Bifurcations of families of 1D-tori in 4D symplectic maps

TL;DR

This paper investigates the bifurcations of 1D-tori in 4D symplectic maps, revealing how these structures change when crossing resonances, with implications for understanding the organization of phase space.

Contribution

It provides a detailed analysis of bifurcations of 1D-tori in 4D symplectic maps using visualization and analytical predictions, highlighting the role of frequency as a bifurcation parameter.

Findings

Bifurcations occur when crossing resonances without external parameter variation.

Emerging families from bifurcations form the skeleton of resonance channels.

Results are consistent with analytical predictions for quasi-periodically forced oscillators.

Abstract

The regular structures of a generic 4D symplectic map with a mixed phase space are organized by one-parameter families of elliptic 1D-tori. Such families show prominent bends, gaps, and new branches. We explain these features in terms of bifurcations of the families when crossing a resonance. For these bifurcations no external parameter has to be varied. Instead, the longitudinal frequency, which varies along the family, plays the role of the bifurcation parameter. As an example we study two coupled standard maps by visualizing the elliptic and hyperbolic 1D-tori in a 3D phase-space slice, local 2D projections, and frequency space. The observed bifurcations are consistent with analytical predictions previously obtained for quasi-periodically forced oscillators. Moreover, the new families emerging from such a bifurcation form the skeleton of the corresponding resonance channel.