstable period-p point (or period-p orbit) of a map5 What are universals of unimodal maps6 What is the universal route to chaos7 Idea behind renormalisation8 What are universal limiting functions in the context of maps9 Name discrete time dynamics analysis methods
10 What is Poincareacute section11 What is Poincareacute map (return map)12 What is Lorenz section
D Kartofelev 1414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Periodically driven damped Dung oscillator
Figure Mechanical system featuring Dung oscillations
The non-autonomous equation of motion has the following form
x x+ x3 + x = F cost (16)
where is the damping coecient F and are the forcing strengthand frequency respectively
DKartofelev YFX1520 17 30
Periodically driven damped Dung oscillator
We introduce the variable exchange y = x and rewrite (16)
(x = y
y = x x3 y + F cost
(17)
The equivalent 3-D system is obtained by applying variable exchangez = t 8
gtlt
gt
x = y
y = x x3 y + F cos z
z =
(18)
The above folds since
z =
Zz dt =
Z dt =
Zdt = t+ C (19)
DKartofelev YFX1520 18 30
If you remove the damping and the external forcing terms the system becomes equivalent to theproblem of the particle in the double-well potential given by potential function
V (x) =1
4x4 minus 1
2x2 (26)
that was presentedstudied in Lecture 5 (go and check it out)
Figure 4 shows three ways you can think graphically about the solution of Duffing oscillator and itsPoincareacute section cf cdf3 nb3
Figure 4 (Left) Snapshots of a trajectory every 2π time units cf Slides 15 16 The 2π-periodicity isa natural choice since the external force is 2π-periodic for ω = 1 (Middle) A trajectory in the three-dimensional Cartesian xyz-space corresponding to Sys (18 slide numbering) (Right) A trajectory in thethree-dimensional space where the z-axis is 2π-periodic and is folded into a closed circle cf Slides 15 16The blue outline shows the Poincareacute section
The following numerical file shows the solution to Duffing oscillator and the quantitatively accuraterenderings of the three different ways to plot the solution of Duffing oscillator shows in Fig 4
Numerics cdf3 nb3Periodically forced damped Duffing oscillator Numerical solution and phase portrait 2π-periodicPoincareacute section of the Duffing oscillator (static)
The following slides show how the Poincareacute section of Duffing oscillator for t ≫ 1 changes as we continu-ously move it along the 2π-periodic time or z-axis as shown in Fig 5
D Kartofelev 1014 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Figure 5 Poincareacute section dynamics and periodic z-axis A moving Poincareacute section featuring a singleintersection point shown with the blue dot
Slides 19 20
Poincare section4 Dung oscillator
-15 -10 -05 00 05 10 15
-05
00
05
10
x(t)
y(t)
Poincareacute section
4See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 19 30
Poincare section dynamics Dung oscillator
No embedded video files in this pdf
DKartofelev YFX1520 20 30
Notice that the Poincareacute section is folding onto itself continuously stretching and the intersectionpoints are being progressively mixed (via re-injection process) within the region defined by theattractor This type of Poincareacute section dynamics is common for strange attractors
The dynamics of the Poincareacute section of Duffing oscillator is explored in the following numerical file
Numerics cdf4 nb4Periodically forced damped Duffing oscillator and dynamic Poincareacute section animation
212 Example Roumlssler attractor
Originally studied by Otto Roumlssler (1940ndash) The system is a product of pure imagination and it exhibitschaotic dynamics by design
D Kartofelev 1114 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 21ndash24
Rossler attractor
Rossler attractor5 is given by (as mentioned in Lecture 9)
8gtlt
gt
x = y x
y = x+ ay
z = b+ z(x c)
(20)
Chaotic solution exists for a = 01 b = 01 c = 14
5See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 21 30
Rossler attractor Poincare sections6
-10 -5 0 50
5
10
15
y(t)
z(t)
Poincareacute section yz-plane
6See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 22 30
The system has only one nonlinearity the zx term in the third equation
Rossler attractor Poincare sections7
-10 -5 0 5 100
5
10
15
20
25
30
x(t)
z(t)
Poincareacute section xz-plane
7See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 23 30
Rossler attractor return map and Poincare sections
Credit Y Maistrenko and R PaskauskasDKartofelev YFX1520 24 30
Slide 24 features the shape of Roumlssler map mdash the Poincareacute map of Roumlssler attractor
The following numerical file was used to calculate the attractor and the Poincareacute sections shown above
Numerics cdf5 nb5Numerical solution of Roumlssler attractor Poincareacute section of Roumlssler attractor
Slides 25 26
Rossler attractor orbit diagram
DKartofelev YFX1520 25 30
Rossler attractor period doubling
No embedded video files in this pdf
DKartofelev YFX1520 26 30
D Kartofelev 1214 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The orbit diagram shown on Slide 25 is calculated using the Roumlssler map shown on Slide 24
The following interactive numerical file shows the period doubling bifurcation happening in the xy-projection of Roumlssler attractor cf Slide 25
Numerics cdf6 nb6Periodic orbits in Roumlssler attractor for varied c value
The following slide shows the 2π-periodic dynamics of the Poincareacute section of Roumlssler attractor
Slide 27Poincare section dynamics Rossler attractor
No embedded video files in this pdf
DKartofelev YFX1520 27 30
Once again notice the folding stretching and mixing of the Poincareacute section cf Slides 19 and20 This type of Poincareacute section dynamics will be explored further in the next weekrsquos lecture
3 Lorenz section
Slide 28
Lorenz section
Section of a sectionDKartofelev YFX1520 28 30
The slide above shows a Poincareacute section of a strange attractor where we sliced the attractor with aplane thereby exposing its cross section If we take a further one-dimensional slice or Lorenz sectionthrough the Poincareacute section we find an infinite set of points separated by gaps of various sizes Inthe next weekrsquos lecture we will look into this discovery and into the notion of Lorenz section in greaterdetail
D Kartofelev 1314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
4 Attractor reconstruction
In applications it is often the case that you are able to measure or are restricted to only measuring onetime-series related to a higher order system mdash one aspect of a problem Letrsquos say you need to analyse theattractor that governs your measured one-dimensional time-series signal s(t) eg show that it is strangeAt first it might seem that there is not enough information But there exist a surprising data analysistechnique known as the attractor reconstruction
For systems governed by an attractor the dynamics in the full higher dimensional phase space canbe reconstructed from measurements of just a single time series s(t) Somehow a single variable carriessufficient information about all the others The method is based on time delays τ For instance define atwo-dimensional vector
983187x(t) = (s(t) s(t+ τ)) (27)
for some time delay τ gt 0 and signal s(t) where 983187x = (x y) Signal s(t) can therefore represent either thevariable x(t) or y(t) The time-series s(t) generates a trajectory 983187x(t) in a two-dimensional phase space thatrepresents its attractorrsquos trajectory You can also considered the attractor in three dimensions by definingthe three-dimensional vector
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)
for time delay τ gt 0 For four-dimensional attractor use
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)
etcThe following interactive numerical file demonstrates attractor reconstruction using three examples First
a sine wave is used to reconstruct its two-dimensional attractor mdash circle Second a quasi-periodic signalis used to reconstruct its three-dimensional attractor mdash torus Third time-series y(t) of Lorenz system isused to reconstruct the three-dimensional Lorenz attractor
Numerics cdf7 nb7Higher-dimensional attractor reconstruction from 1-D time-domain signals Examples sine wavequasi-periodic signal y(t) of Lorenz attractor
Revision questions
1 What are the values of Feigenbaum constants2 What are Feigenbaum constants3 Define superstable fixed point of a map4 Define superstable period-p point (or period-p orbit) of a map5 What are universals of unimodal maps6 What is the universal route to chaos7 Idea behind renormalisation8 What are universal limiting functions in the context of maps9 Name discrete time dynamics analysis methods
10 What is Poincareacute section11 What is Poincareacute map (return map)12 What is Lorenz section
D Kartofelev 1414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Figure 5 Poincareacute section dynamics and periodic z-axis A moving Poincareacute section featuring a singleintersection point shown with the blue dot
Slides 19 20
Poincare section4 Dung oscillator
-15 -10 -05 00 05 10 15
-05
00
05
10
x(t)
y(t)
Poincareacute section
4See Mathematica nb file uploaded to course webpage
DKartofelev YFX1520 19 30
Poincare section dynamics Dung oscillator
No embedded video files in this pdf
DKartofelev YFX1520 20 30
Notice that the Poincareacute section is folding onto itself continuously stretching and the intersectionpoints are being progressively mixed (via re-injection process) within the region defined by theattractor This type of Poincareacute section dynamics is common for strange attractors
The dynamics of the Poincareacute section of Duffing oscillator is explored in the following numerical file
Numerics cdf4 nb4Periodically forced damped Duffing oscillator and dynamic Poincareacute section animation
212 Example Roumlssler attractor
Originally studied by Otto Roumlssler (1940ndash) The system is a product of pure imagination and it exhibitschaotic dynamics by design
D Kartofelev 1114 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 21ndash24
Rossler attractor
Rossler attractor5 is given by (as mentioned in Lecture 9)
8gtlt
gt
x = y x
y = x+ ay
z = b+ z(x c)
(20)
Chaotic solution exists for a = 01 b = 01 c = 14
5See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 21 30
Rossler attractor Poincare sections6
-10 -5 0 50
5
10
15
y(t)
z(t)
Poincareacute section yz-plane
6See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 22 30
The system has only one nonlinearity the zx term in the third equation
Rossler attractor Poincare sections7
-10 -5 0 5 100
5
10
15
20
25
30
x(t)
z(t)
Poincareacute section xz-plane
7See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 23 30
Rossler attractor return map and Poincare sections
Credit Y Maistrenko and R PaskauskasDKartofelev YFX1520 24 30
Slide 24 features the shape of Roumlssler map mdash the Poincareacute map of Roumlssler attractor
The following numerical file was used to calculate the attractor and the Poincareacute sections shown above
Numerics cdf5 nb5Numerical solution of Roumlssler attractor Poincareacute section of Roumlssler attractor
Slides 25 26
Rossler attractor orbit diagram
DKartofelev YFX1520 25 30
Rossler attractor period doubling
No embedded video files in this pdf
DKartofelev YFX1520 26 30
D Kartofelev 1214 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The orbit diagram shown on Slide 25 is calculated using the Roumlssler map shown on Slide 24
The following interactive numerical file shows the period doubling bifurcation happening in the xy-projection of Roumlssler attractor cf Slide 25
Numerics cdf6 nb6Periodic orbits in Roumlssler attractor for varied c value
The following slide shows the 2π-periodic dynamics of the Poincareacute section of Roumlssler attractor
Slide 27Poincare section dynamics Rossler attractor
No embedded video files in this pdf
DKartofelev YFX1520 27 30
Once again notice the folding stretching and mixing of the Poincareacute section cf Slides 19 and20 This type of Poincareacute section dynamics will be explored further in the next weekrsquos lecture
3 Lorenz section
Slide 28
Lorenz section
Section of a sectionDKartofelev YFX1520 28 30
The slide above shows a Poincareacute section of a strange attractor where we sliced the attractor with aplane thereby exposing its cross section If we take a further one-dimensional slice or Lorenz sectionthrough the Poincareacute section we find an infinite set of points separated by gaps of various sizes Inthe next weekrsquos lecture we will look into this discovery and into the notion of Lorenz section in greaterdetail
D Kartofelev 1314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
4 Attractor reconstruction
In applications it is often the case that you are able to measure or are restricted to only measuring onetime-series related to a higher order system mdash one aspect of a problem Letrsquos say you need to analyse theattractor that governs your measured one-dimensional time-series signal s(t) eg show that it is strangeAt first it might seem that there is not enough information But there exist a surprising data analysistechnique known as the attractor reconstruction
For systems governed by an attractor the dynamics in the full higher dimensional phase space canbe reconstructed from measurements of just a single time series s(t) Somehow a single variable carriessufficient information about all the others The method is based on time delays τ For instance define atwo-dimensional vector
983187x(t) = (s(t) s(t+ τ)) (27)
for some time delay τ gt 0 and signal s(t) where 983187x = (x y) Signal s(t) can therefore represent either thevariable x(t) or y(t) The time-series s(t) generates a trajectory 983187x(t) in a two-dimensional phase space thatrepresents its attractorrsquos trajectory You can also considered the attractor in three dimensions by definingthe three-dimensional vector
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)
for time delay τ gt 0 For four-dimensional attractor use
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)
etcThe following interactive numerical file demonstrates attractor reconstruction using three examples First
a sine wave is used to reconstruct its two-dimensional attractor mdash circle Second a quasi-periodic signalis used to reconstruct its three-dimensional attractor mdash torus Third time-series y(t) of Lorenz system isused to reconstruct the three-dimensional Lorenz attractor
Numerics cdf7 nb7Higher-dimensional attractor reconstruction from 1-D time-domain signals Examples sine wavequasi-periodic signal y(t) of Lorenz attractor
Revision questions
1 What are the values of Feigenbaum constants2 What are Feigenbaum constants3 Define superstable fixed point of a map4 Define superstable period-p point (or period-p orbit) of a map5 What are universals of unimodal maps6 What is the universal route to chaos7 Idea behind renormalisation8 What are universal limiting functions in the context of maps9 Name discrete time dynamics analysis methods
10 What is Poincareacute section11 What is Poincareacute map (return map)12 What is Lorenz section
D Kartofelev 1414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
Slides 21ndash24
Rossler attractor
Rossler attractor5 is given by (as mentioned in Lecture 9)
8gtlt
gt
x = y x
y = x+ ay
z = b+ z(x c)
(20)
Chaotic solution exists for a = 01 b = 01 c = 14
5See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 21 30
Rossler attractor Poincare sections6
-10 -5 0 50
5
10
15
y(t)
z(t)
Poincareacute section yz-plane
6See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 22 30
The system has only one nonlinearity the zx term in the third equation
Rossler attractor Poincare sections7
-10 -5 0 5 100
5
10
15
20
25
30
x(t)
z(t)
Poincareacute section xz-plane
7See Mathematica nb file uploaded to the course webpage
DKartofelev YFX1520 23 30
Rossler attractor return map and Poincare sections
Credit Y Maistrenko and R PaskauskasDKartofelev YFX1520 24 30
Slide 24 features the shape of Roumlssler map mdash the Poincareacute map of Roumlssler attractor
The following numerical file was used to calculate the attractor and the Poincareacute sections shown above
Numerics cdf5 nb5Numerical solution of Roumlssler attractor Poincareacute section of Roumlssler attractor
Slides 25 26
Rossler attractor orbit diagram
DKartofelev YFX1520 25 30
Rossler attractor period doubling
No embedded video files in this pdf
DKartofelev YFX1520 26 30
D Kartofelev 1214 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The orbit diagram shown on Slide 25 is calculated using the Roumlssler map shown on Slide 24
The following interactive numerical file shows the period doubling bifurcation happening in the xy-projection of Roumlssler attractor cf Slide 25
Numerics cdf6 nb6Periodic orbits in Roumlssler attractor for varied c value
The following slide shows the 2π-periodic dynamics of the Poincareacute section of Roumlssler attractor
Slide 27Poincare section dynamics Rossler attractor
No embedded video files in this pdf
DKartofelev YFX1520 27 30
Once again notice the folding stretching and mixing of the Poincareacute section cf Slides 19 and20 This type of Poincareacute section dynamics will be explored further in the next weekrsquos lecture
3 Lorenz section
Slide 28
Lorenz section
Section of a sectionDKartofelev YFX1520 28 30
The slide above shows a Poincareacute section of a strange attractor where we sliced the attractor with aplane thereby exposing its cross section If we take a further one-dimensional slice or Lorenz sectionthrough the Poincareacute section we find an infinite set of points separated by gaps of various sizes Inthe next weekrsquos lecture we will look into this discovery and into the notion of Lorenz section in greaterdetail
D Kartofelev 1314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
4 Attractor reconstruction
In applications it is often the case that you are able to measure or are restricted to only measuring onetime-series related to a higher order system mdash one aspect of a problem Letrsquos say you need to analyse theattractor that governs your measured one-dimensional time-series signal s(t) eg show that it is strangeAt first it might seem that there is not enough information But there exist a surprising data analysistechnique known as the attractor reconstruction
For systems governed by an attractor the dynamics in the full higher dimensional phase space canbe reconstructed from measurements of just a single time series s(t) Somehow a single variable carriessufficient information about all the others The method is based on time delays τ For instance define atwo-dimensional vector
983187x(t) = (s(t) s(t+ τ)) (27)
for some time delay τ gt 0 and signal s(t) where 983187x = (x y) Signal s(t) can therefore represent either thevariable x(t) or y(t) The time-series s(t) generates a trajectory 983187x(t) in a two-dimensional phase space thatrepresents its attractorrsquos trajectory You can also considered the attractor in three dimensions by definingthe three-dimensional vector
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)
for time delay τ gt 0 For four-dimensional attractor use
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)
etcThe following interactive numerical file demonstrates attractor reconstruction using three examples First
a sine wave is used to reconstruct its two-dimensional attractor mdash circle Second a quasi-periodic signalis used to reconstruct its three-dimensional attractor mdash torus Third time-series y(t) of Lorenz system isused to reconstruct the three-dimensional Lorenz attractor
Numerics cdf7 nb7Higher-dimensional attractor reconstruction from 1-D time-domain signals Examples sine wavequasi-periodic signal y(t) of Lorenz attractor
Revision questions
1 What are the values of Feigenbaum constants2 What are Feigenbaum constants3 Define superstable fixed point of a map4 Define superstable period-p point (or period-p orbit) of a map5 What are universals of unimodal maps6 What is the universal route to chaos7 Idea behind renormalisation8 What are universal limiting functions in the context of maps9 Name discrete time dynamics analysis methods
10 What is Poincareacute section11 What is Poincareacute map (return map)12 What is Lorenz section
D Kartofelev 1414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
The orbit diagram shown on Slide 25 is calculated using the Roumlssler map shown on Slide 24
The following interactive numerical file shows the period doubling bifurcation happening in the xy-projection of Roumlssler attractor cf Slide 25
Numerics cdf6 nb6Periodic orbits in Roumlssler attractor for varied c value
The following slide shows the 2π-periodic dynamics of the Poincareacute section of Roumlssler attractor
Slide 27Poincare section dynamics Rossler attractor
No embedded video files in this pdf
DKartofelev YFX1520 27 30
Once again notice the folding stretching and mixing of the Poincareacute section cf Slides 19 and20 This type of Poincareacute section dynamics will be explored further in the next weekrsquos lecture
3 Lorenz section
Slide 28
Lorenz section
Section of a sectionDKartofelev YFX1520 28 30
The slide above shows a Poincareacute section of a strange attractor where we sliced the attractor with aplane thereby exposing its cross section If we take a further one-dimensional slice or Lorenz sectionthrough the Poincareacute section we find an infinite set of points separated by gaps of various sizes Inthe next weekrsquos lecture we will look into this discovery and into the notion of Lorenz section in greaterdetail
D Kartofelev 1314 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
4 Attractor reconstruction
In applications it is often the case that you are able to measure or are restricted to only measuring onetime-series related to a higher order system mdash one aspect of a problem Letrsquos say you need to analyse theattractor that governs your measured one-dimensional time-series signal s(t) eg show that it is strangeAt first it might seem that there is not enough information But there exist a surprising data analysistechnique known as the attractor reconstruction
For systems governed by an attractor the dynamics in the full higher dimensional phase space canbe reconstructed from measurements of just a single time series s(t) Somehow a single variable carriessufficient information about all the others The method is based on time delays τ For instance define atwo-dimensional vector
983187x(t) = (s(t) s(t+ τ)) (27)
for some time delay τ gt 0 and signal s(t) where 983187x = (x y) Signal s(t) can therefore represent either thevariable x(t) or y(t) The time-series s(t) generates a trajectory 983187x(t) in a two-dimensional phase space thatrepresents its attractorrsquos trajectory You can also considered the attractor in three dimensions by definingthe three-dimensional vector
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)
for time delay τ gt 0 For four-dimensional attractor use
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)
etcThe following interactive numerical file demonstrates attractor reconstruction using three examples First
a sine wave is used to reconstruct its two-dimensional attractor mdash circle Second a quasi-periodic signalis used to reconstruct its three-dimensional attractor mdash torus Third time-series y(t) of Lorenz system isused to reconstruct the three-dimensional Lorenz attractor
Numerics cdf7 nb7Higher-dimensional attractor reconstruction from 1-D time-domain signals Examples sine wavequasi-periodic signal y(t) of Lorenz attractor
Revision questions
1 What are the values of Feigenbaum constants2 What are Feigenbaum constants3 Define superstable fixed point of a map4 Define superstable period-p point (or period-p orbit) of a map5 What are universals of unimodal maps6 What is the universal route to chaos7 Idea behind renormalisation8 What are universal limiting functions in the context of maps9 Name discrete time dynamics analysis methods
10 What is Poincareacute section11 What is Poincareacute map (return map)12 What is Lorenz section
D Kartofelev 1414 1003211 As of March 7 2020
Lecture notes 11 Nonlinear Dynamics YFX1520
4 Attractor reconstruction
In applications it is often the case that you are able to measure or are restricted to only measuring onetime-series related to a higher order system mdash one aspect of a problem Letrsquos say you need to analyse theattractor that governs your measured one-dimensional time-series signal s(t) eg show that it is strangeAt first it might seem that there is not enough information But there exist a surprising data analysistechnique known as the attractor reconstruction
For systems governed by an attractor the dynamics in the full higher dimensional phase space canbe reconstructed from measurements of just a single time series s(t) Somehow a single variable carriessufficient information about all the others The method is based on time delays τ For instance define atwo-dimensional vector
983187x(t) = (s(t) s(t+ τ)) (27)
for some time delay τ gt 0 and signal s(t) where 983187x = (x y) Signal s(t) can therefore represent either thevariable x(t) or y(t) The time-series s(t) generates a trajectory 983187x(t) in a two-dimensional phase space thatrepresents its attractorrsquos trajectory You can also considered the attractor in three dimensions by definingthe three-dimensional vector
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)
for time delay τ gt 0 For four-dimensional attractor use
983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)
etcThe following interactive numerical file demonstrates attractor reconstruction using three examples First
a sine wave is used to reconstruct its two-dimensional attractor mdash circle Second a quasi-periodic signalis used to reconstruct its three-dimensional attractor mdash torus Third time-series y(t) of Lorenz system isused to reconstruct the three-dimensional Lorenz attractor
Numerics cdf7 nb7Higher-dimensional attractor reconstruction from 1-D time-domain signals Examples sine wavequasi-periodic signal y(t) of Lorenz attractor
Revision questions
1 What are the values of Feigenbaum constants2 What are Feigenbaum constants3 Define superstable fixed point of a map4 Define superstable period-p point (or period-p orbit) of a map5 What are universals of unimodal maps6 What is the universal route to chaos7 Idea behind renormalisation8 What are universal limiting functions in the context of maps9 Name discrete time dynamics analysis methods
10 What is Poincareacute section11 What is Poincareacute map (return map)12 What is Lorenz section
D Kartofelev 1414 1003211 As of March 7 2020
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