Pomona College Math 181

Dynamical Systems

Prof. Richard H. Elderkin

Mondays and Wednesdays, 1:15 p.m., Millikan Laboratory 213

Modern dynamical systems theory originally grew out of the investigation of ordinary differential equations, beginning primarily with the contributions of Poincare’ and Lyapunov around the turn of the twentieth century. More recently, in the late twentieth century, it expanded to include the theories of bifurcations and chaos. Hence this theoretical course has a strong motivation from and use in applications. The theory builds on earlier coursework in linear algebra, differential equations and analysis, so both Math 102 and Math 101 or 131 are prerequisite. The subject also develops a strong geometric flavor which holds its own beauty.

Text: Stability, Instability and Chaos: and Introduction to the Theory of Nonlinear Differential Equations, by Paul Glendinning, originally published by Cambridge University Press, 1994.

Summary list of probable topics:

1. Phase space, flows, limit sets
2. Stability
3. Linear d.e.’s
1. Normal forms
2. Geometry of phase space and invariant manifolds
3. Floquet theory
4. Linearization and Hyperbolicity
1. Stable manifold theorem
2. Structural stability
5. Phase plane
1. Poincare’ index
2. Dulac’s criterion
3. Poincare’-Bendixson Theory
6. Period orbits
1. Return maps
2. More Floquet theory
7. Bifurcation theory
1. Center manifolds
2. Local bifurcations: saddle-node, transcritical, pitchfork and Hopf
3. Period doublings
8. Chaos
1. Characterizations
2. Period three implies chaos
9. Lorenz equations

Course structure:

Class meetings: MW 1:15-2:30

Office hours: M 4:00-5:00 and TWR 2:40-3:45

Basis for grade:

Two “midterm” exams @ 20% each

Final exam = 25%

Homework = 35%