On the coexistence of cycles and chaotic solutions of difference equations with random parameters

The article deals with the difference equations, the right part of which in a certain moment of time depends not only on the value in the previous moment but also on a random parameter from a given Omega manifold. For this stochastic model various dynamic modes of development are studied. It should be mentioned that they differ from the modes of the determined models and describe the processes in real physical systems in a more comprehensive way. We have received the conditions for the unstable cycles existence, fulfilled for all values of a random parameter and fulfilled with probability one, and also conditions under which solutions are chaotic with probability one. It is shown that chaotic solutions exist when an equation with random parameters either has no cycles or all cycles are unstable with probability one. The problem of the coexistence for stochastic cycles of different periods is also investigated.