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A finite-scale geometric growth invariant for chaotic and weakly chaotic dynamics

Author

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  • Vijayan, Vinesh
  • J., Sandhiya Jenifer
  • K., Karpagavalli
  • M., Mohanasundari

Abstract

We introduce a finite-scale geometric growth invariant (FSGI) that quantifies the growth rate of localized sets under time evolution in dissipative dynamical system. Defined at finite time and resolution — without relying on symbolic dynamics or Markov partitions — this invariant converges in uniformly hyperbolic systems to a resolution-dependent plateau, whose logarithmic scaling coefficient equals the Kolmogorov–Sinai entropy. In non-hyperbolic systems, it decays to zero, reflecting absent entropy production, while remaining well-defined at finite scales. Numerical results for the Hénon map and Feigenbaum point confirm these behaviors. Our approach provides a finite-scale geometric characterization of chaos, consistent with classical entropy theory. It also applies to open intermittent systems, where trajectories escape and asymptotic invariants fail, revealing finite-scale signatures of transient weak chaos. We further define a resolution-dependent crossover time that measures the geometric coherence horizon of finite neighborhoods, distinguishing chaotic from weakly chaotic regimes.

Suggested Citation

  • Vijayan, Vinesh & J., Sandhiya Jenifer & K., Karpagavalli & M., Mohanasundari, 2026. "A finite-scale geometric growth invariant for chaotic and weakly chaotic dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 209(P2).
    • Handle: RePEc:eee:chsofr:v:209:y:2026:i:p2:s0960077926006879
    DOI: 10.1016/j.chaos.2026.118546
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