Basic scientific interests are concentrated around problems of qualitative analysis of complex dynamical systems (stability, bifurcations, robustness with respect to perturbation of various kind) in situations when traditional in classic analysis suppositions about smoothness or continuity of dynamical systems under consideration, or about continuity of the state space or the "time component", are not satisfied. Together with M. A. Krasnosel'skii the method of parameter functionalization was developed which allowed to analyze bifurcations of steady states and periodic regimes (a kind of Hopf bifurcation) of non-smooth dynamical systems. Later, with utilization of this method the so-called effect of subfurcation was discovered, i.e. the effect of bifurcating short-living long-periodic regimes of nonsmooth dynamical system in situations when for their smooth analogs the bifurcation of invariant cycles takes place. There were developed essential principles of the theory of stability for the so-called asynchronous systems, i.e. systems describing dynamics of objects updating their states at discrete time instants asynchronously with each other (the typical example of such systems is the computational network). As a bypass result, the algebraic insolubility of the problem of stability analysis for the infinite products of matrices from a finite family was established. Methods of analysis of the dynamics of spatial discretizations for continuous dynamical systems were studied too.
Main publications:
Kozyakin V., Krasnosel'skii M. The method of parameter functionalization in the Hopf bifurcation problem // Nonlinear Analysis, TMA, 1987, 11, 2, 149–161.
