Chaos Induced by Heteroclinic Cycles Connecting Repellers for First-Order Partial Difference Equations

Abstract

This paper studies chaos and chaotification for a class of first-order partial difference equations with a finite or infinite system size by using the theory of heteroclinic cycles connecting repellers. Three criteria of chaos induced by regular and nondegenerate heteroclinic cycles connecting repellers or regular heteroclinic cycles connecting repellers are established, respectively. Especially, one chaotification theorem for general discrete systems in the finite-dimensional space [Formula: see text] or the infinite-dimensional space [Formula: see text] is established, which is also based on heteroclinic cycles connecting repellers. Then, by using this result, two chaotification schemes for first-order partial difference equations are established. For illustrating the existence of chaos and the validity of chaotification schemes, three examples are provided with computer simulations.