Chaotic behaviors of one dimensional wave equations with van der Pol nonlinear boundary conditions

Abstract

For one-dimensional linear wave equations with van der Pol boundary conditions, there have existed several different ways in the literature to characterize the complexity of their solutions. In this paper, we give a new way to characterize the dynamical behaviors of the systems, including almost equicontinuity and Devaney chaos. In order to do this, we consider a one-dimensional system (I, f) and its functional envelope (L1(I,I),Hf) and show that Hf is weakly mixing if and only if f is; Hf is Devaney chaotic if f is weakly mixing; Hf is almost equicontinuous if f has an attracting 2-period orbit. Those abstract results have their own significance and can be applied to the linear wave equations with van der Pol boundary conditions.