Chaotic oscillations of wave equations due to nonlinear boundary condition

In this paper, we consider the chaotic behavior of a one-dimensional wave equation with the nonlinear boundary at x = 1, which causes the energy of the system to fall, while the other boundary is linear, which causes the energy of the system to rise. We study the dynamical behavior of the Riemann invariants of the wave equation and prove the onset of chaos in the sense of exponential growth of total variation in the gradient of the displacement of the wave equation. We also prove that when the parameters satisfy certain conditions, the system has no complex oscillations. Finally, numerical simulations are presented to verify the theoretical outcomes.