II. Towards a Theory of Nonlinear Dynamics & Chaos 3. Dynamics in State Space: 1- & 2- D 4. 3-D State Space & Chaos 5. Iterated Maps 6. Quasi-Periodicity & Chaos 7. Intermittency & Crises 8. Hamiltonian Systems

3.Dynamics in State Space: 1- & 2- D State space / Phase SpaceH. 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. Concepts to be introduced:

State Space Spring obeying Hooke’s law : Degrees of freedom : 1. Classical mechanics (phase space) : number of (q,p) pairs. 2. Dynamical systems (state space) : number of independent variables. Cycle: Closed periodic trajectory

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

One n th order ODE ~ n 1 st order ODEs Mass spring:2 nd order ODE Two 1 st order ODEs

Given u* is a fixed point if Caution: Autonomous version of a non-autonomous system requires special treatment [ u n+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.

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

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. Basin of 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.

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

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

λ = 0

Repells λ = 0 Attracts Saddle points Repells Attracts Conve x Concave

Structural Instability Saddle point is structurally unstable

Taylor series Expansion X 0 = fixed point > 0  X 0 repellor < 0  X 0 node = 0  X 0 s.p. / node / rep. Lyapunov exponent

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

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.

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

2-D State Space Fixed point

Special case λ 1 < 0 λ 2 < 0 λ 1 > 0 λ 2 < 0 λ 1 0 λ 1 > 0 λ 2 > 0

Saddle point at (x,y) = (-π/2, 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. Hyperbolic point: λ  0 1-D saddle point are non-hyperbolic since λ = 0

Brusselator A,B > 0 Fixed points: →

General 2-D Case State variables can be any pair fromEx 3.11 →

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

R < 0R > 0

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

Jacobian Matrix at Fixed Point → → → Jacobian matrix

2-D State Space Tr J < 0Tr J > 0 Δ < 0 Both λ complex, Real part negative Spiral node Both λ complex, Real part positive Spiral repellor Δ > 0 det J > 0 Both λ real & negative node Both λ real & positive repellor Δ > 0 det J < 0 Both λ real & of opposite signs, Saddle point Both λ real & of opposite signs, Saddle point

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

Limit Cycles Limit cycle: closed loop in state space to (from) which nearby trajectories are attracted (repelled). 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 1. fixed point, or 2. limit cycle. Invariant set: region in state space where a trajectory starting in it will remain there forever. Implications: no chaos in 2-D systems. limit cycle in Brusselator. Delayed DE: Topology: Poincare index theorem  vortex DoF =  : IC for t  [-T,0] needed

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

Poincare Map Exercise Fixed point of F : Near P*: Let → = (characteristic / Floquet / Lyapunov) multiplier → Characteristic exponent M < 1Attracting M > 1Repelling M = 1Saddle (rare in 2-D )

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

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

2 fixed points for μ> 0 No Fixed points if μ< 0. nod e 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.

Lifted (Suspended) State Space Repellor ↓ Saddle point Node ↓ Node No fixed point Flow along extra dimension X 2 always towards original axis X 1.

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

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

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

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

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