Chaos of the 2D linear hyperbolic equation with general van der Pol type boundary condition

Abstract

This paper is concerned with the complex dynamics of the initial-boundary value problem of a 2D linear hyperbolic partial differential equation (PDE), where the parameter α that appeared in the general van der Pol type boundary condition is given by α∈R. The whole real line is divided into three intervals of the parameter to discuss the dynamics. The existence of chaos is first established in the sense of the exponential growth of total variation when the parameter locates in the central interval, which allows it to be positive, negative, or zero. By analyzing the chaotic dynamics of the piecewise continuous map induced by the hyperbolic PDE, such a PDE is further rigorously proved to be chaos in the interval of the positive parameter that is to the right of the central interval. Finally, the asymptotic behaviors of the hyperbolic PDE are systematically presented in the rest of the whole real line; more precisely, the hyperbolic PDE is either globally stable or unbounded.