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 |