Chaotic Traveling Wave Solutions in Coupled Chua's Circuits

Coupled arrays of Chua's circuits have been studied for many years. The existence of traveling wave solutions for such system was shown numerically in Perez-Munuzuri et al. (Traveling wave front and its failure in a one-dimensional array of Chua's circuit. Chua's circuit : a paradigm for Chaos. World Scientific, Singapore, pp 336-350, 1993). The existence of periodic traveling wave solutions has been proved recently (Chow et al. in J Appl Anal Comput 3:213-237, 2013). The purpose of this paper is to prove the existence of chaotic traveling wave solutions for such system. Using the method of singular perturbations, we show that the ODE system for the traveling waves can have a heteroclinic loop consisting of two traveling waves moving at the same speed. Moreover, at the equilibrium points of the heteroclinic loop, the dominant eigenvalues of the system are a pair of complex numbers with negative real parts. By a generalization of Shilnikov's theorem of symbolic dynamics, the system can have chaotic behavior near the traveling heteroclinic orbits.