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Abstract: We investigate the local dynamics of a system of oscillators with a large number of elements and with diffusion-type couplings containing a large delay. We isolate critical cases in the stability problem for the zero equilibrium state and show that all of them are infinite-dimensional. Using special infinite normalization methods, we construct quasinormal forms, that is, nonlinear boundary value problems of parabolic type whose nonlocal dynamics determines the behavior of solutions of the original system in a small neighborhood of the equilibrium state. These quasinormal forms contain either two or three spatial variables, which emphasizes the complexity of dynamic properties of the original problem.

Abstract: The local dynamics of coupled identical nonlinear systems of second-order differential equations in a two-dimensional domain is studied. The main assumption is that the number of such equations is quite large. This makes it possible to move to a system with two continuous spatial variables. Critical cases in the problem of stability of the equilibrium state are highlighted. They all are of infinite dimension, i.e., the infinitely many roots of the characteristic equation for the linearized problem tend to the imaginary axis as the natural small parameter tends to zero. Special nonlinear partial differential equations are constructed whose nonlocal dynamics describes the behavior of the initial system in a neighborhood of the equilibrium state, which plays the role of a normal form. It should especially be noted that the constructed partial differential systems contain four spatial variables with boundary conditions for each of them.

Abstract: A logistic delay equation with diffusion, which is important in applications, is studied. It is assumed that all of its coefficients, as well as the coefficients in the boundary conditions, are rapidly oscillating functions of time. An averaged equation is constructed, and the relation between its solutions and the solutions of the original equation is studied. A result on the stability of the solutions is formulated, and the problem of local dynamics in the critical case is studied. An algorithm for constructing the asymptotics of the solutions and an algorithm for studying their stability are proposed. It is important to note that the corresponding algorithm contains both a regular and a boundary layer component. Meaningful examples are given.

The delay logistic equation with rapidly oscillating coefficients is studied. An averaged equation is constructed, and its dynamics is investigated. Algorithms relating the dynamical modes of the original and averaged equations are developed. It is established that the solutions are particularly sensitive to the choice of functions describing the oscillations of the delay coefficient.

The existence of periodic solutions of nonlinear equations generalizing logistic equations with delay is studied. The existence of sets of periodic solutions of two types is established. The stability and asymptotics of periodic solutions under changes of the parameters are studied.

The logistic equation supplemented with a summand characterizing delay is considered. The local and nonlocal dynamics of this equation are studied. For equations with delay, we use the standard Andronov—Hopf bifurcation methods and the asymptotic method developed by the author and based on the construction of special evolution equations defining the local dynamics of the equations containing delay. In addition, we study the existence and methods of constructing the asymptotics of nonlocal relaxation cycles. A comparison of the results obtained with those for the Hutchinson equation and some of its generalizations is given.

Parametrically generated optical patterns are investigated for finite and large-scale transverse aperture lasers. Standing and rotating patterns as well as periodic and chaotic pattern alternations are described in the framework of the amplitude equation formalism. Sensitive dependence on the geometrical size of the system is demonstrated even in the case of large-scale systems

Problems concerning the behavior of solutions of parabolic boundary value problems in a neighborhood of a cycle in near-critical cases are considered. As a consequence of smallness of the diffusion coefficients the critical cases are “infinite-dimensional.” Steady-state regimes in a neighborhood of a cycle are studied by means of the formalism of normalization methods.

Probable dislocation-type nucleation centers of a-martensite crystals with {259}-{3 10 15} habits, which are observed in iron alloys, are selected. It is demonstrated that 30° dislocations of the initial ? phase are most consistent with the wave pattern of the growth of such crystals. A particular dislocation nucleation center corresponds to each martensite crystal with a given morphology.

We study the stability of the solutions of boundary value problems for a certain class of Petrovskii-parabolic systems with sufficiently small diffusion coefficients. The dimension of the set of critical cases in the stability problem turns out to be infinite. We develop an efficient algorithm for studying stability. As examples we consider parabolic boundary value problems with delay and rapidly oscillating coefficients, the problem of parametric resonance under a double-frequency perturbation, and problems with variable leading terms and variable domain of definition. Bibliography: 21 titles.

• developed the methods of local analysis of dynamics. • studied the interaction of waves moving in different directions. • got special evolution nonlinear equations to describe the original equation dynamics. Using asymptotic methods we investigate a local in a steady state neighbourhood behavior of solutions in the Fermi–Pasta–Ulam model. Basing on the formalism of the normalization method we got special nonlinear evolution equations. The dynamics of these equations substantially defines the behavior of solutions of original equations. We made the conclusions about the interaction of waves moving in different directions. [ABSTRACT FROM AUTHOR]

The reform of February 19, 1861, was one of the key events in the history of Russia in the nineteenth century. It laid the groundwork for the vigorous development of capitalism, brought about radical changes in the economic structure of the country, and largely determined the path of political and socioeconomic development for a long time to come. "A capitalist Russia was coming to replace feudal Russia."1