II. Towards a Theory of Nonlinear Dynamics & Chaos

1. II. Towards a Theory of Nonlinear Dynamics & Chaos • Dynamics in State Space: 1- & 2- D • 3-D State Space & Chaos • Iterated Maps • Quasi-Periodicity & Chaos • Intermittency & Crises • Hamiltonian Systems

2. 3. Dynamics in State Space: 1- & 2- D Concepts to be introduced: State space / Phase Space H. Poincare J.W. Gibbs Fixed points ( equilibium / stationary / critical / singular ) points Limit Cycles Stability (attractor) / Instability (repellor) Bifurcations : Change of stability / Birth of f.p. or l.c.

3. State Space • Degrees of freedom : • Classical mechanics (phase space) : • number of (q,p) pairs. • Dynamical systems (state space) : • number of independent variables. Spring obeying Hooke’s law : Cycle: Closed periodic trajectory

4. Systems of 1st Order ODEs Autonomous DoF = n Non-Autonomous Dimension of state space = number of 1st order autonomous ODEs. N-DoF non-autonomous → (N+1)-DoF autonomous Autonomous Non-crossing theorem is applicable only to autonomous systems

5. One nth order ODE ~ n 1st order ODEs Mass spring: 2nd order ODE Two 1st order ODEs

6. Given u* is a fixed point if Caution: Autonomous version of a non-autonomous system requires special treatment [ un+1 = 1  0 ]. • All dynamical systems can be converted to a set of 1st order ODEs. • For some systems this requires DoF = ∞, e.g., • PDEs • integral – differential eqs • memory eqs If the system is dissipative, only a few DoFs will remain active eventually.

7. No-Intersection Theorem • A state space trajectory cannot cross itelf. • 2 distinct state space trajectories cannot intersect in a finite amount of time. Physical implication : Determinism Mathematical origin : Uniqueness solutions of ODE that satisfy the Lipschitz condition (f bounded). • Apparent violations: • Asymptotic intersects. • Projections

8. Dissipative Systems & Attractors • Transients not important in dissipative systems ( long time final states independent of IC ) • Attractor: Region of state space to which some trajectories converge. • Basinof an attractor: Region of state space through which all trajectories converge to that attractor. • Separatrix: Boundary between the basins of two different attractors. • Miscellaneous: • Fractal basin boundaries. • Riddled basins of attraction. • Dimension of the state space.

9. 1-D State Space Evolution eq. : Fixed point: • Types of fixed points in 1-D state spaces: • Nodes / sinks / stable fixed points • Repellors / sources / unstable fixed points • Saddle points

10. Type Determination Let X0 be a fixed point: = characteristic value ( eigenvalue ) of X0 For λ > 0 For λ < 0

11. λ = 0

12. λ = 0 Repells Attracts Convex Saddle points λ = 0 Attracts Repells Concave

13. Structural Instability Saddle point is structurally unstable

14. Taylor series Expansion X0 = fixed point Lyapunov exponent  > 0  X0 repellor • = 0  X0 s.p. / node / rep.  < 0  X0 node

15. Trajectories in 1-D State Space   local behavior Globalbehavior determined by matching fixed point basins. → joining arrows pointing toward (away) from nodes (repellors). • f continuous •  neighboring fixed points cannot be • both nodes or both repellors • saddle points of different types Exercises 3.8- 3,4

16. Bounded systems: • Outermost fixed points must be • nodes or • type I saddle point on the left • type II saddle point on the right. • A node must be on the repelling side of a saddle point.

17. When Is A System Dissipative ? Defining characteristics of a dissipative system : Motion reduced asymptotically to a few active DoFs. Cluster of ICs ( those that lead to fixed points excluded ). C.f., statistical ensembles. System is dissipative near XA if df/dX < 0. e.g., near a node. Divergence theorem

18. 2-D State Space Fixed point

19. Special case λ1 > 0 λ2< 0 λ1 < 0 λ2< 0 λ1 < 0 λ2> 0 λ1 > 0 λ2> 0

20. Saddle point at (x,y) = (-π/2 , 0 )

21. Hyperbolic point: λ 0 • Invariant manifold: • all trajectories along the principal axes of a hyperbolic saddle point. • Stable (invariant) manifold (in-set): • Trajectories heading towards the hyperbolic saddle point. • Unstable (invariant) manifold (out-set): • Trajectories heading away from the hyperbolic saddle point. In-sets & outsets serves as separatrices. 1-D saddle point are non-hyperbolic since λ = 0

22. Brusselator A,B > 0 → Fixed points:

23. General 2-D Case → State variables can be any pair from Ex 3.11

24. Complex Characteristic Values Note: is either real or purely imaginary Spirals, inward if R < 0 (focus) outward if R > 0 Limit cycle if R = 0

25. R < 0 R > 0

26. Dissipation & Divergence Theorem 2-D state space Area :  f < 0  dissipative

27. Jacobian Matrix at Fixed Point → → → Jacobian matrix

28. 2-D State Space

29. Example: The Brusselator Set A = 1 & let B be control parameter : • B < 2, spiral node • 2 < B < 4, spiral repellor (converge to another limit cycle) • B > 4, do exercise 3.14-2.

30. Limit Cycles Limit cycle: closed loop in state space to (from) which nearby trajectories are attracted (repelled).  vortex Invariant set: region in state space where a trajectory starting in it will remain there forever. • Poincare-Bendixson theorem: • Let R be a finite invariant set in a 2-D state space, then any trajectory in it must, as t → ∞, approach a • fixed point , or • limit cycle. Delayed DE: • Implications: • no chaos in 2-D systems. • limit cycle in Brusselator. DoF = : IC for t[-T,0] needed Topology: Poincare index theorem

31. Poincare Sections Poincare section in n-D state space: An (n-1)-D hyper-surface that cuts through the trajectory of a n-D continuous flow and reduces it to a (n-1)-D discrete map. Example: Limit cycle in 2-D state space

32. Poincare Map Exercise 3.16-1 Fixed point of F : Near P*: Let → = (characteristic / Floquet / Lyapunov) multiplier → Characteristic exponent

33. Bifurcation Theory Appendix B Study of changes in the character of fixed points. ( limit cycles are fixed points in Poincare sections ) • 2 types of bifurcation diagrams: • control parameter vs location of fixed point. • control parameter vs characteristic value.

34. Bifurcations in 1-D Normal form δ> 0 repellor δ< 0 node Bifurcation at δ= 0

35. No Fixed points if μ< 0. 2 fixed points for μ> 0 node repellor • For μ= 0, x* = 0 is a saddle point. • For μ> 0, x* = ±μ form repellor-node pair. • μ= 0 is repellor-node bifurcation point. • Other names: saddle-node / tangent / fold bifurcation Note that for μ= 0, x* = 0 is structurally unstable.

36. Lifted (Suspended) State Space Flow along extra dimension X2 always towards original axis X1. Repellor ↓ Saddle point Node ↓ Node No fixed point

37. Bifurcation in 2-D The Brusselator B > 4, λ± real (Transcritical) bifurcation at B = 2. Node → Repellor + limit cycle 4 B  0, λ± complex

38. Normal Form Equations Fixed point at x = 0. Bifurcation at μ = 0. Saddle – node bifurcation: μ> 0 : node at saddle at μ< 0 : no fixed point μ= 0 : bifurcation

39. Transcritical bifurcation: 2 fixed points switching types of stability Pitchfork bifurcation: 1 → 3 fixed points

40. Limit Cycle Bifurcations Spiral-in-spiral-out bifurcation at Re(λ) = 0 Hopf bifurcation: birth of stable limit cycle Poincare section of limit cycle in 2-D → 1-D dynamics Normal form: Polar coordinates:

41. Fixed points: for μ< 0 r* = 0 spiral node for μ> 0 r* = 0 spiral repellor limit cycle, period = 2π Hopf bifurcation at μ= 0 Limit cycle: asymptotic time-dependent behavior of dissipative system.