Chaotic oscillations of the 1D wave equation due to extreme imbalance of self-regulations

In this paper, we mainly consider the chaotic behaviors of the initial-boundary value problems of 1D wave equation on an interval. The boundary conditions at the left and right endpoints have nonlinearities of van der Pol type, both of which cause the total energy of the system to rise and fall within certain bounds. We show that the interactions of these nonlinear boundary conditions can cause chaotic oscillations of the wave equation when the parameters enter a certain regime, which can be taken as a new way to lead to the onset of chaos. We further prove that for any given initial data and parameters, the variables “force” and “velocity” will fall into a bounded region as time increases. Numerical simulations are presented to illustrate the theoretical outcomes.