Generalized separatrix mapping in a system with more than two degrees of freedom.

Abstract

In this work, we investigate a generalization of the separatrix map to higher dimensions, in particular, a three-dimensional (3D) symplectic map derived from the Arnold Hamiltonian, which involves a pendulum and a free rotator coupled through a time-dependent periodic perturbation. This map was originally introduced by Chirikov to study not only the chaotic layer around the separatrix of the main resonance of the system but also the diffusion along this layer, i.e., Arnold diffusion. In two limiting cases in frequency space, it was assumed that this map reduces to the 2D separatrix map, and thus all its known results are applicable, such as the layer half-width and the maximum Lyapunov exponent. Here, we focus on the dynamics of this 3D map, particularly in the frequency domain dominated by low-order resonances between the pendulum frequency near the separatrix and that of the free rotator. By means of numerical and analytical estimates, we show that in this region of the frequency space, the results differ from the expected ones. The maximum Lyapunov exponent and the metric entropy are computed and compared with analytical estimates as well as with results for the standard separatrix map.