Sensitive dependence on initial conditions

It is shown that the property of sensitive dependence on initial conditions in the sense of Guckenheimer follows from the other two more technical parts of one of the most common recent definitions of chaotic systems. It follows that this definition applies to a broad range of dynamical systems, many of which should not be considered chaotic. We investigate the implications of sensitive dependence on initial conditions and its relation to dynamical properties such as rigidity, ergodicity, minimality and positive topological entropy. In tight of these investigations and several examples which we exhibit, we propose a natural family of dynamical systems—x-systems—as a better abstract framework for a general theory of chaotic dynamics.