My major current interest is the application of dynamical systems theory to the study of infinite-dimensional evolutionary problems, especially those given by boundary-value problems for partial differential equations. In many cases, such problems induce on particular spaces of smooth functions infinite-dimensional dynamical systems, whose phase spaces are noncompact with respect to their associated metrics. Compactification of the phase space via different suitable metrics make it possible to reveal and explain a number of intrigue properties of trajectories for the dynamical system in hand, and with them the properties of solutions for the original evolutionary problem. Among these properties are self-structuring and coming out the predictability horizon, which makes the above-mentioned evolutionary problems a useful tool in research into the mathematical mechanisms for self-organization and spatial-temporal chaotization in complex systems.

I continue to be interested in nonlinear continuous time difference equations and differential-difference equations, especially the long-term behavior and asymptotics of their solutions.