Speciality:
01.01.02 (Differential equations, dynamical systems, and optimal control)
E-mail: , ,
Keywords: dynamical systems,
nonautonomous dynamical systems,
topological theory of dynamical systems,
global attractors,
almost periodical systems,
linear differential equations in the Banach spaces.
Subject:
I) Global attractors of nonautonomous dynamical systems. There was developed a general theory of compact global attractors of autonomous and nonautonomous dynamical systems. Different types of attractors were introduced, there is given a description of these classes of attractors and there are established relations between them. The structure of compact global attractor for different classes of systems (linear; homogeneous; C-analytic; the systems with a hyperbolic centre etc) was studied. There was developed the method of Lyapunov's functions for study of compact global attractors of nonautonomous dynamical systems. The conditions of existence of compact global atractors for certain classes of evolutionary equations (quasilinear systems; C-analytic systems; quasihomogeneous systems; monotone systems; systems of Lorenz; Navier–Stokes"s equations etc) are given. There was established an upper semicontinuos dependance of small parameter of compact global attractor.
II) Linear nonautonomous dynamical systems in Banach spaces. Different types of stabilities for linear differential equations with almost periodic coefficients and relations between them were studied. For the linear equations in Banach spaces the well-known theorem of Cameron–Johnson was generalized.
Main publications:
Cheban D. N., “Ogranichennye resheniya lineinykh pochti periodicheskikh sistem differentsialnykh uravnenii”, Izvestiya RAN. Seriya matematicheskaya, 62:3 (1998), 155–174
Cheban D. N., “Uniform exponential stability of linear almost periodic systems in Banach spaces”, Electronic Journal of Differential Equations, 2000, no. 29, 18 p.
Cheban D. N., “Global attractors of nonautonomous quasihomogeneous dynamical systems”, Proceedings of the International Conference on Dynamical Systems and Differential Equations (May 18–21, 2000, Kennesaw, USA), Discrete Contin. Dynam. Systems, 2001, Added Volume, 96–101
Cheban D. N., Kloeden P. E., Schmalfuß B., “Pullback attractors in dissipative nonautonomous differential equations under discretization”, J. Dynam. Differential Equations, 13:1 (2001), 185–213
Cheban D. N., “Global pullback attractors of $\mathbb C$-analytic nonautonomous dynamical systems”, Stochastics and Dynamics, 1:4 (2001), 511–535
