The k -Zero Bifurcation Theorem for m -Parameterized Systems

Given an m-parameterized family of n-dimensional vector fields, with an equilibrium point with linearization of eigenvalue zero with algebraic multiplicity k, with k ≤ m, and geometric multiplicity one, our goal in this paper is to find sufficient conditions for the family of vector fields such that the dynamics on the k-dimensional m-parameterized center manifold around the equilibrium point becomes locally topologically equivalent to a given unfolding. Finally, the result is applied to the study of the Rössler system.