  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 which we derive the necessary conditions for its existence in the
three-dimensional case, formulated using the Lyapunov exponents. We also
describe some phenomenological scenarios for the appearance of
pseudo-hyperbolic attractors of various types in one-parameter families of
three-dimensional diffeomorphisms, we propose new methods for studying such
attractors (in particular, a method of saddle charts and a modified method of
Lyapunov diagrams). We consider also three-dimensional generalized Henon maps
as examples.
