Multiparameter bifurcation and stability of solutions at a double eigenvalue

Abstract

This paper deals with some problems of bifurcation theory for general non-linear eigenvalue prob-lem for 2-dimensional parameter space. An explicit analysis of the bifurcation for 2-dimensional parameter space is done and the structure of the non-trivial solution branches of the bifurcation equation near origin is given. Since the study of the bifurcation problem is closely related to change in the qualitative behaviour of the systems, and to exchange of stability, analysis of the stability of the bifurcating solutions is done here. It is proved that the stability of the bifurcating solutions is de-termined, to the lowest non-vanishing order, by the eigenvalues of the Fréchet derivative of the re-duced bifurcation equation.