Bursts in the chaotic trajectory lifetimes preceding controlled periodic motion

Area of Science:

• Dynamical systems and chaos theory.
• Statistical mechanics and ergodic theory.

Background:

• Understanding the behavior of chaotic systems is crucial for various scientific and engineering applications.
• The average lifetime of a trajectory within a specific region of a chaotic attractor, denoted tau(H), is a key parameter for controlling chaos.
• Deviations from the natural invariant measure can occur, impacting predictability.

Purpose of the Study:

• To investigate the average lifetime [tau(H)] of trajectories entering a small region (H) on a chaotic attractor.
• To analyze how the presence of short periodic orbits influences this lifetime.
• To introduce a formula relating the lifetime deviation to the properties of these periodic orbits.

Main Methods:

• Analysis of chaotic dynamics and trajectory behavior.
• Mathematical formulation of average lifetimes in chaotic systems.
• Derivation of a relationship between lifetime, invariant measure, and periodic orbit properties.

Main Results:

• The average lifetime [tau(H)] deviates significantly from the inverse of the invariant measure [&mgr;(N)(H)(-1)] when region H is visited by a short periodic orbit.
• A novel formula is introduced that quantifies this deviation.
• The formula connects the ratio tau(H)/&mgr;(N)(H)(-1) to the expanding eigenvalue of the relevant short periodic orbit.

Conclusions:

• Short periodic orbits play a critical role in the dynamics of chaotic systems, causing deviations in trajectory lifetimes.
• The introduced formula provides a new tool for understanding and predicting chaotic system behavior.
• This finding has implications for the control and manipulation of chaos in various applications.