Chaotic Vibrations of the One-dimensional Wave Equation Due to a Self-excitation Boundary Condition. Ii. Energy Injection, Period Doubling and Homoclinic Orbits

Consider the initial-boundary value problem of the linear wave equation wtt − wxx = 0 on an interval. The boundary condition at the left endpoint is linear homogeneous, injecting energy into the system, while the boundary condition at the right endpoint has cubic nonlinearity of a van der Pol type. We show that the interactions of these linear and nonlinear boundary conditions can cause chaos to the Riemann invariants (u, v) of the wave equation when the parameters enter a certain regime. Period-doubling routes to chaos and homoclinic orbits are established. We further show that when the initial data are smooth satisfying certain compatibility conditions at the boundary points, the space-time trajectory or the state of the wave equation, which satisfies another type of the van der Pol boundary condition, can be chaotic. Numerical simulations are also illustrated.