The paper deals with topical issues of modern mathematical theory of dynamical chaos and its applications. At present, it is customary to assume that dynamical chaos in finitedimensional smooth systems can exist in three different forms. This is dissipative chaos, the mathematical image of which is a strange attractor; conservative chaos, for which the entire phase space is a large "chaotic sea" with randomly spaced elliptical islands inside it; and mixed dynamics, characterized by the principal inseparability in the phase space of attractors, repellers and conservative elements of dynamics. In the present paper (which opens a cycle of three our papers), elements of the theory of pseudo-hyperbolic attractors of multidimensional maps are presented. Such attractors, as well as hyperbolic ones, are genuine strange attractors, but they allow the existence of homoclinic tangencies. We give a mathematical definition of a pseudo-hyperbolic attractor for the case of multidimensional maps, from 查看全文>>