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arXiv · 2411.18691

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

Tulio Meneghelli de Oliveira·Vinicius Wiggers·Eduardo Scafi·Silvio Zanin·Cesar Manchein·Marcus Werner Beims
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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.

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Tulio Meneghelli de Oliveira, Vinicius Wiggers, Eduardo Scafi, Silvio Zanin, Cesar Manchein, Marcus Werner Beims. 2024-11-27. Lyapunov stability under $q$-dilatation and $q$-contraction of coordinates. https://arxiv.org/abs/2411.18691

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