Chaotic oscillations of 1D wave equation with or without energy-injections

It is interesting and challenging to study chaotic phenomena in partial differential equations. In this paper, we mainly study the problems for oscillations governed by 1D wave equation with general nonlinear feedback control law and energy-conserving or energy-injecting effects at the boundaries. We show that i) energy-injecting effect at the boundary is the necessary condition for the onset of chaos when the nonlinear feedback law is an odd function; ii) chaos never occurs if the nonlinear feedback law is an even function; iii) when one of the two ends is fixed, only the effect of self-regulation at the other end can still cause the onset of chaos; whereas if one of the two ends is free, there will never be chaos for any feedback control law at the other end. In addition, we give a sufficient condition about the general feedback law at one of two ends to ensure the occurrence of chaos. Numerical simulations are provided to demonstrate the effectiveness of the theoretical outcomes.