Numerical computation of origins for Hopf bifurcation in a two-parameter problem

Abstract

We consider nonlinear evolution equations of the form

$$ \frac{{du}}{{dt}} = F(u,\lambda, \gamma)\;\quad F:\text{R}^n \times \text{R} \to \text{R}^n $$

(1)

where λ and γ denote real parameters. In order to obtain a qualitative description of the solution set of (1) as function of the parameters, one can determine the “bifurcation set” by computing paths of turning points, Hopf bifurcation points and (in some cases) regular bifurcation points. This can be done by a continuation procedure applied to a suitable determining system, starting from a known branching point for the one-parameter problem $du/dt = F(u,\lambda,\gamma ^f ) (\gamma ^f \text{fixed)}$ (see e.g. [9,17]).