  A pitchfork bifurcation of an $(m-1)$-dimensional invariant submanifold of a
dynamical system in $\mathbb{R}^m$ is defined analogous to that in
$\mathbb{R}$. 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 $M$. For continuous
dynamical systems, the theorem is proved by transforming it to the discrete
case. Techniques used for proving the theorem involve differential topology and
analysis. The theorem is illustrated by means of a canonical example.
