Chaotic vibration of a two-dimensional wave equation with nonlinear boundary condition

In this paper, we consider the chaotic dynamical behavior of a two-dimensional wave equation due to an energy-injecting boundary condition and an energy dissipation boundary condition. By using the chaotic mapping theory and the method of characteristic, we prove the onset of chaos in the sense of exponential growth of total variation of the 2D wave equation. Moreover, We also prove that the system has not the chaotic oscillations when the parameters enter certain ranges. Numerical examples are provided to verify the effectiveness of our theoretical results.