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

Author

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

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.

Suggested Citation

  • Oliveira, Tulio M. & Wiggers, Vinicius & Scafi, Eduardo & Zanin, Silvio & Manchein, Cesar & Beims, Marcus W., 2025. "Nonlinear stability (q-stability) under dilatation or contraction of coordinates," Chaos, Solitons & Fractals, Elsevier, vol. 194(C).
    • Handle: RePEc:eee:chsofr:v:194:y:2025:i:c:s0960077925002280
    DOI: 10.1016/j.chaos.2025.116215
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