Mathematical theory of dynamical chaos and its applications: Review. Part 2. Spiral chaos of three-dimensional flows

The main goal of the present paper is an explanation of topical issues of the theory of spiral chaos of three-dimensional flows, i.e. the theory of strange attractors associated with the existence of homoclinic loops to the equilibrium of saddle-focus type, based on the combination of its two fundamental principles, Shilnikov’s theory and universal scenarios of spiral chaos, i.e. those elements of the theory that remain valid for any models, regardless of their origin. The mathematical foundations of this theory were laid in the 60th in the famous works of L.P. Shilnikov, and on this subject to date, a lot of important and interesting results have been accumulated. However, these results, for the most part, were related to applications, and, perhaps for this reason, the theory of spiral chaos lacked internal unity – until recently it seemed to consist of separate parts. As it seems for us, the main results of our review allow to fill this gap. So, in the paper we present a fairly complete and illustrative proof of the famous theorem of Shilnikov (1965), describe the main elements of the phenomenological theory of universal scenarios for the emergence of spiral chaos, and also, from a unified point of view, consider a number of three-dimensional models which demonstrate this chaos. They are both the classical models (the systems of Rössler and Arneodo–Coullet–Tresser) and several models known from applications. We discuss advantages of such a new approach to the study of problems of dynamical chaos (including the spiral one), and our recent works devoted to the study of chaotic dynamics of four-dimensional flows and three-dimensional maps show that it is also quite effective. In particular, the next, third, part of the review will be devoted to these results.