Lyapunov stability under $q$-dilatation and $q$-contraction of coordinates

TL;DR

This paper investigates how $q$-dilatation and $q$-contraction transformations affect Lyapunov stability in various dynamical systems, revealing that these transformations generally modify the Lyapunov exponents and influence orbit stability.

Contribution

It introduces a $q$-deformed Jacobian and $q$-derivative approach to analyze Lyapunov stability in both dissipative and conservative systems, providing new insights into stability boundaries and Lyapunov exponent behavior.

Findings

$q$-contraction decreases Lyapunov exponents

$q$-dilatation increases Lyapunov exponents

Transformations affect stability of regular and chaotic orbits

Abstract

This study examines the Lyapunov stability under coordinate $q$-contraction and $q$-dilatation in three dynamical systems: the discrete-time dissipative H\'enon map, and the conservative, non-integrable, continuous-time H\'enon-Heiles and diamagnetic Kepler problems. The stability analysis uses the $q$-deformed Jacobian and $q$-derivative, with trajectory stability assessed for $q > 1$ (dilatation) and $q < 1$ (contraction). Analytical curves in the parameter space mark boundaries of distinct low-periodic motions in the H\'enon map. Numerical simulations compute the maximal Lyapunov exponent across the parameter space, in Poincar\'e surfaces of section, and as a function of total energy in the conservative systems. Simulations show that $q$-contraction ($q$-dilatation) generally decreases (increases) positive Lyapunov exponents relative to the $q = 1$ case, while both transformations tend to increase Lyapunov exponents for regular orbits. Some exceptions to this trend remain unexplained regarding Kolmogorov-Arnold-Moser (KAM) tori stability.