ABSTRACT

This chapter is concerned with discussions about reality of chaos in Hénon mappings. In general, this type of chaos is characterized by the occurrence of dangerous homoclinic tangencies. In this case the corresponding chaotic attractor can be seen as the unifi ed limit set of the whole attracting set of trajectories. This set includes a subset of both chaotic and stable periodic trajectories which have long periods and weak and narrow basins of attraction and stability regions. This is a result of the fact that this attractor is holed by a set of basins of attraction of different periodic orbits. Now, the most interesting results concerned with different shape of the Hénon mappings are presented with some details in this chapter. Indeed, in Section 2.1 several methods on measuring chaos in the Hénon mappings are presented and discussed. In Section 2.2 the most interesting bifurcations phenomena in the Hénon mappings are given. In particular, those related with the existence of tangencies and unusual bifurcation phenomena are discussed in Section 2.2. Section 2.3 deals with the type of chaos in the Hénon mappings, i.e., these maps are quasi-attractors. Section 2.4 and Section 2.5 are concerned with the problem of period-doubling cascades for large perturbations of the Hénon families. Especially, a general theory is presented with many explanations, results and examples.