Chaotic Oscillations of Solutions of the Klein–Gordon Equation Due to Imbalance of Distributed and Boundary Energy Flows

Abstract

Chaos in physical systems governed by nonlinear PDEs have been amply observed and reported. However, rigorous proofs for their occurrences are challenging. In particular, for a second order PDE of hyperbolic type with a van der Pol cubic nonlinearity in one of the boundary conditions and a spatially distributed antidamping term in a linear governing equation, no proof for the onset of chaos was available even though chaos was expected to occur when the antidamping term becomes sufficiently strong. In this paper, we use an operator-factoring technique together with the analogy with the one-dimensional wave equation to prove that for the Klein–Gordon equation chaos occurs for a class of equations and boundary conditions when system parameters enter a certain regime. Chaotic and nonchaotic profiles of solutions are illustrated by computer graphics.