Chaos induced by heteroclinic cycles connecting repellers in complete metric spaces

This paper is concerned with chaos induced by heteroclinic cycles connecting repellers for maps in complete metric spaces. The concepts of heteroclinic cycle connecting fixed points and heteroclinic cycle connecting repellers are introduced. Two classifications of heteroclinic cycles are given: regular and singular; nondegenerate and degenerate. Several criteria of chaos are established for maps in complete metric spaces and metric spaces with the compactness in the sense that each bounded and closed set of the space is compact, respectively, by employing the coupled-expansion theory. They are all induced by regular and nondegenerate heteroclinic cycles connecting repellers. The maps in these criteria are proved to be chaotic in the sense of both Devaney and Li–Yorke. An illustrative example is provided with computer simulations.