Pitchfork bifurcations of invariant manifolds

A pitchfork bifurcation of an ( m − 1 ) -dimensional invariant submanifold of a dynamical system in R m is defined analogous to that in R . Sufficient conditions for such a bifurcation to occur are stated and existence of the bifurcated manifolds is proved under the stated hypotheses. For discrete dynamical systems, the existence of locally attracting manifolds M + and M − , after the bifurcation has taken place is proved by constructing a diffeomorphism of the unstable manifold M. Techniques used for proving the theorem involve differential topology and analysis. The theorem is illustrated by means of a canonical example.