Chaotic systems with absorption

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.