Instability versus nonlinearity in certain nonautonomous oscillators: A critical dynamical transition driven by the...

Area of Science:

• Nonlinear Dynamics
• Mathematical Physics
• Chaos Theory

Background:

• The study examines the equation of motion q+Omega(2)(t)q+alpha/q/(gamma-2)q=0, a model for nonlinear and unstable systems.
• Time-varying frequency Omega(t) introduces exponential instability in the linear counterpart.
• Understanding the interplay between nonlinearity and instability is crucial in various physical phenomena.

Purpose of the Study:

• To analyze the behavior of the given nonlinear dynamical system.
• To identify the dominant mechanisms governing the system's dynamics at different energy levels.
• To explore the relationship between dynamical transitions and phenomena like polaron behavior and chaos.

Main Methods:

• Analysis of the equation of motion under varying energy conditions.
• Investigation of the competition between exponential instability and nonlinear bounding effects.
• Identification of a critical curve in the initial phase space.

Main Results:

• At low energies, exponential instability dominates; at high energies, nonlinear bounding prevails.
• System energy exhibits quasiperiodic peaks with an average recurrence time T(peak).
• A critical curve S(omega) in the initial phase space governs the divergence of T(peak) with a universal critical exponent a=1.

Conclusions:

• The study reveals a critical transition in the system's dynamics based on initial energy.
• This transition is linked to phenomena such as mobile-to-self-trapped polarons and transitions to chaos.
• The findings have implications for problems like the Fermi-Pasta-Ulam problem.