Numerical Methods for Dynamical Systems

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Every odd spring semester

Approximation of the dynamical structure of a differential equation and preservation of dynamical structure under discretization.

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Topic Outline: 
  • Review Material Covered at a Brisk Pace:
    • Basics of dynamical systems: - Autonomous and nonautonomous, existence and uniqueness of solutions, regularity, maps and flows, manifolds and transversality, diffeo/homeomorphisms, fixed points, Lyapunov functions, gradient systems, invariant sets, stable and unstable manifolds, Grobman-Hartman theorem dissipativity, attractors, conditioning and dichotomy
    • Basics of numerical analysis: - Interpolation, Newton's method and root finding, quadrature, solution of initial value problems, error control principles, some topics of numerical linear algebra
  • Computing Fixed Points and Phase Portraits - Linearization, eigenvalues, stability, stable and unstable manifolds
  • Computing Periodic Orbits with Known and Unknown Periods - Computing solutions of boundary value problems, error control, linearization, Floquet theory, stable and unstable manifolds
  • Computation of Bifurcations in Parameter Dependent Systems - Regular paths and bifurcations, bifurcation types for equilibria and for periodic orbits, arc length parametrization, continuation at simple bifurcation points on the same branch, branch switching
  • Computation of Connecting Orbits - Homoclinic and heteroclinic trajectories, boundary conditions at infinity, transversal homoclinics and its implications, between fixed points and periodic solutions
  • Computation of Normal Hyperbolicity - Persistence of invariant manifolds
  • Computation of Invariant Manifolds - Invariant tori, rotation numbers
  • Computation of Hyperbolic Structures - Hyperbolic sets, attractors, Lyapunov exponents, spectrum; estimates of dimensions
  • Preservation of Dynamical Structure Under Discretization - Applications to Hamiltonian systems, dissipative systems, and problems with invariant regions
  • Alternatives to Classic Error Analysis - Backward error analysis, hyperbolic sets, shadowing, modified equations
  • Computation of Topological Invariants