Search

—The asymptotics of spatially inhomogeneous periodic solutions of the complex, spatially distributed Hutchinson equation with periodic boundary conditions are presented. It is shown that such solutions are observable in a numerical experiment.

We considered the process of density wave propagation in the logistic equation with diffusion, such as Fisher–Kolmogorov equation, and arguments deviation. Firstly, we studied local properties of solutions corresponding to the considered equation with periodic boundary conditions using asymptotic methods. It was shown that increasing of period makes the spatial structure of stable solutions more complicated. Secondly, we performed numerical analysis. In particular, we considered the problem of propagating density waves interaction in infinite interval. Numerical analysis of the propagating waves interaction process, described by this equation, was performed at the computing cluster of YarSU with the usage of the parallel computing technology—OpenMP. Computations showed that a complex spatially inhomogeneous structure occurring in the interaction of waves can be explained by properties of the corresponding periodic boundary value problem solutions by increasing the spatial variable changes interval. Thus, the complication of the wave structure in this problem is associated with its space extension.

A system of two logistic equations with delay coupled by delayed control has been considered. It has been shown that, in the case of a fairly large delay control coefficient, the problem of the dynamics of the initial systems has been reduced to investigating the nonlocal dynamics of special families of partial differential equations that do not contain small and large parameters. New interesting dynamic phenomena are discovered based on the results of numerical analysis. Systems of three logistic delay equations with two types of diffusion relations have been considered. Special families of partial differential equations that do not contain small and large parameters have also been constructed for each of these systems. The research results for the dynamic properties of the original equations have been presented. It has been shown that the difference in the dynamics of the considered systems of three equations may be of a fundamental nature.

We consider a second-order linear differential equation of with a small factor at the highest derivative. We study the asymptotic behavior of eigenvalues of the first boundary-value problem (the Dirichlet problem) under the assumption that turning points (points where the coefficient at the first derivative equals to zero) exist. It has been shown that only the behavior of coefficients of the equation in a small neighborhood of the turning points is essential. The main result is a theorem on the limit values of the eigenvalues of the first boundary-value problem.

We consider the problem of the density wave propagation of a logistic equation with the deviation of the spatial variable and diffusion (the Fisher–Kolmogorov equation with the deviation of the spatial variable). The Ginzburg–Landau equation was constructed in order to study the qualitative behavior of the solution near the equilibrium state. We analyzed the profile of the wave equation and found conditions for the appearance of oscillatory regimes. The numerical analysis of the wave propagation shows that, for a fairly small spatial deviation, this equation has a solution similar to that the classical Fisher–Kolmogorov equation. An increase in this spatial deviation leads to the existence of the oscillatory component in the spatial distribution of solutions. A further increase in the spatial deviation leads to the destruction of the traveling wave. This is expressed in the fact that undamped spatiotemporal fluctuations exist in a neighborhood of the initial perturbation. These fluctuations are close to the solution of the corresponding boundary value problem with periodic boundary conditions. Finally, when the spatial deviation is large enough we observe intensive spatiotemporal fluctuations in the whole area of wave propagation.

The local dynamics of systems of difference and singularly perturbed differential-difference equations is studied in the neighborhood of a zero equilibrium state. Critical cases in the problem of stability of its state of equilibrium have infinite dimension. Special nonlinear evolution equations, which act as normal forms, are set up. It is shown that their dynamics defines the behavior of solutions to the initial system.

We consider the scalar complex equation with spatially distributed control. Its dynamical properties are studied by asymptotic methods when the control coefficient is either sufficiently large or sufficiently small and the function of distribution is either almost symmetric or significantly nonsymmetric relative to zero. In all cases we reduce original equation to quasinormal form – the family of special parabolic equations, which do not contain big and small parameters, which nonlocal dynamics determines the behaviour of solutions of the original equation.

Dynamics of laser with time-variable delayed feedback is analyzed in the neighborhood of the equilibrium. For the system, averaged over a rapidly variable, we obtain parameters at which the stationary state is stable. Stabilization of the stationary state due to modulation of the delay can be successful (unsuccessful) in domains adjacent to super (sub-) critical Hopf bifurcation boundaries. In a vicinity of the bifurcation points, stable and unstable periodic solutions are asymptotically described in dependence on the modulation frequency.

Asymptotic solutions of parabolic boundary value problems are studied in a neighborhood of both an equilibrium state and a cycle in near-critical cases which can be considered as infinite-dimensional due to small values of the diffusion coefficients. Algorithms are developed to construct normalized equations in such situations. Principle difference between bifurcations in two-dimensional and one-dimensional spatial systems is demonstrated.

In this paper the local dynamics of systems of nonlinear PDEs with small diffusion is studied. The main feature of these systems lies in the fact that the dimension of a critical case in the stability problem for an equilibrium state is equal to infinity. Algorithms that reduce the initial problem to the analysis of nonlocal dynamics of special evolution equations playing the role of normal forms are developed.

We consider the problem of density wave propagation in a logistic equation with delay and diffusion (Fisher-Kolmogorov equation with delay). A Ginzburg-Landau equation was constructed in order to study the qualitative behavior of the solution near the equilibrium state. The numerical analysis of wave propagation shows that for a sufficiently small delay this equation has a solution similar to the solution of a classical Fisher-Kolmogorov equation. The delay increasing leads to existence of the oscillatory component in spatial distribution of solutions. A further increase of delay leads to destruction of the traveling wave. That is expressed in the fact that undamped spatio-temporal fluctuations exist in a neighborhood of the initial perturbation. These fluctuations are close to the solution of the corresponding boundary value problem with periodic boundary conditions. Finally, when the delay is sufficiently large we observe intensive spatio-temporal fluctuations in the whole area of wave propagation.

This paper considers the behavior of solutions from the neighborhood of an equilibrium state of nonlinear two-component parabolic problems with diffusion matrixes of one or two eigenvalues to zero. It has been shown that problems related to stability have infinite dimension. Reported here is the development of an algorithm that constructs universal families of nonlinear boundary-value problems which do not contain small parameters and whose nonlocal dynamics describes local dynamics of original boundary-value problems. In addition, an exhaustive set of universal systems for two-component parabolic equations is presented. It is concluded that a hyper multistability phenomenon is one characteristic of these systems.

Pattern formation described by differential-difference equations with diffusion is investigated. It is shown that an arbitrarily small diffusion induces space-time turbulence just at the instability threshold of the homogeneous stationary solution. We prove this property by deriving a complex Ginzburg-Landau equation on the basis of normal form analysis. Well above threshold, such turbulent structures give way to synchronized states ordered by spirals and targets. This secondary instability can be understood with an asymptotic method representing the system as a cellular automaton network.