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.
Here are some Mathematica notebooks discussed in class.