Many faces of stickiness in Hamiltonian systems
Leonid A Bunimovich1, Luz V Vela-Arevalo
1School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA. bunimovh@math.gatech.edu
Chaos (Woodbury, N.Y.)
|July 5, 2012
Summary
This study differentiates between internal and external stickiness in Hamiltonian systems using billiard examples. It highlights how elliptic and parabolic KAM-islands exhibit distinct stickiness behaviors.
Area of Science:
- Physics
- Dynamical Systems
- Chaos Theory
Background:
- Hamiltonian systems exhibit complex dynamics, including phenomena like stickiness.
- Stickiness refers to the tendency of trajectories to remain in a region of phase space for extended periods.
- Understanding stickiness is crucial for characterizing the long-term behavior of dynamical systems.
Purpose of the Study:
- To differentiate between internal and external stickiness in Hamiltonian systems.
- To investigate the role of KAM tori and KAM-islands in stickiness phenomena.
- To compare the stickiness abilities of elliptic and parabolic KAM-islands.
Main Methods:
- Visual analysis of billiard systems as a model for Hamiltonian dynamics.
- Distinction between stickiness within chaotic seas (internal) and near KAM tori (external).
- Classification of KAM-islands into elliptic and parabolic types.
Main Results:
- Demonstrated a clear difference between internal and external stickiness.
- Identified two types of KAM-islands: elliptic and parabolic.
- Showcased that elliptic and parabolic KAM-islands possess different stickiness characteristics.
Conclusions:
- The type and location of structures (KAM tori, KAM-islands) significantly influence stickiness in Hamiltonian systems.
- The distinction between internal and external stickiness provides a more nuanced understanding of chaotic dynamics.
- Further research into the specific mechanisms of stickiness in different KAM-island types is warranted.
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