Chaotic Behaviors Of One-Dimensional Wave Equations With Van Der Pol Boundary Conditions Containing A Source Term

For one-dimensional wave equations with the van der Pol boundary conditions, there have been several different ways in the literature to characterize the complexity of their solutions. However, if the right-end van der Pol boundary condition contains a source term, then a considerable technical difficulty arises as to how to describe the complexity of the system. In this paper, we take advantage of a topologically dynamical method to characterize the dynamical behaviors of the systems, including sensitivity, transitivity and Li-Yorke chaos. For this end, we consider a system (I,f(infinity)) induced by a sequence of continuous maps and its functional envelope (L1(I,I),H-infinity), and show that, under some considerable condition, H-infinity is transitive if and only if f(infinity) is weakly mixing of order 3; H-infinity is Li-Yorke chaotic and sensitive if f(infinity) is strongly mixing. Those abstract results have their own significance and can be applied to such kind of equations.