Chaotic Dynamic of a Symmetric Tree-Shaped Wave Network

The chaotic dynamic behavior of a symmetric tree-shaped network of wave equations described by a system of partial differential equations is considered. The nonlinearities of van der Pol type are proposed at three boundary endpoints that can cause the total energy of the system to rise and fall within certain bounds. At the interconnected point of the wave equations, the energy is injected into the system through an anti-damping velocity feedback. We show that when the parameters satisfy certain conditions, the snapback repeller is existence and the system is chaotic. Finally, we give some numerical simulations to illustrate the theoretical outcomes.