Chaotic oscillations of the Klein-Gordon equation with distributed energy pumping and van der Pol boundary regulation and distributed time-varying coefficients

Abstract

Consider the Klein-Gordon equation with variable coefficients, a van der Pol cubic nonlinearity in one of the boundary conditions and a spatially distributed antidamping term, we use a variable-substitution technique together with the analogy with the 1-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.

Keywords

• Chaotic Oscillations
• Klein-Gordon equation
• distributed energy pumping
• van der Pol boundary regulation