3-D State Space & Chaos

• Fixed Points, Limit Cycles, Poincare Sections

• Routes to Chaos– Period Doubling– Quasi-Periodicity– Intermittency & Crises– Chaotic Transients & Homoclinic Orbits

• Homoclinic Tangles & Horseshoes• Lyapunov Exponents

Heuristics

No chaos in 1- & 2-D state space

Chaos: nearby trajectories diverge exponentially for short times

Restrictions:

• orbits bounded

• no intersection

• exponential divergence

Can’t be all satisfied in 1- or 2-D

Strange attractor

0 td t d e λ= Lyapunov exponent

Chaotic attractor

Chaos is interesting only in systems with attractors.Counter-example: ball perched on hill top.

Routes to Chaos

Surprises• Ubiquity of chaotic behavior• Universality of routes to chaos

Except for solitons, there are no general method for solving non-linear ODEs.

Asymptotic motion

Regular ( stationary / periodic ) Chaotic

Known Types of Transitions to Chaos

• Local bifurcations (involves 1 limit cycle)– Period doubling– Quasi-periodicity– Intermittency:

• Type I ( tangent bifurcation intermittency )• Type II ( Hopf bifurcation intermittency )• Type III ( period-doubling bifurcation intermittency )

• On-off intermittency

• Global bifurcations ( involves many f.p. or l.c. )– Chaotic transients– Crises

A system can possess many types of transitions to chaos.

3-D Dynamical Systems

1 2 3, ,i ix f x x x 1,2,3i autonomous

x f x

2-D system with external t-dependent force :

1 1 1 2

2 2 1 2

, ,

, ,

x f x x t

x f x x t

1 1 1 2 3

2 2 1 2 3

3

, ,

, ,

1

x f x x x

x f x x x

x

~

Ex 4.4-1. van der Pol eq.

21 cosx x x x F t

Fixed Points in 3-D

1 1 1

1 2 3

2 2 2

1 2 3

3 3 3

1 2 3

f f f

x x x

f f fJ

x x x

f f f

x x x

3 2 0p q r 3 0x ax b

2 3

2 3

b as

s = Discriminant

Index of a fixed point = # of Reλ > 0 = dim( out-set )

1

3x p 21

33

a q p 312 9 27

27b p q p r

→

1/ 3

2

bA s

1/ 3

2

bB s

1or 3

2x A B A B A B i

A+B (A-B)i Roots

s < 0

A = B* complex

real real 3 real

s = 0

A = B real real 0 3 real (2 equal)

s > 0

A B real real imaginary

1 real, 2 complex

index = 0

index = 0

index = 3

index = 3

S.P. Bif

Poincare Sections

Poincare sections:

• Autonomous n-D system: (n-1)-D transverse plane.

• Periodically driven n-D system: n-D transverse plane.

( stroboscopic portrait with period of driven force )

transverse plane

Non- transverse plane

Trajectory is on surface of torus in 3-D state space of equivalent autonomous system.

Phase : [0 , 2π)

Poincare section = surface of constant phase of force

Limit cycle (periodic) → single point in Poincare sectionSubharmonics of period T = N Tf → N points in Poincare section

Periodically driven 2-D system ( non-autonomous )

Approach to a limit cycle

Caution: Curve connecting points P0 , P1 , P2 etc, is not a trajectory.

Limit Cycles

Assume: uniqueness of solution to ODEs

→ existence of Poincare map function 1

1 , ,n n n nj j k jx F x x F x 1, ,j k

Fixed point ~ limit cycle:

1* *, , * *j j k jx F x x F x

Floquet matrix: *

i

j

FJM

x

x x

1 1

1

1 *

k

k k

k

F F

x x

F F

x x

x x

Characteristic values :: stabilityBut, F usually can’t be obtained from the original differential eqs.

Floquet multipliers = Eigenvalues of JM = Mj

Dissipative system: det 1jj

JM M

1*n n nj j j j jd y y M d

Mj < 0 alternation

Yj = coordinate along the

jth eigenvector of JM

( Not allowed in 2-D systems due to the non-crossing theorem )

fixed point Floquet Multiplier Cycle

Node |Mj| < 1 j Limit cycle

Repellor |Mj| > 1 j Repelling cycle

Saddle mixed Saddle cycle

3-D case:

Circle denotes |M| = 1

Ex 4.6-1

Quasi-Periodicity

T2 can be represented in 3-D state space :

1

2

3

sin cos

cos

sin sin

r R

r

r R

x R r t t

x r t

x R r t t

21

22

0

0

x x

y y

→ 4-D state space

2 2 21 1

2 2 22 2

x x E

y y E

→ trajectories on torus T2

System with 2 frequencies:

r

R

rational

r

R

irrational

• Commensurate

• Phase-locked

• Mode-locked

• Incommensurate

• quasi-periodic

• Conditionally periodic

• Almost periodic

Neither periodic, nor chaotic

Routes to Chaos I: Period-Doubling

Flip bifurcation:

all |M| < 1 (Limit cycle) → One M < -1 (period doubling)

( node ) ( 1 saddle + 2 nodes )

There’s no period-tripling, quadrupling, etc. See Chap 5.

Routes to Chaos II: Quasi-Periodicity

Hopf bifurcation:

spiral node → Limit cycle

Ruelle-Takens scenario :

2 incommensurate frequencies ( quasi-periodicity )

→ chaos

Landau turbulence: Infinite series of Hopf bifurcations

21

r r r

Details in Chap 6

Routes to Chaos III: Intermittency & Crises

Details in Chap 7

Intermittency:

periodic motion interspersed with irregular bursts of chaos

Crisis:

Sudden disappearance / appearance / change of the size of basin of chaotic attractor.

Cause: Interaction of attractor with unstable f.p. or l.c.

Routes to Chaos IV: Chaotic Transients & Homoclinc Orbits

Global bifurcation:

• Crises: Interaction between chaotic attractor & unstable f.p / l.c.

• sudden appearance / disappearance of attractor.

• Chaotic transients:

Interaction of trajectory with tangles near saddle cycle(s).

• not marked by changes in f.p. stability

→ difficult to analyse.

• most important for ODEs, e.g. Lorenz model.

• Involves homoclinic / heteroclinic orbits.

Homoclinic Connection

Saddle Cycle

Poincare section

Critical theorem:

The number of intersects between the in-sets & out-sets of a saddle point in the Poincare section is 0 or ∞.

See E.A.Jackson,

Perspectives of Nonlinear Dynamics

Poincare section

Non-Integrable systems: Homoclinic tangle

Integrable systems: Homoclinic connection

Heteroclinic Connection (Integrable systems)

Heteroclinic Tangle (Non-Integrable

systems)

Lorenz Eqs

Sil’nikov Chaos:

• 3-D: Saddle point with characteristic values

a, -b + i c, -b - i c a,b,c real, >0

• → 1-D outset, 2-D spiral in-set.

• If homoclinic orbit can form & a > b, then chaos occurs for parameters near homoclinic formation.

• Distinction: chaos occurs before formation of homoclinic connection.

Homoclinic Tangles & Horseshoes

• Stretching along WU.

• Compressing along WS.

• Fold-back

• → Horseshoe map

Smale-Birkoff theorem:

Homoclinic tangle ~ Horseshoe map

Details in Chap 5

Experiment: Fluid mixing.

2-D flow with periodic perturbation,

dye injection near hyperbolic point.

Lyapunov Exponents & Chaos

Quantify chaos:

• distinguish between noise & chaos.

• measure degree of chaoticity.

Chapters:

• 4: ODEs

• 5: iterated map

• 9,10: experiment

x f x 0

0 0

x

d ff x x x

d x

x(t), x0(t) = trajectories with nearby starting points x(0), x0(0).

0s x x = distance between the trajectories

For all x(t) near x0(t):

0 0s x x f x f x 0x

d fs

d x 00 x ts t s e

0

0

x

d fx

d x = Lyapunov exponent

at x0.

→

0x = average over x0 on same trajectory.

n-D system:

0 s x x 0i i is x x

x f x 0

0 0 x

f x x x f

i ix f x 0

0 0i

i j jj j

ff x x

x

x

x

0 0 s x x f x f x 0

x

s f

0 0i i i i is x x f f x x0

ij

j j

fs

x

x

Let ua be the eigenvector of J(x0) with eigenvalue λa(x0).

0i j jj

J s x

0 J x s

0a a au x u→ 00 a t

a at e xu u

Chaotic system: at least one positive average λa = <λa (x0) >.

0

0i

i jj

fJ

x

x

x

Behavior of a cluster of ICs

0 exp ii

V t V t 1

ii

dV

V dt

f

Dissipative system:

0ii

3-D ODE: • One <λ> must be 0 unless the attractor is a fixed point.

H.Haken, Phys.Lett.A 94,71-4 (83)• System dissipative → at least one <λ> must be negative.• System chaotic → one <λ> positive.

Hyperchaos: More than one positive <λ>.

Signs of λs Type of attractor

-, -, - Fixed point

0, -, - Limit cycle

0, 0, - Quasi-periodic torus

+, 0, - Chaotic

Spectra of Lyapunov exponents in 3-D state space

Cautionary Tale

Choatic → <λ> > 0 converse not necessarily true.

Pseudo-chaos:

On outsets of saddle point <λ> > 0 for short time

Example: pendulum

sing

uL

u

Saddle point: Θ=πg

L

0 03.00 0.

3.02 0.0724

u

same E