University of Kansas, Department of Mathematics

Math 881 (Numerical Dynamical Systems)

Fall 2009

Lecture notes will be made available to attending students.

Outline of material to be covered:

• Basics of dynamical systems. Maps and flows. Linear systems. Smoothness wrt ICs/parameters. Stability of equilibria. Lyapunov functions. Invariant sets. Stable and unstable manifolds. Hadamard graph transform.
• Some numerical analysis. Newton's method & root finding; Broyden's update. Gauss-Newton method. Path following principles.
• Parameter dependent systems. Regular paths and bifurcations of equilibria. Genericity and codimension one bifurcations.
• How to compute? Finding bifurcation points: monitor functions. Continuation of equilibria, branch switching. Computational issues: bordered systems, bialternate product. Two parameter continuation. Bifurcations in systems with special symmetries.
• Periodic solutions and Floquet theory. Boundary value problems (BVPs) and their conditioning.
• How to compute? Solution of BVPs: shooting, collocation and error control. Periodic orbits and extra constraints: known/unknown period, linearization, computation of Floquet multipliers.
• Parameter dependent systems. Computation of periodic orbits from Hopf points. Continuation and bifurcations for periodic orbits.
• Connecting orbits. Homoclinic and heteroclinic trajectories between fixed points. Existence, stability. Their computation and continuation: truncation of BVPs on infinite intervals, projected BCs.
• Invariant manifolds. Computation of stable and unstable manifolds of equilibria. Normal hyperbolicity: persistence of invariant manifolds. Invariant tori.
• Stability: Lyapunov exponents and spectrum of a system. Theory and computational approaches.