2009/10/31 by Christian S. Rodrigues, Alessandro P. S. de Moura, Celso Grebogi
Mathematics · Physics and Astronomy · #Amplitude #Classical mechanics #Exponential decay #Exponential function #Fractal #Geometry #Hamiltonian (control theory) #Hamiltonian system #Mathematical analysis #Mathematics #Physics #Power law #Quadratic equation #Quantum chaos and dynamical systems #Quantum mechanics #Random walk #Scattering #Scientific Research and Discoveries #Statistical physics #Theoretical and Computational Physics #nlin.CD
paper · pdf · doi:10.1103/physreve.82.026211
published as Physical Review E 82, 026211 (2010) · 6 pages, 6 figures - Up to date with corrections suggested by referees
openalex publication_date 2010/08/27 · arxiv created 2010/08/30 · arxiv updated 2010/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A great number of physical processes are described within the context of Hamiltonian scattering. Previous studies have rather been focused on trajectories starting outside invariant structures, since the ones starting inside are expected to stay trapped there forever. This is true though only for the deterministic case. We show however that, under finitely small random fluctuations of the field, trajectories starting inside Kolmogorov-Arnold-Moser (KAM) islands escape within finite time. The nonhyperbolic dynamics gains then hyperbolic characteristics due to the effect of the random perturbed field. As a consequence, trajectories which are started inside KAM curves escape with hyperboliclike time decay distribution, and the fractal dimension of a set of particles that remain in the scattering region approaches that for hyperbolic systems. We show a universal quadratic power law relating the exponential decay to the amplitude of noise. We present a random walk model to relate this distribution to the amplitude of noise, and investigate these phenomena with a numerical study applying random maps.