This course will be the first semester of a graduate level introduction to differential equations and dynamical systems with emphasis on qualitative, geometrical methods for nonlinear systems. It will be followed in the spring by Math 8502 -- Dynamical Systems and Differential Equations, which will explore further topics.

Understanding the modern theory of dynamical systems requires a lot of ideas from many parts of mathematics. I hope to cover both the differential equations theory itself and these background ideas in a way which can be understood not only by math graduate students but by any mathematically inclined student with a solid knowledge of linear algebra, advanced calculus and elementary differential equations.

Here are some of the topics I would like to cover in the first semester: basic existence and uniqueness theory, flows and flow boxes, invariant sets, flows on manifolds, alpha and omega limit sets, flows in the plane, Poincare-Bendixson theory, index theory, Poincare maps, rotation numbers, variational equations, theory of linear systems, dynamics near equilibria including Hartman's theorem and the stable manifold theorem. 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 |

"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 |

Based on several homework assignments throughout the semester. No exams. The link leads PDF versions of the homework assignments so far.

Homework | 100 % |