Math 243 A,B,C, Syllabus


Ordinary Differential Equations, Dynamical Systems and Bifurcation Theory

I.

Existence, Uniqueness and Stability

1.

The local existence theorem

2.

Global solutions and stability

3.

Absorbing sets and Milnor's theorem

II.

The Geometry of Phase Space

1.

Vector fields and flows

2.

Equivalence of flows

III.

Linear Systems and Hyperbolicity

1.

Solutions of linear systems

2.

Classification of hyperbolic stationary solutions

IV.

Maps and Diffeomorphisms

1.

One-dimensional maps and the Poincaré map

2.

The Hartmann-Grobmann and the invariant manifold theorems

3.

Invariant manifold theorems for maps

4.

Classification of flows and maps

V.

Hyperbolic Structure and Chaos

1.

Horseshoe maps and symbolic dynamics

2.

The Birkoff-Smale homoclinic theorem

3.

The Melnikov method

VI.

Center Manifolds

1.

The center manifold theorem

2.

The attractiveness of the center manifold

3.

Example, the Lorentz equations

VII.

Bifurcation Theory and Normal Forms

1.

The saddle-node, transcritical and pitchfork bifurcations

2.

Poincaré-Birkoff normal forms

3.

The Hopf and period-doubling bifurcations

VIII.

The Feigenbaum Period-Doubling Cascade

1.

The quadradic map

2.

Scaling behavior

3.

Renormalization theory

IX.

The Ruelle-Takens Cascade

1.

Quasi-periodic orbits and twist maps

2.

The instability of the three-torus

3.

The strange attractor

X.

Global Phenomena

1.

Slow-manifolds, Fenischel-Jones-Kopell theory

2.

Extended attractors

3.

Example, the Ginzburg-Landau equation