Bifurcation Theory

Abstract

Many problems in mathematics, and its applications to theoretical physics, chemistry, and biology, lead to a problem of the form

$$ f(\lambda ,\,x)\, = \,0, $$

(13.1)

where f is an operator on R x B₁ into B₂, with B₁ B₂ Banach spaces. For example, (13.1) could represent a system of differential or integral equations, depending on a parameter λ. We are interested in the structure of the solution set; namely, the set

$$ {f^{ - 1}}(0)\, = \,\{ (\lambda ,\,x)\, \in \,R\, \times \,{B_1}\,:\,f(\lambda ,\,x)\, = \,0\} .$$

(13.2)

In particular, we seek conditions on f in order that we can determine when a solution (λ, x̄) of (13.1) lies on a “curve” of solutions (λ, x(λ)), at least locally; i.e., for |λ — λ| < ε. We may also inquire as to when (λ̄, x̄) lies on several solution curves, (λ, x₁(λ)), (λ, x₂(λ)),….