HOW OFTEN ARE CHAOTIC SADDLES NONHYPERBOLIC

In this paper, we numerically investigate the fraction of nonhyperbolic parameter values in chaotic dynamical systems. By a nonhyperbolic parameter value we mean a parameter value at which there are tangencies between some stable and unstable manifolds. The nonhyperbolic parameter values are important because the dynamics in such cases is especially pathological. For example, near each such parameter value, there is another parameter value at which there are infinitely many coexisting attractors. In particular, Newhouse and Robinson proved that the existence of one nonhyperbolic parameter value typically implies the existence of an interval ('a Newhouse interval') of nonhyperbolic parameter values. We numerically compute the fraction of nonhyperbolic parameter values for the Henon map in the parameter range where there exist only chaotic saddles (i.e., nonattracting invariant chaotic sets). We discuss a theoretical model which predicts the fraction of nonhyperbolic parameter values for small Jacobians. Two-dimensional diffeomorphisms with similar chaotic saddles may arise in the study of Poincare return map for physical systems. Our results suggest that (1) nonhyperbolic chaotic saddles are common in chaotic dynamical systems; and (2) Newhouse intervals can be quite large in the parameter space.