Isochronous and period-doubling diagrams for symplectic maps of the plane

TL;DR

This paper enhances the visualization of stability and bifurcation structures in symplectic maps of the plane by extending Hénon's framework with period-doubling diagrams and employing chaos indicators to analyze dynamical regimes.

Contribution

It introduces a combined isochronous and period-doubling diagram approach for symplectic maps, extending Hénon's original framework, and applies modern chaos indicators for detailed dynamical analysis.

Findings

Effective differentiation of regular and chaotic regimes.

Identification of twistless orbits and bifurcations.

Enhanced understanding of bounded motion regions.

Abstract

Symplectic mappings of the plane serve as key models for exploring the fundamental nature of complex behavior in nonlinear systems. Central to this exploration is the effective visualization of stability regimes, which enables the interpretation of how systems evolve under varying conditions. While the area-preserving quadratic H\'enon map has received significant theoretical attention, a comprehensive description of its mixed parameter-space dynamics remain lacking. This limitation arises from early attempts to reduce the full two-dimensional phase space to a one-dimensional projection, a simplification that resulted in the loss of important dynamical features. Consequently, there is a clear need for a more thorough understanding of the underlying qualitative aspects. This paper aims to address this gap by revisiting the foundational concepts of reversibility and associated symmetries, first explored in the early works of G.D. Birkhoff. We extend the original framework proposed by H\'enon by adding a period-doubling diagram to his isochronous diagram, which allows to represents the system's bifurcations and the groups of symmetric periodic orbits that emerge in typical bifurcations of the fixed point. A qualitative and quantitative explanation of the main features of the region of parameters with bounded motion is provided, along with the application of this technique to other symplectic mappings, including cases of multiple reversibility. Modern chaos indicators, such as the Reversibility Error Method and the Generalized Alignment Index, are employed to distinguish between various dynamical regimes in the mixed space of variables and parameters. These tools prove effective in differentiating regular and chaotic dynamics, as well as in identifying twistless orbits and their associated bifurcations.