Continuation methods: introduction

The goal of these methods[Kuz],[Govaerts],[Rabinowitz],[Mei],[Keller] is to find solutions $x\in\mathbb R^n$ to nonlinear equations

$$$\mathbb R^n\ni F(x,p) = 0 \quad\tag{E}$$$

as function of a real parameter $p$. Given a known solution $(x_0,p_0)$, we can, under reasonable assumptions, continue it by computing a 1d curve of solutions $\gamma = (x(s),p(s))_{s\in I}$ passing through $(x_0,p_0)$.

For the sequel, it is convenient to use the following formalism [Kuz]

1. prediction of the next point
2. correction
3. step size control.

Natural continuation

We just use this simple continuation method to give a trivial example of the formalism. Knowing $(x_0, p_0)$, we form the predictor $(x_0, p_0+ds)$ for some $ds$ and use it as a guess for a Newton corrector applied to $x\to F(x, p_0+ds)$. The corrector is thus the newton algorithm.

This continuation method is continuation(prob, Natural(), options).

Usage

You should almost never use this predictor for computations. It fails at turning points, is not adaptive, ...

Step size control

Each time the corrector phase failed, the step size $ds$ is halved. This has the disadvantage of having lost Newton iterations (which costs time) and imposing small steps (which can be slow as well). To prevent this, the step size can be controlled internally with the idea of having a constant number of Newton iterations per point. This is in part controlled by the aggressiveness factor a in ContinuationPar.

References

• Kuz

Kuznetsov, Elements of Applied Bifurcation Theory.

• Govaerts

Govaerts, Numerical Methods for Bifurcations of Dynamical Equilibria; Allgower and Georg, Numerical Continuation Methods

• Rabinowitz

Rabinowitz, Applications of Bifurcation Theory; Dankowicz and Schilder, Recipes for Continuation

• Mei

Mei, Numerical Bifurcation Analysis for Reaction-Diffusion Equations

• Keller

Keller, Lectures on Numerical Methods in Bifurcation Problems