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.

中文翻译：

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非线性边界条件引起的波动方程混沌振荡

在本文中，我们考虑了非线性边界在 x = 1 处的一维波动方程的混沌行为，这导致系统的能量下降，而另一个边界是线性的，这导致系统的能量下降上升。我们研究了波动方程的黎曼不变量的动力学行为，并在波动方程位移梯度的总变化的指数增长意义上证明了混沌的开始。我们还证明了当参数满足一定条件时，系统没有复杂的振荡。最后，通过数值模拟来验证理论结果。