Transient Chaos in Coupled Oscillators with Shape Deformable Potential

Abstract

The dynamics of a perturbed system consisting of a particle embedded in a strongly nonlinear potential V(ϕ; r, η) whose shape can be varied continuously as a function of r in the range -1 < r < 1, and coupled to an harmonic oscillator is analyzed. The perturbations are made of the coupling and damping forces. When they are removed, the subsystem formed by the anharmonic oscillator contains saddle points connected by homoclinic loops. Thus, the whole unperturbed system has saddle-centre points whose stable and unstable manifolds coincide in three-dimensional manifold. We concentrate our analysis of the perturbed system in the regions near these saddle-centre points. First, the Melnikov theory is used to investigate the presence of horseshoes chaos in the dynamics. Due to some discrepancies of the Melnikov theory, we next rederive the boundary between the regions of regular and irregular motions in the parameters space by using regular perturbation expansion. It is found here that for small values of the natural pulsation ω of the linear oscillator, complicated behavior intensifies in the system as the absolute value of the shape parameter r approaches zero. On the contrary, chaotic motion intensifies according as r decreases from values close to 1 to values close to -1, for large ω. The numerical analysis, which includes computation of maximal Lyapunov exponent, bifurcation diagrams and Poincaré sections, shows that the system exhibits transient stochastic behavior, but ultimately settles down on a simple set which is either a fixed point or a limit cycle. The dependence of this transient behavior on the system parameters agrees qualitatively well with the analytical predictions.