### John Guckenheimer

## Algorithms for Bifurcations

Dynamical systems theory elucidates geometric structure in the solution
of ordinary differential equations. Bifurcation theory describes the
changes that occur in these structures as parameters change. Algorithms
based upon classical numerical methods and mathematical theory
facilitate the visualization of dynamical systems. This chapter reviews
the phenomena described by the theory and surveys the algorithms that
are used to compute them. Examples are used throughout. Software
packages that incorporate algorithms for studying dynamical systems are
discussed and their performance is evaluated.

Tentative Outline:

- Computation of trajectories
- Existence and uniqueness
- Numerical integration methods: explicit RK methods
- Taxonomy of numerical integration algoithms
- Stopping conditions
- Shadowing and approximation for systems
- Rigorous computation

- Invariant sets
- Examples and classification of invariant sets
- Periodic orbit algorithms
- Algorithms for quasiperiodic invariant sets
- Horseshoes and chaos
- Statistical properties of chaotic sets

- Local bifurcations
- Singularity theory and transversality
- Continuation
- Classification of bifurcations
- Defining equations and detection

- Numerical issues and software packages
- Global bifurcations (??)

*
John Guckenheimer *

Jan 16 1998