Numerical detection of bifurcation point in the curve

Abstract

We are presenting a numerical method which detects the presence and position of a bifurcation simplex , the regular $(k+1)$-dimensional simplex, which may be considered as fat bifurcation in the curve of zeroes of the $C^1$ map $f:{\mathbb R}^{k+1}\to{\mathbb R}^k$. On the other hand the bifurcation simplex appears in the neighbourhood of the bifurcation point, meaning that we have the method to locate the bifurcation point as well. The method does not require any estimation of the derivative of the function $f$ and refers to the values of the map $f$ only in the vertices of certain triangulation. The bifurcation simplex is detected by change of the Brouwer degree value of the restriction of the map $f$ to the appropriate $k$-simplex. This publication is co-financed by the European Union as part of the European Social Fund within the project Center for Applications of Mathematics.