Lecture 11: Feigenbaum’s analysis of period doubling ...dima/YFX1520/LectureNotes_11.pdf · The...

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

D Kartofelev 114 1003211 As of March 7 2020


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)



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)



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)




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)


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)


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



Slides 8 9

Feigenbaumrsquos analysis renormalisation

f(xR0) f2(xR1)

(a) (b) (c)



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)



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



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




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


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




The solution shown above is taken from the following numerical file


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



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


Discrete time dynamics analysis

Construction of a Poincare map (return map recurrence map)


Poincare section periodic forcing3

3Credit J Thompson H Stewart 1986 Nonlinear dynamics and chaos

geometrical methods for engineers and scientists Chichester UK Wiley


Poincare section periodic forcing


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



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



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)


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



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



Poincare section dynamics Dung oscillator

No embedded video files in this pdf


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



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


Rossler attractor Poincare sections6

-10 -5 0 50

5

10

15

y(t)

z(t)

Poincareacute section yz-plane



The system has only one nonlinearity the zx term in the third equation


-10 -5 0 5 100

5

10

15

20

25

30

x(t)

z(t)

Poincareacute section xz-plane



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


Rossler attractor period doubling





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



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



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



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)


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)





xlowast = xm (5)



f(x r) = x

x

= xm and f0(x

r) = 0 ) r = 2 (3)


n+1 =|f 00(x

r)|2

2n +O(3n) (4)







983063Taylor series

about xn = xlowast


f prime(xlowast)



(7)










2η2n +O(η3n) =


η2n +O(η3n) (11)




2η2n +O(η3n) (12)




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






f2(x

r) = x (x = xm) ) r = 1 +

p5 (5)

f2(x

r)0=

d

dx [f(f(x r))] = f


r) = 0 (6)











limn1

Rn1

Rn= (7)





p5 323607 period-2

r2 = 1 +p6 344949 R2 349856 period-4


r1 3569945672 R1 = r1 356994567 period-21


limn1

rn1 rn2

rn rn1= lim

n1

Rn1

Rn= (8)









Slides 8 9


f(xR0) f2(xR1)

(a) (b) (c)




2x

crarr R1

(a) (b) (c)






983060 (14)




α2 R2

983060

asymp α3f8983059 x

α3 R3

983060


α4 R4

983060


αn Rn

983060

(15)


limnrarrinfin

αnf (2n)983059 x

αn Rn

983060= g0(x) (16)





αnf (2n)983059 x

αn Rn+i

983060 (17)





983060=


983064= αf2

983059xα Rinfin

983060= g(x) (18)


g(x) = αg2983059xα

983060 (19)



gprime(0) = 0 (20)

Also we can set

g(0) =


983064= g(0 Rinfin) = const (21)

tog(0) = 1 (22)



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)



Slides 10 11



g(x) = crarr g(gx

crarr

) = crarr g

2x

crarr

(9)


crarr =1

g(1) (10)


g(x) = 1 + ax2 + bx

4 + cx6 + dx

8 + ex10 + (11)





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


Logistic map










Slide 12



g(x) = 1 + ax2 + bx

4 + cx6 + dx

8 +O(x10) (14)


crarr =1

g(1) 250316 (001 error) (15)

-04 -02 00 02 04000204060810

xn

x x+1 g(x)

Logistic map







Reading suggestion









Slides 13ndash16

Poincare section





















x x+ x3 + x = F cost (16)





(x = y

y = x x3 y + F cost

(17)


gtlt

gt

x = y


z =

(18)


z =

Zz dt =

Z dt =

Zdt = t+ C (19)



V (x) =1

4x4 minus 1

2x2 (26)










Slides 19 20


-15 -10 -05 00 05 10 15

-05

00

05

10

x(t)

y(t)














Slides 21ndash24

Rossler attractor


8gtlt

gt

x = y x

y = x+ ay

z = b+ z(x c)

(20)





-10 -5 0 50

5

10

15

y(t)

z(t)






-10 -5 0 5 100

5

10

15

20

25

30

x(t)

z(t)









Slides 25 26
















3 Lorenz section

Slide 28

Lorenz section








983187x(t) = (s(t) s(t+ τ)) (27)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)




Revision questions







xlowast = xm (5)



f(x r) = x

x

= xm and f0(x

r) = 0 ) r = 2 (3)


n+1 =|f 00(x

r)|2

2n +O(3n) (4)







983063Taylor series

about xn = xlowast


f prime(xlowast)



(7)










2η2n +O(η3n) =


η2n +O(η3n) (11)




2η2n +O(η3n) (12)




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






f2(x

r) = x (x = xm) ) r = 1 +

p5 (5)

f2(x

r)0=

d

dx [f(f(x r))] = f


r) = 0 (6)











limn1

Rn1

Rn= (7)





p5 323607 period-2

r2 = 1 +p6 344949 R2 349856 period-4


r1 3569945672 R1 = r1 356994567 period-21


limn1

rn1 rn2

rn rn1= lim

n1

Rn1

Rn= (8)









Slides 8 9


f(xR0) f2(xR1)

(a) (b) (c)




2x

crarr R1

(a) (b) (c)






983060 (14)




α2 R2

983060

asymp α3f8983059 x

α3 R3

983060


α4 R4

983060


αn Rn

983060

(15)


limnrarrinfin

αnf (2n)983059 x

αn Rn

983060= g0(x) (16)





αnf (2n)983059 x

αn Rn+i

983060 (17)





983060=


983064= αf2

983059xα Rinfin

983060= g(x) (18)


g(x) = αg2983059xα

983060 (19)



gprime(0) = 0 (20)

Also we can set

g(0) =


983064= g(0 Rinfin) = const (21)

tog(0) = 1 (22)



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)



Slides 10 11



g(x) = crarr g(gx

crarr

) = crarr g

2x

crarr

(9)


crarr =1

g(1) (10)


g(x) = 1 + ax2 + bx

4 + cx6 + dx

8 + ex10 + (11)





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


Logistic map










Slide 12



g(x) = 1 + ax2 + bx

4 + cx6 + dx

8 +O(x10) (14)


crarr =1

g(1) 250316 (001 error) (15)

-04 -02 00 02 04000204060810

xn

x x+1 g(x)

Logistic map







Reading suggestion









Slides 13ndash16

Poincare section





















x x+ x3 + x = F cost (16)





(x = y

y = x x3 y + F cost

(17)


gtlt

gt

x = y


z =

(18)


z =

Zz dt =

Z dt =

Zdt = t+ C (19)



V (x) =1

4x4 minus 1

2x2 (26)










Slides 19 20


-15 -10 -05 00 05 10 15

-05

00

05

10

x(t)

y(t)














Slides 21ndash24

Rossler attractor


8gtlt

gt

x = y x

y = x+ ay

z = b+ z(x c)

(20)





-10 -5 0 50

5

10

15

y(t)

z(t)






-10 -5 0 5 100

5

10

15

20

25

30

x(t)

z(t)









Slides 25 26
















3 Lorenz section

Slide 28

Lorenz section








983187x(t) = (s(t) s(t+ τ)) (27)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)




Revision questions






2η2n +O(η3n) (12)




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






f2(x

r) = x (x = xm) ) r = 1 +

p5 (5)

f2(x

r)0=

d

dx [f(f(x r))] = f


r) = 0 (6)











limn1

Rn1

Rn= (7)





p5 323607 period-2

r2 = 1 +p6 344949 R2 349856 period-4


r1 3569945672 R1 = r1 356994567 period-21


limn1

rn1 rn2

rn rn1= lim

n1

Rn1

Rn= (8)









Slides 8 9


f(xR0) f2(xR1)

(a) (b) (c)




2x

crarr R1

(a) (b) (c)






983060 (14)




α2 R2

983060

asymp α3f8983059 x

α3 R3

983060


α4 R4

983060


αn Rn

983060

(15)


limnrarrinfin

αnf (2n)983059 x

αn Rn

983060= g0(x) (16)





αnf (2n)983059 x

αn Rn+i

983060 (17)





983060=


983064= αf2

983059xα Rinfin

983060= g(x) (18)


g(x) = αg2983059xα

983060 (19)



gprime(0) = 0 (20)

Also we can set

g(0) =


983064= g(0 Rinfin) = const (21)

tog(0) = 1 (22)



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)



Slides 10 11



g(x) = crarr g(gx

crarr

) = crarr g

2x

crarr

(9)


crarr =1

g(1) (10)


g(x) = 1 + ax2 + bx

4 + cx6 + dx

8 + ex10 + (11)





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


Logistic map










Slide 12



g(x) = 1 + ax2 + bx

4 + cx6 + dx

8 +O(x10) (14)


crarr =1

g(1) 250316 (001 error) (15)

-04 -02 00 02 04000204060810

xn

x x+1 g(x)

Logistic map







Reading suggestion









Slides 13ndash16

Poincare section





















x x+ x3 + x = F cost (16)





(x = y

y = x x3 y + F cost

(17)


gtlt

gt

x = y


z =

(18)


z =

Zz dt =

Z dt =

Zdt = t+ C (19)



V (x) =1

4x4 minus 1

2x2 (26)










Slides 19 20


-15 -10 -05 00 05 10 15

-05

00

05

10

x(t)

y(t)














Slides 21ndash24

Rossler attractor


8gtlt

gt

x = y x

y = x+ ay

z = b+ z(x c)

(20)





-10 -5 0 50

5

10

15

y(t)

z(t)






-10 -5 0 5 100

5

10

15

20

25

30

x(t)

z(t)









Slides 25 26
















3 Lorenz section

Slide 28

Lorenz section








983187x(t) = (s(t) s(t+ τ)) (27)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)




Revision questions












limn1

Rn1

Rn= (7)





p5 323607 period-2

r2 = 1 +p6 344949 R2 349856 period-4


r1 3569945672 R1 = r1 356994567 period-21


limn1

rn1 rn2

rn rn1= lim

n1

Rn1

Rn= (8)









Slides 8 9


f(xR0) f2(xR1)

(a) (b) (c)




2x

crarr R1

(a) (b) (c)






983060 (14)




α2 R2

983060

asymp α3f8983059 x

α3 R3

983060


α4 R4

983060


αn Rn

983060

(15)


limnrarrinfin

αnf (2n)983059 x

αn Rn

983060= g0(x) (16)





αnf (2n)983059 x

αn Rn+i

983060 (17)





983060=


983064= αf2

983059xα Rinfin

983060= g(x) (18)


g(x) = αg2983059xα

983060 (19)



gprime(0) = 0 (20)

Also we can set

g(0) =


983064= g(0 Rinfin) = const (21)

tog(0) = 1 (22)



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)



Slides 10 11



g(x) = crarr g(gx

crarr

) = crarr g

2x

crarr

(9)


crarr =1

g(1) (10)


g(x) = 1 + ax2 + bx

4 + cx6 + dx

8 + ex10 + (11)





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


Logistic map










Slide 12



g(x) = 1 + ax2 + bx

4 + cx6 + dx

8 +O(x10) (14)


crarr =1

g(1) 250316 (001 error) (15)

-04 -02 00 02 04000204060810

xn

x x+1 g(x)

Logistic map







Reading suggestion









Slides 13ndash16

Poincare section





















x x+ x3 + x = F cost (16)





(x = y

y = x x3 y + F cost

(17)


gtlt

gt

x = y


z =

(18)


z =

Zz dt =

Z dt =

Zdt = t+ C (19)



V (x) =1

4x4 minus 1

2x2 (26)










Slides 19 20


-15 -10 -05 00 05 10 15

-05

00

05

10

x(t)

y(t)














Slides 21ndash24

Rossler attractor


8gtlt

gt

x = y x

y = x+ ay

z = b+ z(x c)

(20)





-10 -5 0 50

5

10

15

y(t)

z(t)






-10 -5 0 5 100

5

10

15

20

25

30

x(t)

z(t)









Slides 25 26
















3 Lorenz section

Slide 28

Lorenz section








983187x(t) = (s(t) s(t+ τ)) (27)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)




Revision questions





Slides 8 9


f(xR0) f2(xR1)

(a) (b) (c)




2x

crarr R1

(a) (b) (c)






983060 (14)




α2 R2

983060

asymp α3f8983059 x

α3 R3

983060


α4 R4

983060


αn Rn

983060

(15)


limnrarrinfin

αnf (2n)983059 x

αn Rn

983060= g0(x) (16)





αnf (2n)983059 x

αn Rn+i

983060 (17)





983060=


983064= αf2

983059xα Rinfin

983060= g(x) (18)


g(x) = αg2983059xα

983060 (19)



gprime(0) = 0 (20)

Also we can set

g(0) =


983064= g(0 Rinfin) = const (21)

tog(0) = 1 (22)



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)



Slides 10 11



g(x) = crarr g(gx

crarr

) = crarr g

2x

crarr

(9)


crarr =1

g(1) (10)


g(x) = 1 + ax2 + bx

4 + cx6 + dx

8 + ex10 + (11)





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


Logistic map










Slide 12



g(x) = 1 + ax2 + bx

4 + cx6 + dx

8 +O(x10) (14)


crarr =1

g(1) 250316 (001 error) (15)

-04 -02 00 02 04000204060810

xn

x x+1 g(x)

Logistic map







Reading suggestion









Slides 13ndash16

Poincare section





















x x+ x3 + x = F cost (16)





(x = y

y = x x3 y + F cost

(17)


gtlt

gt

x = y


z =

(18)


z =

Zz dt =

Z dt =

Zdt = t+ C (19)



V (x) =1

4x4 minus 1

2x2 (26)










Slides 19 20


-15 -10 -05 00 05 10 15

-05

00

05

10

x(t)

y(t)














Slides 21ndash24

Rossler attractor


8gtlt

gt

x = y x

y = x+ ay

z = b+ z(x c)

(20)





-10 -5 0 50

5

10

15

y(t)

z(t)






-10 -5 0 5 100

5

10

15

20

25

30

x(t)

z(t)









Slides 25 26
















3 Lorenz section

Slide 28

Lorenz section








983187x(t) = (s(t) s(t+ τ)) (27)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)




Revision questions







983060=


983064= αf2

983059xα Rinfin

983060= g(x) (18)


g(x) = αg2983059xα

983060 (19)



gprime(0) = 0 (20)

Also we can set

g(0) =


983064= g(0 Rinfin) = const (21)

tog(0) = 1 (22)



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)



Slides 10 11



g(x) = crarr g(gx

crarr

) = crarr g

2x

crarr

(9)


crarr =1

g(1) (10)


g(x) = 1 + ax2 + bx

4 + cx6 + dx

8 + ex10 + (11)





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


Logistic map










Slide 12



g(x) = 1 + ax2 + bx

4 + cx6 + dx

8 +O(x10) (14)


crarr =1

g(1) 250316 (001 error) (15)

-04 -02 00 02 04000204060810

xn

x x+1 g(x)

Logistic map







Reading suggestion









Slides 13ndash16

Poincare section





















x x+ x3 + x = F cost (16)





(x = y

y = x x3 y + F cost

(17)


gtlt

gt

x = y


z =

(18)


z =

Zz dt =

Z dt =

Zdt = t+ C (19)



V (x) =1

4x4 minus 1

2x2 (26)










Slides 19 20


-15 -10 -05 00 05 10 15

-05

00

05

10

x(t)

y(t)














Slides 21ndash24

Rossler attractor


8gtlt

gt

x = y x

y = x+ ay

z = b+ z(x c)

(20)





-10 -5 0 50

5

10

15

y(t)

z(t)






-10 -5 0 5 100

5

10

15

20

25

30

x(t)

z(t)









Slides 25 26
















3 Lorenz section

Slide 28

Lorenz section








983187x(t) = (s(t) s(t+ τ)) (27)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)




Revision questions









Slide 12



g(x) = 1 + ax2 + bx

4 + cx6 + dx

8 +O(x10) (14)


crarr =1

g(1) 250316 (001 error) (15)

-04 -02 00 02 04000204060810

xn

x x+1 g(x)

Logistic map







Reading suggestion









Slides 13ndash16

Poincare section





















x x+ x3 + x = F cost (16)





(x = y

y = x x3 y + F cost

(17)


gtlt

gt

x = y


z =

(18)


z =

Zz dt =

Z dt =

Zdt = t+ C (19)



V (x) =1

4x4 minus 1

2x2 (26)










Slides 19 20


-15 -10 -05 00 05 10 15

-05

00

05

10

x(t)

y(t)














Slides 21ndash24

Rossler attractor


8gtlt

gt

x = y x

y = x+ ay

z = b+ z(x c)

(20)





-10 -5 0 50

5

10

15

y(t)

z(t)






-10 -5 0 5 100

5

10

15

20

25

30

x(t)

z(t)









Slides 25 26
















3 Lorenz section

Slide 28

Lorenz section








983187x(t) = (s(t) s(t+ τ)) (27)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)




Revision questions







Slides 13ndash16

Poincare section





















x x+ x3 + x = F cost (16)





(x = y

y = x x3 y + F cost

(17)


gtlt

gt

x = y


z =

(18)


z =

Zz dt =

Z dt =

Zdt = t+ C (19)



V (x) =1

4x4 minus 1

2x2 (26)










Slides 19 20


-15 -10 -05 00 05 10 15

-05

00

05

10

x(t)

y(t)














Slides 21ndash24

Rossler attractor


8gtlt

gt

x = y x

y = x+ ay

z = b+ z(x c)

(20)





-10 -5 0 50

5

10

15

y(t)

z(t)






-10 -5 0 5 100

5

10

15

20

25

30

x(t)

z(t)









Slides 25 26
















3 Lorenz section

Slide 28

Lorenz section








983187x(t) = (s(t) s(t+ τ)) (27)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)




Revision questions








x x+ x3 + x = F cost (16)





(x = y

y = x x3 y + F cost

(17)


gtlt

gt

x = y


z =

(18)


z =

Zz dt =

Z dt =

Zdt = t+ C (19)



V (x) =1

4x4 minus 1

2x2 (26)










Slides 19 20


-15 -10 -05 00 05 10 15

-05

00

05

10

x(t)

y(t)














Slides 21ndash24

Rossler attractor


8gtlt

gt

x = y x

y = x+ ay

z = b+ z(x c)

(20)





-10 -5 0 50

5

10

15

y(t)

z(t)






-10 -5 0 5 100

5

10

15

20

25

30

x(t)

z(t)









Slides 25 26
















3 Lorenz section

Slide 28

Lorenz section








983187x(t) = (s(t) s(t+ τ)) (27)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)




Revision questions






Slides 19 20


-15 -10 -05 00 05 10 15

-05

00

05

10

x(t)

y(t)














Slides 21ndash24

Rossler attractor


8gtlt

gt

x = y x

y = x+ ay

z = b+ z(x c)

(20)





-10 -5 0 50

5

10

15

y(t)

z(t)






-10 -5 0 5 100

5

10

15

20

25

30

x(t)

z(t)









Slides 25 26
















3 Lorenz section

Slide 28

Lorenz section








983187x(t) = (s(t) s(t+ τ)) (27)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)




Revision questions





Slides 21ndash24

Rossler attractor


8gtlt

gt

x = y x

y = x+ ay

z = b+ z(x c)

(20)





-10 -5 0 50

5

10

15

y(t)

z(t)






-10 -5 0 5 100

5

10

15

20

25

30

x(t)

z(t)









Slides 25 26
















3 Lorenz section

Slide 28

Lorenz section








983187x(t) = (s(t) s(t+ τ)) (27)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)




Revision questions













3 Lorenz section

Slide 28

Lorenz section








983187x(t) = (s(t) s(t+ τ)) (27)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)




Revision questions








983187x(t) = (s(t) s(t+ τ)) (27)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ)) (28)


983187x(t) = (s(t) s(t+ τ) s(t+ 2τ) s(t+ 3τ)) (29)




Revision questions




Lecture 11: Feigenbaum’s analysis of period doubling ...dima/YFX1520/LectureNotes_11.pdf · The...

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Transcript of Lecture 11: Feigenbaum’s analysis of period doubling ...dima/YFX1520/LectureNotes_11.pdf · The...