EE222 - Spring’16 - Lecture 2 Notes11 Licensed under a Creative CommonsAttribution-NonCommercial-ShareAlike4.0 International License.Murat Arcak

January 21 2016

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

ee222 - spring’16 - 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

]

ee222 - spring’16 - 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

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.

ee222 - spring’16 - 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.

ee222 - spring’16 - 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: