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Изучается модель с запаздыванием полупроводникового лазера. Исследована устойчивость состояния равновесия, выделены бифуркационные значения параметров. Как оказалось, возникающие критические случаи имеют бесконечную размерность. В ситуациях, когда значения параметров близки к критическим, построены уравнения первого приближения для определения асимптотических разложений амплитуд решений. Эти уравнения имеют вид нелинейных краевых задач параболического типа, содержащих в ряде случаев интегральные слагаемые в нелинейности. Приведены асимптотические формулы, связывающие решения исходной модели и построенных краевых задач.

We study the dynamics of the closed chain of a large number of coupled lasers. The coupling between elements is supposed to be unidirectional. The distributed integro-differential model is proposed which takes into account the delay due to the optoelectronic conversion of signals. The bifurcation value of the coupling coefficient is obtained, at which the stationary state of elements in the chain becomes unstable. It is shown that the critical case has infinite dimension if the number of elements in the chain tends to infinity. A two-dimensional complex Ginzburg-Landau equation with convection is obtained as a quasi-normal form. We get the homogeneous periodic solutions of the quasi-normal which correspond to inhomogeneous traveling waves in a distributed model. Such solutions can be interpreted as phase-synchronized regimes in the chain of coupled lasers.

Рассматриваются замкнутые в кольца цепочки уравнений Ван-дер-Поля и цепочки из систем двух уравнений Ван-дер-Поля первого порядка. Предполагается, что связи являются однородными и количество элементов цепочек достаточно велико. Естественным образом осуществляется переход к зависящим непрерывно от пространственной переменной функциям. Исследуется поведение при $t\to\infty$ всех решений таких цепочек с достаточно малыми по норме начальными условиями. Выделены критические случаи в задаче об устойчивости и показано, что все они имеют бесконечную размерность. Построены специальные нелинейные краевые задачи параболического типа, не содержащие малых параметров, которые играют роль нормальных форм. Их локальная динамика определяет поведение решений исходных краевых задач с двумя пространственными переменными. Сформулированы условия, при которых динамические свойства обеих цепочек близки друг к другу. Установлено, что в ряде случаев динамика цепочек систем уравнений Ван-дер-Поля оказывается существенно сложнее и разнообразнее по сравнению с динамикой цепочек уравнений Ван-дер-Поля второго порядка.

Рассматривается модернизированное уравнение Камассы-Холма с периодическими краевыми условиями. Квадратичные нелинейности в этом уравнении существенно отличаются от нелинейностей в классическом уравнении Камассы-Холма, но обладают всеми его в некотором смысле основными свойствами. Исследуются так называемые нерегулярные решения, т. е. такие, которые являются быстро осциллирующими по пространственной переменной. Исследуется вопрос о построении асимптотики периодических по времени решений и более сложных решений, главные части асимптотических разложений которых являются многочастотными. Изучаются задача о возможности компактной записи этих асимптотических разложений и вопрос о сведении задачи о построении главных членов асимптотик к анализу решений специальных нелинейных краевых задач. Показано, что это возможно лишь для классического уравнения Камассы-Холма.

Рассматривается семейство отображений, которые используются при численном моделировании логистического уравнения с запаздыванием. Это уравнение находит широкое применение в задачах математической экологии. Вместе с тем, представленные отображения сами по себе могут служить моделями динамики популяций, поэтому их изучение представляет значительный интерес. В работе сопоставляются свойства траекторий данных отображений и исходного уравнения с запаздыванием. Показано, что поведение решений отображений может быть достаточно сложным, в то время как логистическое уравнение с запаздыванием имеет лишь устойчивое состояние равновесия или цикл.

Рассмотрено важное для приложений логистическое уравнение с запаздыванием и диффузией. Предполагается, что все его коэффициенты, а также коэффициенты в граничных условиях являются быстро осциллирующими функциями времени. Построено усредненное уравнение и рассмотрен вопрос о связи между его решениями и решениями исходного уравнения. Сформулирован результат об устойчивости решений и изучена задача о локальной динамике в критическом случае. Предложен алгоритм построения асимптотики решений и алгоритм исследования устойчивости. Важно отметить, что соответствующий алгоритм содержит как регулярные, так и погранслойные составляющие. Приведены содержательные примеры. Библиография: 23 названия.

—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 analyze the dynamics of a class B laser with optoelectronic delayed feedback and periodic modulation of the delay time. For the delay-differential model, we describe asymptotically spike regimes with various properties, namely, with inter-spikes intervals greater than delay, less delay, and a mixed type. Sets of initial conditions in an infinite-dimensional phase space are determined for each type of pulsed solutions. The dynamics of each regime is described by the dynamics of a certain finite-dimensional mapping. By computing the maps we obtain the modulation parameters at which the chaotic spiking of desired structure is realized. We demonstrate bistability of chaotic spiking, their intermittence and merge.

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.

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.

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 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.

A new asymptotic method for investigating complex relaxation oscillations in a system with delay is offered. By applying it, we can reduce the problem of predator–prey system dynamics to that of one-dimensional maps analysis. Some conclusions of a biological nature based on asymptotic analysis are made.

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.

Synchronization and synergetic effects in coupled systems play a significant role in nature and artificial networks. They are under permanent investigation in view of various applications, see for example [55] and references therein. The detailed review of coupled lasers dynamics is presented in [56]. In this chapter, using above elaborated asymptotic method, we study relaxation oscillations in coupled systems whose elements are whole lasers and laser cavity modes. Both spiking and rectangular pulsing are considered in such systems.

In this chapter we consider dynamics of a laser with non-linear intracavity element, whose loss can be controlled by the density of the generated radiation. An example of such an element is Pockels cell. The scheme of the device is given in Fig. 4.1.

Low-frequency relaxation oscillations are well-known phenomenon in a laser without additional elements.

Lasing control by means of pumping current modulation via the feedback (FB) circuit has been first applied in laser diodes [46, 49, 50] and inCO2 lasers [47, 48]. Such optoelectronic control can stabilize the stationary generation as well as the generation of short pulses following in period correlated with the delay time in the FB circuit. The scheme of the device is given in Fig. 3.1.

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.

A local dynamics for class-B laser systems with optoelectronic feedback is considered in the case of a very long time delay. We determine the conditions for the bifurcation of infinite dimension. It is shown that the (ordinary) delay system can be described by a systematically derived complex Ginzburg–Landau equation without delay terms. The complicated quasi-periodic dynamics near equilibrium can be demonstrated analytically on the basis of space–time representation.

A local dynamics for class B laser systems with optoelectronic and optical time-delayed feedback is considered. In detail, we determine the conditions for Hopf bifurcation and multiple bifurcations. The complicated quasiperiodic dynamics near equilibrium is demonstrated analytically.

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.

We present a theoretical analysis of the dynamics of lasers with delayed optical feedback where the time delay is periodically varied due to vibration of external mirror. Parameter regions have found of stable generation, periodic and quasiperiodic oscillations in dependence on characteristics of vibration.

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.