EE C222/ME C237 - Spring'18 - Lecture 2 Notes: Murat Arcak January 22 2018

EE C222/ME C237 - Spring’18 - Lecture 2 Notes1 1
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January 22 2018
Essentially Nonlinear Phenomena Continued
1. Finite escape time
2. Multiple isolated equilibria
3. Limit cycles: Linear oscillators exhibit a continuum of periodic

orbits; e.g., every circle is a periodic orbit for ẋ = Ax where
" #
0 −β
A= (λ1,2 = ∓ jβ).
β 0
In contrast, a limit cycle is an isolated periodic orbit and can occur

only in nonlinear systems.
harmonic limit cycle

oscillator
Example: van der Pol oscillator

iL iR
C v̇C = −i L + vC − v3C
+
Li̇ L = vC C vC L
−
i R = −vC + v3C
vC
vC ∧ ∧ "negative resistance"
iL
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):
ẋ = σ(y − x )
ẏ = rx − y − xz
ż = xy − bz.
Chaotic behavior with σ = 10, b = 8/3, r = 28:

30 50
45
20
40
35
10
30
0
y(t) z 25
20
-10
15
10
-20
-30 0
0 10 20 30 40 50 60 -20 -15 -10 -5 0 5 10 15 20
t x
• For continuous-time, time-invariant systems, n ≥ 3 state vari-

ables required for chaos.
n = 1: x (t) monotone in t, no oscillations:
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 and
n ≥ 2 is enough for chaos (Homework problem).
• For discrete-time systems, n = 1 is enough (we will see an
example in Lecture 5).
Planar (Second Order) Dynamical Systems

Chapter 2 in both Sastry and Khalil
Phase Portraits of Linear Systems: ẋ = Ax
• Distinct real eigenvalues

" #
−1 λ1 0
T AT =
0 λ2
In z = T −1 x coordinates:
ż1 = λ1 z1 , ż2 = λ2 z2 .
The equilibrium is called a node when λ1 and λ2 have the same
sign (stable node when negative and unstable when positive). It is
called a saddle point when λ1 and λ2 have opposite signs.
z2 z2 z2
z1 z1 z1
stable unstable
node node saddle
λ1 < λ2 < 0 λ1 > λ2 > 0 λ2 < 0 < λ1
• Complex eigenvalues: λ1,2 = α ∓ jβ

" #
α −β
T −1 AT =
β α
ż1 = αz1 − βz2 ṙ = αr

→ polar coordinates →
ż2 = αz2 + βz1 θ̇ = β
z2 z2 z2
z1 z1 z1
stable unstable
focus focus center
α<0 α>0 α=0
The phase portraits above assume β > 0 so that the direction of
rotation is counter-clockwise: θ̇ = 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 are
qualitatively similar to the phase portraits of their linearization. Ac-
cording to the Hartman-Grobman Theorem (below) a “continuous
deformation” maps one phase portrait to the other.
h
x∗
Hartman-Grobman Theorem: If x ∗ is a hyperbolic equilibrium of

ẋ = f ( x ), x ∈ Rn , then there exists a homeomorphism2 z = h( x ) defined 2
a continuous map with a continuous
in a neighborhood of x ∗ that maps trajectories of ẋ = f ( x ) to those of inverse
∂f
ż = Az where A , ∂x ∗ .
x=x
The hyperbolicity condition can’t be removed:

Example:
ẋ1 = − x2 + ax1 ( x12 + x22 ) ṙ = ar3
=⇒
ẋ2 = x1 + ax2 ( x11 + x22 ) θ̇ = 1
" #
∗ ∂ f 0 −1
x = (0, 0) A= =
∂x x= x∗ 1 0
There is no continuous deformation that maps the phase portrait of

the linearization to that of the original nonlinear model:
ẋ = Ax ẋ = f ( x )
(a > 0)
Periodic Orbits in the Plane
Bendixson’s Theorem: For a time-invariant planar system
ẋ1 = f 1 ( x1 , x2 ) ẋ2 = f 2 ( x1 , x2 ),
∂f ∂f
if ∇ · f ( x ) = ∂x1 + ∂x2 is not identically zero and does not change
1 2
sign in a simply connected region D, then there are no periodic orbits
lying entirely in D.
Proof: By contradiction. Suppose a periodic orbit J lies in D. Let S f (x) n( x )

denote 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: •x
Z ZZ S
f ( x ) · n( x )d` = ∇ · f ( x )dx. J
J S
| {z } | {z }
=0 6= 0
Example: ẋ = Ax, x ∈ R2 can have periodic orbits only if

Trace( A) = 0, e.g., " #
0 −β
A= .
β 0
Example:
ẋ1 = x2
ẋ2 = −δx2 + x1 − x13 + x12 x2 δ>0
∂ f1 ∂f
∇ · f (x) = + 2 = x12 − δ
∂x1 ∂x2
√
Therefore, no periodic orbit can lie entirely in the region x1 ≤ − δ
√ √
where ∇ · f ( x ) ≥ 0, or − δ ≤ x1 ≤ δ where ∇ · f ( x ) ≤ 0, or
√
x1 ≥ δ where ∇ · f ( x ) ≥ 0.
not possible: x2
x1
√ √
x1 = − δ x1 = δ
x2
possible:
x1
√ √
x1 = − δ x1 = δ

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EE C222/ME C237 - Spring’18 - Lecture 2 Notes1 1

Licensed under a Creative Commons

Essentially Nonlinear Phenomena Continued

1. Finite escape time

2. Multiple isolated equilibria

3. Limit cycles: Linear oscillators exhibit a continuum of periodic

In contrast, a limit cycle is an isolated periodic orbit and can occur

harmonic limit cycle

Example: van der Pol oscillator

4. Chaos: Irregular oscillations, never exactly repeating.

Chaotic behavior with σ = 10, b = 8/3, r = 28:

• For continuous-time, time-invariant systems, n ≥ 3 state vari-

n = 2: Poincaré-Bendixson Theorem (to be studied in Lecture 3)

Planar (Second Order) Dynamical Systems

Phase Portraits of Linear Systems: ẋ = Ax

• Distinct real eigenvalues

λ1 < λ2 < 0 λ1 > λ2 > 0 λ2 < 0 < λ1

• Complex eigenvalues: λ1,2 = α ∓ jβ

ż1 = αz1 − βz2 ṙ = αr

Phase Portraits of Nonlinear Systems Near Hyperbolic Equilibria

hyperbolic equilibrium: linearization has no eigenvalues on the imagi-

Hartman-Grobman Theorem: If x ∗ is a hyperbolic equilibrium of

The hyperbolicity condition can’t be removed:

There is no continuous deformation that maps the phase portrait of

Periodic Orbits in the Plane

Bendixson’s Theorem: For a time-invariant planar system

Proof: By contradiction. Suppose a periodic orbit J lies in D. Let S f (x) n( x )

Example: ẋ = Ax, x ∈ R2 can have periodic orbits only if

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