Three-Dimensional HÉnon-like Maps and Wild Lorenz-like attractors

We discuss a rather new phenomenon in chaotic dynamics connected with the fact that some three-dimensional diffeomorphisms can possess wild Lorenz-type strange attractors. These attractors persist for open domains in the parameter space. In particular, we report on the existence of such domains for a three-dimensional Hénon map (a simple quadratic map with a constant Jacobian which occurs in a natural way in unfoldings of several types of homoclinic bifurcations). Among other observations, we have evidence that there are different types of Lorenz-like attractor domains in the parameter space of the 3D Hénon map. In all cases the maximal Lyapunov exponent, Λ1, is positive. Concerning the next Lyapunov exponent, Λ2, there are open domains where it is definitely positive, others where it is definitely negative and, finally, domains where it cannot be distinguished numerically from zero (i.e. |Λ2| < ρ, where ρ is some tolerance ranging between 10−5 and 10−6). Furthermore, several other types of interesting attractors have been found in this family of 3D Hénon maps.