Bifurcations via singular value decompositions
The paper deals with an equation (1) \(F(a,\lambda)=0\) where \(\lambda\in {\mathbb{R}}\), \(a\in {\mathbb{R}}^ m\), F: \({\mathbb{R}}^ m\times {\mathbb{R}}\to {\mathbb{R}}^ m\) and \(\lambda\) has a meaning of a bifurcation parameter. The authors, considering the equation (1) as an equation (2) \(f(x)=0\) where \(f:{\mathbb{R}}^{m+1}\to {\mathbb{R}}\), \(x=(a,\lambda)\), present a method of applying a singular value decomposition to solve the equation (2) at any case: if x is a regular point, a turning point or a bifurcation point. The local analysis of solutions is provided with the help of Lyapunov- Schmidt reduction via the singular value decomposition to decompose the equation (2) into two equations, an auxiliary equation and a bifurcation equation. A method for determining all the tangent vectors at a bifurcation point is proposed. The global numerical analysis of solutions of (2) gives a numerical method of going from one solution of (2) to another solution. The main results are illustrated by an example.
scientific article; zbMATH DE number 4104271
scientific article