Evidence for a new period-doubling sequence in four-dimensional symplectic maps

Period-doubling bifurcations and islets of stability in two-degree-of-freedom Hamiltonian systems

In this paper, we show that the destruction of the main Kolmogorov-Arnold-Moser (KAM) islands in two-degree-of-freedom Hamiltonian systems occurs through a cascade of period-doubling bifurcations. We calculate the corresponding Feigenbaum constant and the accumulation point of the period-doubling sequence. By means of a systematic grid search on exit basin diagrams, we find the existence of numerous very small KAM islands ("islets") for values below and above the aforementioned accumulation point. We study the bifurcations involving the formation of islets and we classify them in three different types. Finally, we show that the same types of islets appear in generic two-degree-of-freedom Hamiltonian systems and in area-preserving maps.

Global structure of regular tori in a generic 4D symplectic map

For the case of generic 4d symplectic maps with a mixed phase space, we investigate the global organization of regular tori. For this, we compute elliptic 1-tori of two coupled standard maps and display them in a 3d phase-space slice. This visualizes how all regular 2-tori are organized around a skeleton of elliptic 1-tori in the 4d phase space. The 1-tori occur in two types of one-parameter families: (α) Lyapunov families emanating from elliptic-elliptic periodic orbits, which are observed to exist even far away from them and beyond major resonance gaps, and (β) families originating from rank-1 resonances. At resonance gaps of both types of families either (i) periodic orbits exist, similar to the Poincaré-Birkhoff theorem for 2d maps, or (ii) the family may form large bends. In combination, these results allow for describing the hierarchical structure of regular tori in the 4d phase space analogously to the islands-around-islands hierarchy in 2d maps.