Phase-coupled nonlinear dynamical systems, stability, first integrals, and Liapunov exponents

Abstract

The stability of a class of coupled identical autonomous systems of first-order nonlinear ordinary differential equations is investigated. These couplings play a central role in controlling chaotic systems and can be applied in electronic circuits and laser systems. As applications we consider a coupled van der Pol equation and a coupled logistic map. When the uncoupled system admits a first integral we study whether a first integral exists for the coupled system. Gradient systems and the Painlevé property are also discussed. Finally, the relation of the Liapunov exponents of the uncoupled and coupled systems are discussed.