# Differential Equations

Math 118 A/B is an introductory theoretical course in ordinary differential equations with emphasis on the ``modern'' theory of dynamical systems. Throughout this course, we will make regular use of symbolic and numerical tools (using mainly Mathematica) in the new Computational and Applied Mathematics Lab. Note that this is not a Numerical Analysis class, i.e., we will learn how to use the computer as a tool to facilitate the rigorous analysis of differential equations, rather than study numerical methods per se.

Syllabus (118A, 118B)
Schedule and Homework Assignments
Computer Projects
Class Mailing List Archive (118A, 118B)
Differential Equations on the Web

### List of Topics

Basic concepts:
Phase space and phase flows, vector fields, orbits, omega-limit, first integrals, initial value problems, explicit solutions.
Existence and uniqueness:
Contraction mapping theorem, Picard iterations, Lipschitz condition.
Linear systems:
Exponential of an operator, determinant of the exponential, complexification, equations with constant coefficients, equations with coefficients that have a limit, equations with periodic coefficients, variation of constants.
Stability:
Asymptotic stability vs. Lyapunov stability; Stability by linearization, Stability by the method of Lyapunov functions, stable and unstable manifolds.
Periodic Solutions:
Poincaré-Bendixon theorem, stability of periodic solutions.
Miscellaneous Topics:
Perturbation theory, method of averaging, bifurcations, attractors.
Chaos:
Strange attractors, Lorenz system, discrete dynamical systems.
Hamiltonian dynamics:
Variational principles, Hamiltonian functions, basic notions of differential geometry, evolution of a volume element, Liouville's theorem.
KAM theory:
Invariant tori, quasi-periodic motions, Hamiltonian perturbation theory.