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.

Automated summary

In this paper, we investigate the destruction of the main Kolmogorov-Arnold-Moser (KAM) islands in two-degree-of-freedom Hamiltonian systems. We find that this destruction occurs through a cascade of period-doubling bifurcations. By conducting a systematic grid search, we identify numerous very small KAM islands (islets) below and above a certain accumulation point. We also study the bifurcations involved in the formation of these islets and classify them into three different types. Furthermore, we demonstrate that similar types of islets appear in generic two-degree-of-freedom Hamiltonian systems and area-preserving maps.