## Numerical Continuation

In this chapter we give an overview of numerical methods for analyzing the solution behavior of the dynamical system

(1)      x'(t) = f (x,\alpha),      x, f(*,*) \in Rn.

where, depending on the context, \alpha denotes one or more parameters. Throughout we assume that f is as smooth as necessary.

The emphasis in this chapter is on numerical continuation methods, as opposed to simulation methods. 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 state or a periodic solution, then continuation methods become very effective in determining the dependence of this solution on the parameter \alpha. Moreover, continuation techniques can also be used when the solutions are asymptotically unstable. Knowledge of unstable solutions is often critical in the understanding of the global dynamics of a system.

We first review basic continuation techniques for following stationary and periodic solutions to (1). We also describe algorithms for detecting codimension 1 bifurcations, namely folds and Hopf bifurcations, and methods for locating branch points. Branch switching techniques are also described.

Once a codimension 1 bifurcation has been located, it can be followed in two parameters, that is, with \alpha \in R2 in Equation (1). We give details on this for the case of folds, Hopf bifurcations, period-doubling bifurcations and torus bifurcations. We also describe techniques for the detection of codimension 2 bifurcations, including the cusp, and the Bogdanov-Takens, degenerate Hopf, zero-Hopf, and double-Hopf bifurcations. Efficient numerical linear algebra is an important issue in the design of algorithms for following such singularities.

Particular attention is paid to connecting orbits, especially homoclinic orbits. Basic numerical techniques for the computation and continuation of such orbits are outlined. We also describe algorithms for the detection of higher codimension homoclinic bifurcations.

We conclude with a number of applications that illustrate the utility of the numerical techniques in some representative applications.

Wolf-Jürgen Beyn, Alan Champneys, Eusebius Doedel, Willy Govaerts, Yuri A. Kuznetsov, and Björn Sandstede
July 17 1998