Numerical Analysis of Bifurcations

This paper is a brief survey of numerical methods for computing bi-furcations of generic families of dynamical systems. Emphasis is placed upon algorithms that reeect the structure of the underlying mathematical theory while retaining numerical eeciency. Signiicant improvements in the computational analysis of dynamical systems are to be expected from more reliance of geometric insight coming from dynamical systems theory. Finite dimensional dynamical systems arise as models in many settings, including electrical networks, chemical reactors, neuronal networks and multi-body mechanical systems. In all of these examples models are often assembled from component subsystems that are well characterized. Nonetheless, determining the behavior of such a system from is still a substantial challenge. Depending upon context this challenge manifests itself as one of two diierent problems. If we are building artiicial systems, we have the problem of synthesis or design. A set of speciications are presented to us along with a kit of components from which systems can be assembled. Our goal is to build a system that performs according to the speciications. If we are studying natural systems, then we have the analog of the engineer's problem of system identiication. We want to determine which components the whole system was assembled from by observing it. We assert that the insights coming from advances in dynamical systems theory during the past decades provide an intellectual substrate for enhancing our ability to solve these two problems. The realization of this opportunity requires that we develop better computational