Time Delay Control and Frequency Splitting in the forced Kadomtsev-Petviashvili Equation

Abstract

A time delay control is applied to the forced Kadomtsev-Petviashvili (KP) equation Using an appropriate perturbation method, we derive nonlinear equations describing amplitude and phase of the response and discuss in some detail external force-response and frequency-response curves for the fundamental resonance. For the uncontrolled system, we find a frequency splitting, a second frequency aappears in addition to the forcing one. Saddle-center bifurcation, jumps and hysteresis phenomena are observed together with closed orbits of the slow flow equations. There are stable two-period quasi-periodic modulated motion for the KP equation with amplitudes depending on the initial conditions. Subsequently, we study the controlled system finding sufficient conditions for a periodic behavior. We can accomplish a successful control because the amplitude peak of the fundamental resonance can be reduced and the saddle-center bifurcations and two-period quasi-periodic motions can be removed by adequate choices for delay and feedback gains.