Chaos of the coupled Klein-Gordon like equations with superlinear boundary conditions

Abstract

The complicated dynamics is considered for a system of coupled generalized Klein-Gordon equations with two symmetric superlinear boundary conditions. The coupled equations studied here becomes coupled damped Klein-Gordon equations or telegraph equations or wave equations at a certain parameter value. The system is reduced to a discrete iteration of a 2D map differing from the unary map induced by the non-coupled hyperbolic PDEs. In this article, the approach of snapback repeller is applied to establish the sufficient condition on the Li-Yorke chaos of the system. Moreover, by characterizing the asymptotic behavior of the 2D map, the global stability of the system is obtained under a certain range of parameter. In particular, the obtained dynamics not only extends the chaotic results on coupled wave equations with van der Pol type boundary conditions, but also gives the criterion of global stability.