Islands of accelerator modes and homoclinic tangles

Area of Science:

• Dynamical systems theory
• Nonlinear dynamics
• Statistical mechanics

Background:

• Phase space structures, specifically resonant and tangle islands, are key to understanding dynamical systems.
• The distinction and relationship between these island types are crucial for analyzing system behavior.

Purpose of the Study:

• To differentiate between resonant islands and tangle islands based on phase space structure.
• To investigate the nature of accelerator modes in the standard and web maps.
• To elucidate the role of these islands in anomalous transport.

Main Methods:

• Theoretical analysis of phase space structures in near-integrable systems.
• Mathematical proof to establish the distinctness of island categories.
• Demonstration of accelerator modes belonging to the tangle island category.

Main Results:

• Resonant islands and tangle islands represent two distinct sets in the near-integrable limit.
• Accelerator modes of the standard map and web map are definitively classified as tangle islands.
• Stickiness to tangle islands can induce anomalous transport, even from the ergodic component.

Conclusions:

• The definitions of resonant and tangle islands are not generally equivalent.
• Tangle islands, particularly accelerator modes, play a significant role in transport phenomena.
• Understanding island dynamics is essential for predicting anomalous transport in complex systems.