Li-Yorke chaos of wave equations with linear boundary conditions under the weak topology

Abstract

The initial and boundary value problem for a one-dimensional wave equation on a Hilbert space is investigated. The Dirichlet, Neumann and Robin boundary conditions are systematically analyzed. When the Hilbert space is equipped with a weak topology that induced by the bounded linear functionals, the initial and boundary value problems have been rigorously proven to exhibit Li-Yorke chaos. The existence of a pair of conjugate pure imaginary eigenvalues in the linear operator induced from the wave equation is demonstrated to cause weak Li-Yorke chaos. However, they show stability when the weak topology is replaced by the norm topology. This interesting discovery reveals a remarkable phenomenon that the topology of the infinite-dimensional space is a crucial factor determining chaos of the linear hyperbolic PDEs. Furthermore, it illustrates that the infinite-dimensional dynamical systems governed by PDEs can be simple in form and still have chaotic complexity.