1.
Existence and Uniqueness
of local solutions to dx/dt = F (t, x) where F (t, x)
is Lipschitz in x.
2.
Extending local
solutions to global solutions.
3.
Continuous dependence of
solutions on initial conditions and parameters.
4.
Basic theory for linear
systems (existence, uniqueness, superposition, variation of parameters).
5.
Study of dx/dt =
Ax. The matrix exponential. Stability.
6.
Applications and some
practice with numerical software.
7.
Autonomous systems:
omega limit sets, properties of orbits, equilibrium solutions, Hartman-Grobman
Theorem, invariant sets, minimal sets, linearized stability, specialization to
2-d, and using numerical software.
8.
Lyapunov’s second
method, global stability.
9.
Periodic solutions,
limit cycles, Poincaré-Bendixson Theorem (proof and applications).
Main Text:
Jane Cronin: Differential Equations;
Introduction and Qualitative Theory.
Supplementary texts:
Fred Brauer and John Nohel: The Qualitative
Theory of Ordinary Differential Equations; an
Introduction.
Paul Waltman: A Second Course in Elementary Differential
Equations.