Detecting and Computing Bifurcation Points

Abstract

Exploring nonlinear phenomena has become a major challenge in physics, chemistry, biology, engineering, medicine and social science. We consider nonlinear problems of the form

$$\frac{{\partial u}}{{\partial t}} = G(u,\lambda ) $$

((3.1))

where G : X x R ^p → Y is a “smooth” mapping and λ ∈ R ^p represents various control parameters, e.g. Reynolds number, catalyst, temperature, density, initial or final products, etc. Bifurcation theory studies how solutions of (3.1) and their stability change as the parameter λ varies. A point (u ₀, λ ₀) in X x R ^p is called a bifurcation point if it satisfies (3.1) and in all neighborhoods of (u ₀, λ ₀) the problem (3.1) has at least two different (stationary or time-dependent) solution branches. A bifurcation problem is generic if for all G in a small neighborhood of G, the problem

$$\frac{{\partial u}}{{\partial t}} = \tilde G(u,\lambda ) $$

has the same number of solution branches with similar stability behavior as (3.1) in the neighborhood of (u ₀, λ ₀).