Saddle-node bifurcations for hyperbolic sets

Abstract

Hyperbolic sets are robust under perturbations: they persist on an open set of the parameter space. In this paper we investigate the boundary of this open set. Generalizing the theory of fixed points we define saddle-node bifurcations for hyperbolic sets K with one-dimensional unstable directions. In this bifurcation the geometrical splitting of the tangent space is preserved but the expansion in the unstable direction degenerates near a periodic orbit. The compact set K can be followed on a closed half-space bounded by a codimension-one manifold \mathcal{O}^0. On \mathcal{O}^0 the saddle-node bifurcation occurs. On one side of \mathcal{O}^0, K is hyperbolic and on the other side, it has disappeared.