Math 118 A/B
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, omegalimit, 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:

PoincaréBendixon 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, quasiperiodic motions, Hamiltonian
perturbation theory.
Last modified: 1998/03/23
Marcel Oliver
(oliver@math.uci.edu)