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Of all publications in the section: 7
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Article
Krasnoselskii A., Kloeden P., Rachinskii D. I. et al. Discrete and Continuous Dynamical Systems - Series B. 2013. Vol. 18. No. 2. P. i-iii.

The paper is the obituary of Alexei V. Pokrovskii (02.06.1948 - 01.09.2010), this volume of the journal was dedicated to Alexei Pokrovskii.

Added: Mar 10, 2014
Article
Blank M. Discrete and Continuous Dynamical Systems - Series B. 2013. Vol. 18. No. 2. P. 313-329.

One of the main paradigms of the theory of weakly interacting chaotic systems is the absence of phase transitions in generic situation. We propose a new type of multicomponent systems demonstrating in the weak interaction limit both collective and independent behavior of local components depending on fine properties of the interaction. The model under consideration is related to dynamical networks and sheds a new light to the problem of synchronization under weak interactions.

Added: Nov 26, 2014
Article
Vladimirov A. Discrete and Continuous Dynamical Systems - Series B. 2013. Vol. 18. No. 2. P. 565-573.

We prove that the sweeping process on a “regular” class of convex sets is equicontinuous. Classes of polyhedral sets with a given finite set of normal vectors are regular, as well as classes of uniformly strictly convex sets. Regularity is invariant to certain operations on classes of convex sets such as intersection, finite union, arithmetic sum and affine transformation.

Added: Jan 30, 2013
Article
Veretennikov A. Discrete and Continuous Dynamical Systems - Series B. 2013. Vol. 18. No. 2. P. 523-549.
Added: Oct 18, 2014
Article
Krasnosel'skii Alexander M., O'Grady E., Pokrovskii A. et al. Discrete and Continuous Dynamical Systems - Series B. 2013. Vol. 18. No. 2. P. 467-482.

We consider a scalar fast differential equation which is periodically driven by a slowly varying input. Assuming that the equation depends on scalar parameters, we present simple sufficient conditions for the existence of a periodic canard solution, which, within a period, makes n fast transitions between the stable branch and the unstable branch of the folded critical curve. The closed trace of the canard solution on the plane of the slow input variable and the fast phase variable has n portions elongated along the unstable branch of the critical curve. We show that the length of these portions and the length of the time intervals of the slow motion separated by the short time intervals of fast transitions between the branches are controlled by the parameters.

Added: Feb 11, 2013
Article
Chepyzhov V. V., Ilyin A., Zelik S. Discrete and Continuous Dynamical Systems - Series B. 2017. Vol. 22. No. 5. P. 1835 -1855.

We consider the damped and driven two-dimensional Euler equations in the plane with weak solutions having finite energy and enstrophy. We show that these (possibly non-unique) solutions satisfy the energy and enstrophy equality. It is shown that this system has a strong global and a strong trajectory attractor in the Sobolev space H1. A similar result on the strong attraction holds in the spaces H1 ∪ {u: ∥ curl u ∥ Lp < ∞} for p ≥ 2.

Added: May 18, 2017
Article
Chepyzhov V. V. Discrete and Continuous Dynamical Systems - Series B. 2015. Vol. 20. No. 3. P. 811-832.

We construct the trajectory attractor AΣ for the non-autonomous dissipative 2d Euler systems with periodic boundary conditions that contain time dependent dissipation terms −r(t)u such that 0<α≤r(t)≤β, for t≥0. External forces g(x,t),x∈T2,t≥0, also depend on time. The corresponding non-autonomous dissipative 2d Navier--Stokes systems with the same terms −r(t)u and g(x,t) and with viscosity ν>0 also have the trajectory attractor AνΣ. Such systems model large-scale geophysical processes in atmosphere and ocean. We prove that AνΣ→AΣ as viscosity ν→0+ in the corresponding metric space. Moreover, we establish the existence of the minimal limit AminΣ⊆AΣ of the trajectory attractors AνΣ as ν→0+. Every set AνΣ is connected. We prove that AminΣ is a connected invariant subset of AΣ. The problem of the connectedness of the trajectory attractor AΣ itself remains open.

Added: Mar 25, 2016