Nonlinear stability (q-stability) under dilatation or contraction of coordinates

Tulio M. Oliveira, Vinicius Wiggers, Eduardo Scafi, Silvio Zanin, Cesar Manchein and Marcus W. Beims

Chaos, Solitons & Fractals, 2025, vol. 194, issue C

Abstract: This study examines the nonlinear stability of trajectories under coordinate contraction and dilatation in three dynamical systems: the discrete-time dissipative Hénon map, and the conservative, non-integrable, continuous-time Hénon–Heiles and diamagnetic Kepler problems. The nonlinear stability analysis uses the q-deformed Jacobian and q-derivative, with trajectory stability assessed for q>1 (dilatation) and q<1 (contraction). It is shown that q-deformed Jacobian adds nonlinear terms to the linear Lyapunov stability analysis, and is named here as q-stability. Analytical curves in the parameter space mark boundaries of distinct low-periodic motions in the Hénon map. Numerical simulations compute the maximal Lyapunov exponent across the parameter space, in Poincaré surfaces of section, and as a function of total energy in the conservative systems. Simulations show that contraction (dilatation) of coordinates generally decreases (increases) q-stability exponent when compared to the q=1 case with positive Lyapunov exponents. Dilatation and contraction tend to increase the q-stability exponent for Lyapunov stable orbits. Some exceptions to this trend remain unexplained regarding Kolmogorov–Arnold–Moser (KAM) tori stability.

Keywords: q-exponent; Lyapunov exponent; Hamiltonian systems; Dissipative systems; Dilatation/contraction stability (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:chsofr:v:194:y:2025:i:c:s0960077925002280

DOI: 10.1016/j.chaos.2025.116215

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