This is the second semester of a graduate level introduction to differential equations and dynamical systems with emphasis on qualitative, geometrical methods for nonlinear systems.
Topics for the second semester include: linearization, Hartman's theorem; bifurcations of equilibria; existence, continuation and stability of periodic orbits; Poincare' maps and discrete dynamical systems; homoclinic points and homoclinic chaos; one-dimensional maps; circle maps and rotation numbers; introduction to ergodic theory.
In addition a good deal of time will be spent on interesting examples, mostly from classical mechanics, to illustrate and apply the theory.
Although no book is required, here are some recommended references.
| "Ordinary Differential Equations", by V.I. Arnold |
| "Geometrical Methods in the Theory of Ordinary Differential Equations", by V.I. Arnold |
| "Ordinary Differential Equations with Applications", by C. Chicone |
| "Dynamical Systems: Stability, Symbolic Dynamics and Chaos", by Clark Robinson |
| "An Introduction to Ergodic Theory", by Peter Walters |
| "Ordinary Differential Equations", by Jack Hale |
| "Ordinary Differential Equations", by Philip Hartman |
| "Differential Equations, Geometric Theory", by S. Lefschetz |
| "Theory of Ordinary Differential Equations", by E. Coddington and N. Levinson |
| "Stability Theory of Dynamical Systems", by Bhatia and Szego |
Based on several homework assignments throughout the semester. No exams. The link leads PDF versions of the homework assignments so far.
| Homework | 100 % |
Here are some Mathematica notebooks discussed in class.
| Poincare map for the forced pendulum |
| Arnold tongues for Hill's equation |