Chaotic oscillations of one-dimensional coupled wave equations with mixed energy transports

In this paper, we consider the chaotic behavior of one-dimensional coupled wave equations with mixed partial derivative linear energy transport terms. The van der Pol-type symmetric nonlinearities are proposed at two boundary endpoints, which cause the energy of the coupled system to rise and fall within certain ranges. At the interconnected point of the two wave equations, the energy is injected into the system through a middle-point velocity feedback. We prove that when the parameters satisfy some conditions, the coupled wave equations have snapback repellers which can make the whole system chaos in the sense of Li-Yorke. Numerical simulations are presented to verify the theoretical results.