# APMA 2190: Nonlinear Dynamical Systems: Theory and Applications (Fall 2010)

## Class information

Instructor | Björn Sandstede |

Office | Room 325 in 182 George Street |

Phone | 863-2815 |

Email | bjorn_sandstede@brown.edu |

Office Hours | Tuesdays 4-5, Thursdays 11-12, and by appointment |

Class meetings | Tuesdays and Thursdays 2:30-3:50 in Wilson Hall 204 |

## Project reports

A PDF file that contains all project reports is posted below:

## Course material

This course provides a rigorous introduction to ordinary differential equations, studied from a dynamical-systems viewpoint. Motivated by examples from ecology, chemistry, mechanics, and physics, we will study the existence and uniqueness of solutions and the dynamical behavior near equilibria and periodic orbits.

- Existence and uniqueness, dependence on parameters
- Dynamical systems, flows and maps, orbits
- Linear equations and Floquet theory
- Stable and unstable invariant manifolds for equilibria and periodic orbits
- Planar systems: Poincare--Bendixson theorem
- Hamiltonian and gradient systems
- Center manifolds
- Normal forms
- Bifurcation theory for equilibria including Hopf bifurcations

The following topics will be covered in Spring:

- Melnikov method
- Horsehoes, and chaotic dynamics
- Averaging
- Geometric singular perturbation theory
- Lyapunov-Schmidt reduction

## Literature

No textbook is required. Most of the material covered in the course can be found in

- C Chicone:
*Ordinary differential equations with applications*, Springer

Other books that contain some or all of the material covered in the course are

- S-N Chow and J Hale:
*Methods of bifurcation theory*, Springer
- H Amann:
*Ordinary differential equations*, de Gryuter
- EA Coddington and N Levinson:
*Theory of ordinary differential equations*, McGraw-Hill
- P Hartman:
*Ordinary differential equations*, Wiley