Bifurcation

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 \(\lambda \) .