On the definition of chaos

Five definitions of chaos used in discrete dynamical systems are reviewed. Their invariance with respect to topological conjugacy is proved. It is shown that the so-called shadowing property is the most basic of the five definitions: it implies chaos in the sense of coin-tossing, chaos in the sense of Devaney, chaos in the sense of Li-Yorke and chaos in the sense of Bernoulli. The following examples are treated or indicated: Torus automorphisms, snap-back repellors, transversal homoclinic points. In addition, extensions to nonautonomous discrete dynamical systems with implications for nonperiodic differential equations are described.