Bifurcation Theory

Summary

In this chapter we investigate “qualitative changes” in parametrized families of RDS and call these studies “stochastic bifurcation theory”.

The first problem is to develop a mathematical formalization (called D-bifurcation) of “qualitative change” which is connected to the stability of an RDS under an invariant measure (i. e. to the Lyapunov exponents) and reduces to the deterministic definition of bifurcation in the absence of noise (Subsect. 9.2.1). We propose as a first task to study the branching of new invariant measures from a family of reference measures at a parameter value at which the reference measure has a vanishing Lyapunov exponent. We also discuss an older concept on the level of densities in state space (called P-bifurcation) and discuss its relation with D-bifurcation (Subsects. 9.2.2 and 9.5.1).

As the theory of stochastic bifurcation is still in its infancy, we proceed mainly by way of instructive examples.

Sect. 9.3 treats explicitly solvable one-dimensional examples of stochastic transcritical and pitchfork bifurcation. In Subsect. 9.3.4 we prove a general criterion for a pitchfork bifurcation in dimension one using the theory of random attractors, to which we give a brief introduction.

Sect. 9.4 is devoted to the study of the prototypical noisy Duffing-van der Pol oscillator. We report on the state of the art as far as rigorous results are concerned and also present numerical findings supporting conjectures about the “correct” scenario of stochastic Hopf bifurcation.

Sect. 9.5 is devoted to the only available general condition (due to Baxendale) for a D-bifurcation out of the fixed point x = 0 (and an associated P-bifurcation of the new branch of measures) in a family of SDE in ℝ^d.