Melnikov's criteria, parametric control of chaos, and stationary chaos occurrence in systems with asymmetric potential subjected to multiscale type excitation

We consider the problems of chaos and parametric control in nonlinear systems under an asymmetric potential subjected to a multiscale type excitation. The lower bound line for horseshoes chaos is analyzed using the Melnikov's criterion for a transition to permanent or transient nonperiodic motions, complement by the fractal or regular shape of the basin of attraction. Numerical simulations based on the basins of attraction, bifurcation diagrams, Poincare sections, Lyapunov exponents, and phase portraits are used to show how stationary dissipative chaos occurs in the system. Our attention is focussed on the effects of the asymmetric potential term and the driven frequency. It is shown that the threshold amplitude vertical bar gamma(c)vertical bar of the excitation decreases for small values of the driven frequency omega and increases for large values of omega. This threshold value decreases with the asymmetric parameter alpha and becomes constant for sufficiently large values of alpha. gamma(c) has its maximum value for asymmetric load in comparison with the symmetric load. Finally, we apply the Melnikov theorem to the controlled system to explore the gain control parameter dependencies. (C) 2011 American Institute of Physics. [doi:10.1063/1.3650699]