Chaotic Oscillations of 1D Wave Equation Due to a Generalized Nonlinear Energy-Decay Boundary Condition

The study of spatio-temporal chaos in a system governed by PDEs is interesting but challenging. For the past two decades, the interactions of energy-injection and self-regulation are a practical approach to generate chaos in the system governed by 1D wave equation. In this paper, we introduce a different way to ensure the onset of chaos. More specifically, we consider the initial-boundary value problem described by 1D wave equation wtt - wxx = 0 on an interval. The boundary condition at the left endpoint is linear homogeneous, injecting energy into the system, while the boundary condition at the other side has generalized nonlinearity that causes the energy to decay. We show that the interactions of these linear and generalized nonlinear boundary conditions can generate chaos when some parameter enters a certain regime.