On weak spreadability of distributed-parameter systems and its achievement via linear--quadratic control techniques

In this paper, the concept of weak spreadability is introduced.It constitutes an extension of the idea developed by El Jai& Kassara. In the case of linear distributed systems weconsider quadratic control techniques with a conveniently penalizedcriterion which makes the system weakly spreadable. The approachis outlined for a convection—diffusion system, and theresults of a numerical study are also included to illustratethe main features of the considered problem.     

New methods for proving the existence and stability of periodic solutions in singularly perturbed delay systems

We carry out a detailed analysis of the existence, asymptotics, and stability problems for periodic solutions that bifurcate from the zero equilibrium state in systems with large delay. The account is based on a specific meaningful example given by a certain scalar nonlinear second-order differential-difference equation that is a mathematical model of a single-circuit RCL oscillator with delay in a feedback loop.     

Attractors of the sine-Gordon equation in the field of a quasiperiodic external force

The well-known sine-Gordon equation, supplemented with small damping and small quasiperiodic external force, is studied under the zero Dirichlet boundary conditions at the endpoints of a finite interval. The main assumption is that all frequencies of the external force are in 1:1 resonance with certain eigenfrequencies of the unperturbed equation; i.e., the socalled fundamental multifrequency resonance is observed. It is shown that in this case, by an appropriate choice of the parameters of the external force, one can make it so that the boundary value problem has a stable invariant torus of any finite dimension that bifurcates from zero on any preassigned finite set of spatial modes. It is also shown (by numerical analysis) that in a number of cases the above-mentioned torus coexists with a chaotic attractor.     

New methods for proving the existence and stability of periodic solutions in singularly perturbed delay systems

We carry out a detailed analysis of the existence, asymptotics, and stability problems for periodic solutions that bifurcate from the zero equilibrium state in systems with large delay. The account is based on a specific meaningful example given by a certain scalar nonlinear second-order differential-difference equation that is a mathematical model of a single-circuit RCL oscillator with delay in a feedback loop. Original Russian Text ? A.Yu. Kolesov, E.F. Mishchenko, N.Kh. Rozov, 2007, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2007, Vol. 259, pp. 106–133.     

Chaotic buffering and its mathematical models

We propose a general idea of obtaining different chains of coupled oscillators with chaotic buffering. As examples, we consider chains of diffusively coupled generalized cubic Schrödinger equations and nonlinear telegraph equations. We also give examples of systems that have an infinite-dimensional chaotic attractor.     

New aminoalkyl and vinyl derivatives of pyridine and quinoline

New bases have been obtained by the Mannich reaction from 2-methyl-5-vinylpyridine and from 2,4-dimethylquinoline. Decomposition of quaternary salts of these bases has yielded previously unknown vinyl derivatives of pyridine and quinoline.