Chaotic Vibration of the Wave Equation with Nonlinear Feedback Boundary Control: Progress and Open Questions

Abstract

The study of chaotic phenomena in partial differential equations is a challenging subject. In this paper, we survey the recent progress in the study of chaotic vibration of the linear wave equation with nonlinear boundary feedback control law. We show that when there is linear energy injection at one end of the boundary and the self-regulating or van der Pol nonlinearity at the other end of the boundary, chaos occurs as a reconciliation between linear instability and nonlinear self-regulation when the parameters enter a certain regime. A list of open problems is also posed.