Dynamics of driven coupled oscillators with shape deformable potential

Abstract

We introduce in this paper a model of driven coupled nonlinear oscillators that are suited for the description of physical systems which may undergo structural changes or shape distorsion in certain situations. The model is characterized by a potential V RP( φ; r) whose shape can vary continuously as a function of r in the range | r|<1. The dynamics of the model is carefully studied, both analytically and numerically. The analytical analysis is effected in the framework of the Melnikov theory. It reveals that the dynamics depends non trivially on the shape parameter r, proving the importance of deformable potential in the description of real physical systems. In addition, the Melnikov theory indicates that the coupling tends to suppress chaos in the model (in the sense that it raises up its onset), comparatively to the dynamics of the uncoupled oscillators of the model. Our numerical investigation which involves computation of maximal Lyapunov exponents, Poincaré sections and bifurcation diagrams shows a rich spectrum of dynamical behavior including periodic, quasiperiodic and chaotic states. The comparison of analytical and numerical results shows that the Melnikov theory predicts slighly low the onset of chaos for our model, but captures very well the qualitative dynamics of the system.