Bifurcation at multiple eigenvalues and stability of bifurcating solutions

Abstract

AbstractIf K is a bounded linear operator from the real Banach space U into the real Banach space V and ƒ:U×R→V has the value zero at (0, 0), the existence and linear stability of the equilibrium solutions of the dynamical system K dudt = ƒ(u, α) which are close to the origin in U×R are studied. It is assumed that ƒu(0, 0): U → V is a Freholm operator of index zero. The only restriction on the dimension of the null space of ƒu(0, 0) and the order of vanishing, at (0, 0), of ƒ restricted to the null space of Dƒ(0,0):U×R→V, is that they both be finite positive integers. The main result gives conditions under which the equation, which determines the equilibrium solutions in a neighborhood of the origin, also determines the stability of these equilibrium solutions