Lyapunov instability versus relaxation time in two coupled oscillators

Area of Science:

• Nonlinear dynamics
• Statistical mechanics
• Oscillatory systems

Background:

• Understanding the relationship between system energy and dynamic properties like relaxation time and chaos is crucial.
• Coupled oscillator systems are fundamental models in physics, appearing in diverse phenomena from molecular vibrations to celestial mechanics.

Purpose of the Study:

• To investigate the interplay between relaxation time and the largest Lyapunov exponent in a system of two coupled oscillators, with one being harmonic.
• To elucidate the underlying mechanisms responsible for unexpected trends in these dynamic measures as a function of total energy.

Main Methods:

• Analysis of a two-coupled oscillator system, including one harmonic oscillator.
• Examination of the system's parameter space to identify regions with specific dynamic behaviors.
• Investigation of the role of Kolmogorov-Arnold-Moser (KAM) tori and dispersion relations.

Main Results:

• Contrary to common expectations, both the largest Lyapunov exponent and relaxation time were found to increase with total energy over a broad parameter range.
• This phenomenon is linked to the emergence of KAM tori above a critical energy threshold, which reduces the overall chaotic fraction of phase space.
• The dispersion relation was identified as a key factor enabling this behavior.

Conclusions:

• The study reveals a non-intuitive relationship between energy, chaos, and relaxation in coupled oscillator systems.
• The findings highlight the importance of considering topological structures like KAM tori in phase space when analyzing system dynamics.
• This research provides insights applicable to understanding energy damping in complex systems, such as nuclear giant resonances.