On the principle of reduced stability

The “Principle of Reduced Stability” says that the stability of bifurcating stationary or periodic solutions is given by the finite dimensional bifurcation equation obtained by the method of Lyapunov-Schmidt. To be more precise, the linearized stability is governed by the linearization of the bifurcation equation about the bifurcating branch of solutions and in particular by the signs of the real parts of the perturbation of the eigenvalues along this branch. This principle is true for simple eigenvalue bifurcation whereas it may be false for higher dimensional bifurcation equations. A condition for the validity of that principle is given. A counterexample shows that it cannot be dropped in general.