Bifurcation

Abstract

Consider the family of differential equations $\dot{u} = f(u, \lambda ), \;\; u\in {{\mathbb {R}}}^n,\;\; \lambda \in {{\mathbb {R}}}.$ In case $f(u_0, \lambda _0) = 0$, the differential equation with parameter value $\lambda =\lambda _0$ has a rest point at $u_0$ and the linearized system at this point is given by $\dot{W} = f_u(u_0, \lambda _0) W.$ If the eigenvalues of the linear transformation $f_u(u_0, \lambda _0):{\mathbb {R}}^n\mapsto {\mathbb {R}}^n$ are all nonzero, then the transformation is invertible, and by an application of the implicit function theorem there is a curve $\lambda \mapsto \beta (\lambda )$ in ${\mathbb {R}}^n$ such that $\beta (\lambda _0) = u_0$ and $f(\beta (\lambda ), \lambda ) \equiv 0$. In other words, for each $\lambda$ in the domain of $\beta$, the point $\beta (\lambda )\in {\mathbb {R}}^n$ corresponds to a rest point for the member of the family (11.1) at the parameter value $\lambda$.