Different types of critical behavior in conservatively coupled Hénon maps

Abstract

We study the dynamics of two conservatively coupled Hénon maps at different levels of dissipation. It is shown that the decrease of dissipation leads to changes in the structure of the parameter plane and the scenarios of transition to chaos compared to the case of infinitely strong dissipation. Particularly, the Feigenbaum line becomes divided into several fragments. Some of these fragments have critical points of different types, namely, of C and H type, as their terminal points. Also the mechanisms of formation of these Feigenbaum line ruptures are described.

Attractors of a Weakly Dissipative System Allowing Transition to the Stochastic Web in the Conservative Limit

“…However, a certain choice of the path on the parameter plane allows one to observe on the border of chaos the scaling laws typical of the period-doubling cascade in conservative systems, with different scaling constants [29]. This type of critical behavior on the border of chaos is often called a Hamiltonian critical point, or a critical point of H-type, and can be found in a two-dimensional map as a phenomenon of codimension 2 [23,24,31], as it happens in the system under consideration despite the presence of the nonlinear dissipation [18,32]. Let us remember here that the quadratic function can be considered as the simplest power series expansion of the harmonic function cos x.…”