Measuring the transition between nonhyperbolic and hyperbolic regimes in open Hamiltonian systems

TL;DR

This paper investigates how KAM islands influence unpredictability in open Hamiltonian systems, revealing that fluctuations in basin entropy signal transitions between nonhyperbolic and hyperbolic regimes, with implications for understanding chaotic scattering.

Contribution

It demonstrates that basin entropy fluctuations can identify the transition between nonhyperbolic and hyperbolic regimes in open Hamiltonian systems, highlighting the role of KAM islands.

Findings

Fluctuations in basin entropy occur in nonhyperbolic regimes due to KAM islands.

Increased energy causes particles to escape faster, reducing fractal basin boundaries.

Basin entropy fluctuations serve as indicators of the system's hyperbolic or nonhyperbolic nature.

Abstract

We show that the presence of KAM islands in nonhyperbolic chaotic scattering has deep implications on the unpredictability of open Hamiltonian systems. When the energy of the system increases the particles escape faster. For this reason the boundary of the exit basins becomes thinner and less fractal. Hence, we could expect a monotonous decrease in the unpredictability as well as in the fractal dimension. However, within the nonhyperbolic regime, fluctuations in the basin entropy have been uncovered. The reason is that when increasing the energy, both the size and geometry of the KAM islands undergo abrupt changes. These fluctuations do not appear within the hyperbolic regime. Hence, the fluctuations in the basin entropy allow us to ascertain the hyperbolic or nonhyperbolic nature of a system. In this manuscript we have used continuous and discrete open Hamiltonian systems in order to show the relevant role of the KAM islands on the unpredictability, and the utility of the basin entropy to analyze this kind of systems.