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

## 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 R**^{n}.
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 R**^{2} 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