EE C222/ME C237 - Spring'18 - Lecture 2 Notesee222/sp18/LectureNotes/Lecture2.pdf · ee c222/me...
Transcript of EE C222/ME C237 - Spring'18 - Lecture 2 Notesee222/sp18/LectureNotes/Lecture2.pdf · ee c222/me...
EE C222/ME C237 - Spring’18 - Lecture 2 Notes11 Licensed under a Creative CommonsAttribution-NonCommercial-ShareAlike4.0 International License.Murat Arcak
January 22 2018
Essentially Nonlinear Phenomena Continued
1. Finite escape time
2. Multiple isolated equilibria
3. Limit cycles: Linear oscillators exhibit a continuum of periodicorbits; e.g., every circle is a periodic orbit for x = Ax where
A =
[0 −β
β 0
](λ1,2 = ∓jβ).
In contrast, a limit cycle is an isolated periodic orbit and can occuronly in nonlinear systems.
limit cycleharmonicoscillator
Example: van der Pol oscillator
CvC = −iL + vC − v3C
LiL = vC
iL
LC+
−vC
iR
vC
iR = −vC + v3C
"negative resistance"∧∧
iL
vC
ee c222/me c237 - spring’18 - lecture 2 notes 2
4. Chaos: Irregular oscillations, never exactly repeating.
Example: Lorenz system (derived by Ed Lorenz in 1963 as a sim-plified model of convection rolls in the atmosphere):
x = σ(y− x)
y = rx− y− xz
z = xy− bz.
Chaotic behavior with σ = 10, b = 8/3, r = 28:
0 10 20 30 40 50 60-30
-20
-10
0
10
20
30
-20 -15 -10 -5 0 5 10 15 200
5
10
15
20
25
30
35
40
45
50
x
z
t
y(t)
• For continuous-time, time-invariant systems, n ≥ 3 state vari-ables required for chaos.n = 1: x(t) monotone in t, no oscillations:
x
f (x)
n = 2: Poincaré-Bendixson Theorem (to be studied in Lecture 3)guarantees regular behavior.
• Poincaré-Bendixson does not apply to time-varying systems andn ≥ 2 is enough for chaos (Homework problem).
• For discrete-time systems, n = 1 is enough (we will see anexample in Lecture 5).
Planar (Second Order) Dynamical SystemsChapter 2 in both Sastry and Khalil
Phase Portraits of Linear Systems: x = Ax
• Distinct real eigenvalues
T−1 AT =
[λ1 00 λ2
]
ee c222/me c237 - spring’18 - lecture 2 notes 3
In z = T−1x coordinates:
z1 = λ1z1, z2 = λ2z2.
The equilibrium is called a node when λ1 and λ2 have the samesign (stable node when negative and unstable when positive). It iscalled a saddle point when λ1 and λ2 have opposite signs.
z1z1z1
z2z2z2
λ1 < λ2 < 0 λ1 > λ2 > 0 λ2 < 0 < λ1
stablenode
unstablenode saddle
• Complex eigenvalues: λ1,2 = α∓ jβ
T−1 AT =
[α −β
β α
]z1 = αz1 − βz2
z2 = αz2 + βz1→ polar coordinates →
r = αr
θ = β
z1z1z1
z2z2z2
stablefocus
unstablefocus center
α < 0 α > 0 α = 0
The phase portraits above assume β > 0 so that the direction ofrotation is counter-clockwise: θ = β > 0.
Phase Portraits of Nonlinear Systems Near Hyperbolic Equilibria
hyperbolic equilibrium: linearization has no eigenvalues on the imagi-nary axis
Phase portraits of nonlinear systems near hyperbolic equilibria arequalitatively similar to the phase portraits of their linearization. Ac-cording to the Hartman-Grobman Theorem (below) a “continuousdeformation” maps one phase portrait to the other.
ee c222/me c237 - spring’18 - lecture 2 notes 4
x∗h
Hartman-Grobman Theorem: If x∗ is a hyperbolic equilibrium ofx = f (x), x ∈ Rn, then there exists a homeomorphism2 z = h(x) defined 2 a continuous map with a continuous
inversein a neighborhood of x∗ that maps trajectories of x = f (x) to those of
z = Az where A , ∂ f∂x
∣∣∣x=x∗
.
The hyperbolicity condition can’t be removed:
Example:
x1 = −x2 + ax1(x21 + x2
2)
x2 = x1 + ax2(x11 + x2
2)=⇒
r = ar3
θ = 1
x∗ = (0, 0) A =∂ f∂x
∣∣∣∣x=x∗
=
[0 −11 0
]
There is no continuous deformation that maps the phase portrait ofthe linearization to that of the original nonlinear model:
(a > 0)x = Ax x = f (x)
Periodic Orbits in the Plane
Bendixson’s Theorem: For a time-invariant planar system
x1 = f1(x1, x2) x2 = f2(x1, x2),
if ∇ · f (x) = ∂ f1∂x1
+ ∂ f2∂x2
is not identically zero and does not changesign in a simply connected region D, then there are no periodic orbitslying entirely in D.
ee c222/me c237 - spring’18 - lecture 2 notes 5
Proof: By contradiction. Suppose a periodic orbit J lies in D. Let Sdenote the region enclosed by J and n(x) the normal vector to J at x.Then f (x) · n(x) = 0 for all x ∈ J. By the Divergence Theorem:
JS
• x
f (x) n(x)
∫J
f (x) · n(x)d`︸ ︷︷ ︸= 0
=∫∫
S∇ · f (x)dx︸ ︷︷ ︸6= 0
.
Example: x = Ax, x ∈ R2 can have periodic orbits only if
Trace(A) = 0, e.g.,
A =
[0 −β
β 0
].
Example:
x1 = x2
x2 = −δx2 + x1 − x31 + x2
1x2 δ > 0
∇ · f (x) =∂ f1
∂x1+
∂ f2
∂x2= x2
1 − δ
Therefore, no periodic orbit can lie entirely in the region x1 ≤ −√
δ
where ∇ · f (x) ≥ 0, or −√
δ ≤ x1 ≤√
δ where ∇ · f (x) ≤ 0, orx1 ≥
√δ where ∇ · f (x) ≥ 0.
x1 = −√
δ
x1 = −√
δ
x1 =√
δ
x1 =√
δ
x1
x1
x2
x2
not possible:
possible: