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Chaotic systems with absorption.

Eduardo G Altmann1, Jefferson S E Portela, Tamás Tél

  • 1Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany.

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|October 22, 2013
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Summary
This summary is machine-generated.

We present a dynamical-system approach to understand absorption in chaotic systems. This method reveals how absorption impacts multifractality and introduces a new formula for escape rate calculations.

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Area of Science:

  • Dynamical systems theory
  • Chaos theory
  • Statistical physics

Background:

  • Absorption phenomena in chaotic systems are crucial in fields like optics and acoustics.
  • Existing models often do not fully account for the interplay between absorption, return times, and multifractality.

Purpose of the Study:

  • To develop a dynamical-system approach for describing absorption in chaotic systems.
  • To derive a general formula for the escape rate and generalize existing dimension formulas.

Main Methods:

  • Introduction of an operator formalism.
  • Analysis of conditionally invariant measure.
  • Multifractal analysis of dimension spectra.
  • Numerical simulations in a cardioid billiard.

Main Results:

  • A general formula for the escape rate (κ) was derived using the system's invariant measure.
  • Increased multifractality was observed when absorption and return times were considered, compared to standard dimension spectra D(q).
  • A generalized Kantz-Grassberger formula was established, relating the dimension D(1) to escape rate, Lyapunov exponent, average return time, and reflection rate.

Conclusions:

  • The dynamical-system approach effectively describes absorption in chaotic systems.
  • The study highlights the significant impact of absorption and return times on the multifractal properties of chaotic systems.
  • The generalized formula provides a more comprehensive understanding of dimension scaling in systems with absorption.