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 Restrictions: orbits bounded no intersection exponential divergence Can’t be all satisfied in 1- or 2-D Strange attractor λ= Lyapunov exponent Chaotic attractor Chaos is interesting only in systems with attractors. Counter-example: ball perched on hill top.

4. 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

5. 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.

6. 3-D Dynamical Systems autonomou s 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 ) → A+B(A-B)iRoots s < 0A = B* complexreal 3 real s = 0A = B realreal03 real (2 equal) s > 0 A  B real realimaginary1 real, 2 complex

8. index = 0 index = 3 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. 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 T f → N points in Poincare section Periodically driven 2-D system ( non-autonomous )

11. Approach to a limit cycle Caution: Curve connecting points P 0, P 1, P 2 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 = M j Dissipative system: M j < 0  alternation Y j = coordinate along the j th eigenvector of JM ( Not allowed in 2-D systems due to the non-crossing theorem )

14. fixed pointFloquet MultiplierCycle Node |M j | < 1  j Limit cycle Repellor |M j | > 1  j Repelling cycle SaddlemixedSaddle cycle 3-D case: Circle denotes |M| = 1 Ex 4.6-1

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

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 Ruelle-Takens scenario : 2 incommensurate frequencies ( quasi-periodicity ) → chaos Landau turbulence: Infinite series of Hopf bifurcations Details in Chap 6

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

25. Heteroclinic Connection (Integrable systems) Heteroclinic Tangle (Non- 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 W U. Compressing along W S. 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), x 0 (t) = trajectories with nearby starting points x(0), x 0 (0). = distance between the trajectories For all x(t) near x 0 (t): = Lyapunov exponent at x 0. → = average over x 0 on same trajectory.

30. n-D system: Let u a be the eigenvector of J(x 0 ) with eigenvalue λ a (x 0 ). → Chaotic system: at least one positive average λ a =.

31. Behavior of a cluster of ICs Dissipative system: 3-D ODE: One must be 0 unless the attractor is a fixed point. H.Haken, Phys.Lett.A 94,71-4 (83) System dissipative → at least one must be negative. System chaotic → one positive. Hyperchaos: More than one positive.

32. Signs of λsType of attractor -, -, -Fixed point 0, -, -Limit cycle 0, 0, -Quasi-periodic torus +, 0, -Chaotic Spectra of Lyapunov exponents in 3-D state space

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