Chaotic orbits obeying one isolating integral in a four dimensional map

TL;DR

This study confirms that chaotic orbits obeying one isolating integral exist in a four-dimensional map, showing similar phase space structures as in lower-dimensional systems, with implications for long-term dynamical behavior.

Contribution

The paper extends previous findings to a four-dimensional map, demonstrating the persistence of phase space features and orbit behaviors over extended time intervals.

Findings

Chaotic orbits obeying one integral are bounded by regular orbits.

Phase space contains subspaces where chaotic orbits do not enter, occupied by regular orbits.

Features observed in lower-dimensional systems are also present in the four-dimensional map.

Abstract

We have recently presented strong evidence that chaotic orbits that obey one isolating integral besides energy exist in a toy Hamiltonian model with three degrees of freedom and are bounded by regular orbits that isolate them from the Arnold web. The interval covered by those numerical experiments was equivalent to about one million Hubble times in a galactic context. Here we use a four dimensional map to confirm our previous results and to extend that interval fifty times. We show that, at least within that interval, features found in lower dimension Hamiltonian systems and maps are also present in our study, e.g., within the phase space occupied by a chaotic orbit that obeys one integral there are subspaces where that orbit does not enter and are, instead, occupied by regular orbits that, if tori, bound other chaotic orbits obeying one integral and, if cantori, produce stickiness. We argue that the validity of our results might exceed the time intervals covered by the numerical experiments.