Advanced Nonlinear Dynamics and Chaos (18.386J/2.037J)

Homework Assignments

Introductory slides (PDF)

This graduate course discusses advanced topics from the modern theory of nonlinear dynamical systems. Applications will primarily be selected from mechanics and fluid dynamics. Prerequisites: 18.385J/2.036J or equivalent.

Topics include:

• Normally hyperbolic invariant manifolds

  Crash course on manifolds; existence, persistence, and smoothness; geometric singular perturbation theory

• Global bifurcations

  Homoclinic bifurcations; higher-dimensional Melnikov methods; Shilnikov orbits

• The internal structrure of chaos

  Symbolic dynamics, Bernoulli shift map, subshifts of finite type, higher-dimensional chaos

• Hamiltonian dynamical systems

  Canonical and noncanonical Hamiltonians, symplectic geometry, conservation properties, phase space geometry

• Integrable and near-integrable systems

  KAM theory, existence and persistence of invariant tori

• Introduction to infinite-dimensional dynamical systems

  Attractors, inertial manifolds; Hamiltonian and dissipative examples

The course will draw topics and applications from several sources, including the following textbooks:

Guckenheimer, J., and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

Arnold, V. I., Mathematical Methods of Classical Mechanics

Chicone, C., Ordinary Differential Equations with Applications

These books will be on reserve in Baker Library. There is no required textbook.

Grading: Based on homework.