Hierarchical structures in the phase space and fractional kinetics: I. Classical systems

Area of Science:

• Physics
• Nonlinear Dynamics
• Statistical Mechanics

Background:

• Hamiltonian chaotic dynamics is typically non-ergodic due to the presence of invariant 'islands' within the 'stochastic sea'.
• The boundaries of these islands exhibit 'stickiness,' leading to highly erratic particle trajectories and a multifractal space-time structure.
• These complexities complicate the analysis of chaotic transport phenomena.

Purpose of the Study:

• To describe the anomalous properties of chaotic transport using fractional kinetics.
• To investigate how a hierarchical structure of islands influences chaotic dynamics.
• To explore various consequences of this complex chaotic behavior.

Main Methods:

• Application of fractional kinetics to model chaotic transport.
• Analysis of systems with hierarchically structured islands.
• Investigation of phenomena such as Poincare recurrences and characteristic transport exponents.

Main Results:

• Fractional kinetics provides a framework to describe the erratic, multifractal nature of chaotic transport.
• Hierarchical island structures lead to more transparent anomalous transport properties.
• Observed consequences include non-universal transport, log periodicity, and 'chaos erasing'.

Conclusions:

• The non-ergodic nature of Hamiltonian chaos, characterized by sticky islands, can be effectively modeled using fractional kinetics.
• Hierarchical island structures reveal anomalous transport properties and lead to phenomena like log periodicity.
• This approach offers insights into complex chaotic dynamics and transport mechanisms.