Math 118 A/B
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.
- Schedule and Homework Assignments
- Computer Projects
- Class Mailing List Archive
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
- 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.
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
- Miscellaneous Topics:
Perturbation theory, method of averaging,
Strange attractors, Lorenz system, discrete dynamical
- 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
Last modified: 1998/03/23