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.