Chapter 4 Numerical continuation, and computation of normal forms

Abstract

This chapter describes numerical continuation methods for analyzing the solution behavior of the dynamical system. Time-integration of a dynamical system gives much insight into its solution behavior. However, once a solution type has been computed—for example, a stationary solution (equilibrium) or a periodic solution (cycle)—then continuation methods become very effective in determining the dependence of this solution on the parameter α. Once a co-dimension-1 bifurcation has been located, it can be followed in two parameters—that is, with α e ℝ 2 . However, in many cases, detection of higher co-dimension bifurcations requires computation of certain normal forms for equations restricted to center manifolds at the critical parameter values. Pseudo-arclength continuation method allows the continuation of any regular solution, including folds. Geometrically, it is the most natural continuation method. The periodic solution continuation method is very suitable for numerical computations, and it is not difficult to establish the Poincare continuation with the help of it.