Mathematical theory of dynamical chaos and its applications: Review. Part 1. Pseudohyperbolic attractors

We consider important problems of modern theory of dynamical chaos and its applications. At present, it is customary to assume that in the finite-dimensional smooth dynamical systems three fundamentally different forms of chaos can be observed. This is the dissipative chaos, whose mathematical image is a strange attractor; the conservative chaos, for which the whole phase space is a large «chaotic sea» with elliptical islands randomly disposed within it; and the mixed dynamics which is characterized by the principle inseparability, in the phase space, of attractors, repellers and orbits with conservative behavior.
In the first part of this series of our works, we present some elements of the theory of pseudohyperbolic attractors of multidimensional maps. Such attractors, the same as hyperbolic ones, are genuine strange attractors, however, they allow homoclinic tangencies. We also give a description of phenological scenarios of the appearance of pseudohyperbolic attractors of various types for one parameter families of three-dimensional diffeomorphisms, and, moreover, consider some examples of such attractors in three-dimensional orientable and nonorientable Hénon maps.
In the second part, we will give a review of the theory of spiral attractors. Such type of strange attractors are very important and are often observed type in dynamical systems. The third part will be dedicated to mixed dynamics – a new type of chaos which is typical, in particulary, for (time) reversible systems i.e. systems which are invariant with respect to some changes of coordinates and time reversing. It is well known that such systems occur in many problems of mechanics, electrodynamics, and other areas of natural sciences.