Nonisotropic Spatiotemporal Chaotic Vibration of the Wave equation due to Mixing Energy Transport and a van der Pol Boundary Condition

The imbalance of the boundary energy flow due to energy injection at one end and a nonlinear van der Pol boundary condition at the other end of the spatial one-dimensional interval can cause chaotic vibration of the linear wave equation [Chen et al., 1998b, 1998c]. However, such chaotic vibration is isotropic with respect to space and time because the two associated families of characteristics both propagate with the same speed and, thus, the “strength of chaos” is the same along both x and t directions. In this paper, we show that by including a mixed partial derivative linear energy transport term in the wave equation, nonlinearity in the van der Pol boundary condition can also cause chaotic vibration (without energy injection from the other end). Two new families of characteristics now travel with different speeds, leading to strong mixing of waves and nonisotropic spatiotemporal chaos. Parameter range for the route to chaos, including period-doubling, homoclinic orbits and Cantor-like invariant sets is classified. Numerical simulations of chaotic space-time profiles are also illustrated.