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Records and Occupation Time Statistics for Area-Preserving Maps.

Roberto Artuso1,2, Tulio M de Oliveira3, Cesar Manchein3

  • 1Dipartimento di Scienza e Alta Tecnologia and Center for Nonlinear and Complex Systems, Via Valleggio 11, 22100 Como, Italy.

Entropy (Basel, Switzerland)
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Summary
This summary is machine-generated.

Deterministic systems can mimic random processes. This study shows the standard map exhibits diffusive transport and random walk behavior, while the triangle map displays anomalous transport and statistics, compatible with generalized laws.

Keywords:
area-preserving mapsinfinite ergodicityrecord statistics

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Area of Science:

  • Physics
  • Dynamical Systems
  • Statistical Mechanics

Background:

  • Characterizing deterministic systems exhibiting stochastic-like features is a key challenge in dynamics.
  • Transport properties in deterministic systems on non-compact phase spaces are widely studied, including normal and anomalous transport.

Purpose of the Study:

  • To investigate transport properties, records statistics, and occupation time statistics in two area-preserving maps: the Chirikov-Taylor standard map and the Casati-Prosen triangle map.
  • To compare the behavior of these maps with known results for random walks and explore potential generalized laws.

Main Methods:

  • Numerical investigation of area-preserving maps (Chirikov-Taylor standard map and Casati-Prosen triangle map).
  • Analysis of transport properties, records statistics, and occupation time statistics.

Main Results:

  • For the standard map with a chaotic sea, diffusive transport, records statistics, and occupation time fractions align with simple symmetric random walks.
  • The triangle map exhibits anomalous transport and similar anomalies in its records statistics.
  • Occupation time statistics and persistence probabilities in the triangle map are consistent with a generalized arcsine law and transient dynamics.

Conclusions:

  • Deterministic systems can exhibit complex behaviors, including diffusive and anomalous transport, mirroring stochastic processes.
  • The study confirms and extends findings for the standard map, while revealing anomalous behaviors in the triangle map consistent with generalized statistical laws.