My major current interest is the application of dynamical systems theory
to the study of infinitedimensional evolutionary problems, especially
those given by boundaryvalue problems for partial differential equations.
In many cases, such problems induce on particular spaces of smooth functions
infinitedimensional 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 selfstructuring and coming out
the predictability horizon, which makes the abovementioned evolutionary
problems a useful tool in research into the mathematical mechanisms for
selforganization and spatialtemporal chaotization in complex systems.
I continue to be interested in nonlinear continuous time difference
equations and differentialdifference equations, especially the longterm
behavior and asymptotics of their solutions.
