3-D State Space & Chaos

1. 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

2. Heuristics No chaos in 1- & 2-D state space Chaos: nearby trajectories diverge exponentially for short times λ= Lyapunov exponent • Restrictions: • orbits bounded • no intersection • exponential divergence Can’t be all satisfied in 1- or 2-D Strange attractor Chaotic attractor Chaos is interesting only in systems with attractors. Counter-example: ball perched on hill top.

4. Routes to Chaos Except for solitons, there are no general method for solving non-linear ODEs. Surprises • Ubiquity of chaotic behavior • Universality of routes to chaos Asymptotic motion Regular ( stationary / periodic )  Chaotic

5. Known Types of Transitions to Chaos A system can possess many 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

6. 3-D Dynamical Systems autonomous 2-D system with external t-dependent force : ~ Ex 4.4-1. van der Pol eq.

7. Fixed Points in 3-D → s = Discriminant Index of a fixed point = # of Reλ > 0 = dim( out-set )

8. index = 0 index = 3 index = 3 index = 0 S.P.  Bif

9. 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

10. Periodically driven 2-D system ( non-autonomous ) 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 section Subharmonics of period T = N Tf → N points in Poincare section

11. Approach to a limit cycle Caution: Curve connecting points P0 , P1 , P2 etc, is not a trajectory.

12. Limit Cycles Assume: uniqueness of solution to ODEs → existence of Poincare map function Fixed point ~ limit cycle: Floquet matrix: Characteristic values :: stability But, F usually can’t be obtained from the original differential eqs.

13. Floquet multipliers = Eigenvalues of JM = Mj Yj = coordinate along the jth eigenvector of JM Mj < 0  alternation ( Not allowed in 2-D systems due to the non-crossing theorem ) Dissipative system:

14. 3-D case: Circle denotes |M| = 1 Ex 4.6-1

15. Quasi-Periodicity System with 2 frequencies: → trajectories on torus T2 → 4-D state space T2 can be represented in 3-D state space :

16. Commensurate • Phase-locked • Mode-locked • Incommensurate • quasi-periodic • Conditionally periodic • Almost periodic Neither periodic, nor chaotic

17. 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.

18. Routes to Chaos II: Quasi-Periodicity Hopf bifurcation: spiral node → Limit cycle Details in Chap 6 Landau turbulence: Infinite series of Hopf bifurcations Ruelle-Takens scenario : 2 incommensurate frequencies ( quasi-periodicity ) → chaos

19. 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.

20. 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.

21. 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 ∞.

22. See E.A.Jackson, Perspectives of Nonlinear Dynamics

24. Integrable systems: Homoclinic connection Non-Integrable systems: Homoclinic tangle Poincare section

25. Heteroclinic Tangle (Non-Integrable systems) Heteroclinic Connection (Integrable systems)

26. 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.

27. Homoclinic Tangles & Horseshoes • Stretching along WU. • Compressing along WS. • Fold-back • → Horseshoe map Smale-Birkoff theorem: Homoclinic tangle ~ Horseshoe map Details in Chap 5

28. Experiment: Fluid mixing. 2-D flow with periodic perturbation, dye injection near hyperbolic point.

29. Lyapunov Exponents & Chaos • Quantify chaos: • distinguish between noise & chaos. • measure degree of chaoticity. • Chapters: • 4: ODEs • 5: iterated map • 9,10: experiment x(t), x0(t) = trajectories with nearby starting points x(0), x0(0). = distance between the trajectories For all x(t) near x0(t): → = Lyapunov exponent at x0. = average over x0 on same trajectory.

30. n-D system: Let ua be the eigenvector of J(x0) with eigenvalue λa(x0). → Chaotic system: at least one positive averageλa = <λa (x0) >.

31. Behavior of a cluster of ICs Dissipative system: • 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>.

32. Spectra of Lyapunov exponents in 3-D state space

33. Cautionary Tale Choatic → <λ> > 0 converse not necessarily true. Pseudo-chaos: On outsets of saddle point <λ> > 0 for short time Saddle point: Θ=π Example: pendulum