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A piecewise-smooth second-order singularly perturbed differential equation whose right-hand side is a nonlinear function with a discontinuity on some curve is investigated. This is a new class of problems in the case where the degenerate equation has a multiple root on the left-hand side of the curve which separates the domain and an isolated root on the right-hand side of that curve. The asymptotics of a solution with an internal layer near a point on the discontinuous curve and the transition point is constructed. The method to construct the internal layer function is proposed. The behavior of the solution in the internal layer consisting of four zones essentially differs from the case of isolated roots. For sufficiently small parameter values, the existence of a smooth solution with an internal layer from the multiple root of the degenerate equation to the isolated root in the neighborhood of a point on the discontinuous curve is proved. The method can be shown to be effective in the given example.

We consider a boundary value problem for an ordinary singularly perturbed second-order differential equation whose right-hand side is a nonlinear function with a discontinuity along some curve that is independent of the small parameter. For this problem, we study the existence of a smooth solution with steep gradient in a neighborhood of some point lying on this curve. The point itself and an asymptotic representation for the solution are to be determined. The existence theorem is proved by the method of matching asymptotic expansions. To this end, we use theorems on existence of solutions of boundary value problems for singularly perturbed equations and methods for constructing asymptotic approximations to these solutions.

We consider a singularly perturbed boundary value problem for a second-order ordinary differential equation with nonlinear right-hand side containing functions of delayed argument. We prove the existence of a solution with a transition layer that has a more sophisticated structure than the ones studied before and construct a uniform asymptotic approximation to this solution with respect to a small parameter. Vasil’eva’s method is used when constructing the asymptotic approximation, while the existence theorem is proved by combining the matching method and the asymptotic differential inequality method. Conditions for the existence of a solution with monotone internal transition and boundary layers are stated. An example illustrating the class of problems studied here is given.

A Type of second-order Dirichlet problems of Tikhonov system with piecewise-continuous right hand side is studied. By using the multiscale theory and the theory of contrast structures, a first-order continuous, uniform and effective asymptotic solution of the problem is constructed. Existence of the solution is proved and the remainder is estimated. An illustrative example for explaining this method is also given.

A singularly perturbed boundary-value problem for a nonlinear stationary equation of reaction-diffusion-advection type is studied. A new class of problems with discontinuous advective and reactive terms is considered. The existence of contrast structures in problems of this type is proved, and an asymptotic approximation of the solution with an internal transition layer of arbitrary order of accuracy is obtained.

A singularly perturbed boundary value problem for a second-order quasilinear ordinary differential equation is studied. We consider a new class of problems in which the nonlinearities experience discontinuities, which leads to the appearance of sharp transition layers in a neighborhood of the points of discontinuity. The existence of solutions is proved, and their asymptotic expansion with an internal transition layer is constructed.

A spatio-temporal model of megacity development that treats the megacity as an active medium is presented. From our point of view, it is advisable to consider the process of urban ecosystem development from the standpoint of the theory of autowave self-organization in active media. According to this concept, the urban ecosystem is considered as interacting with each other&rsquo, s natural and anthropogenic subsystems with significant heterogeneity of areas affected by human intervention and urban geobiocoenoses. The model is based on the general principles of active medium dynamics, therefore, it is universal for any object to be considered an active medium. The only difference when using the model to predict the development of urban ecosystems in countries with different socio-economic and political prerequisites is the variety of parameters included in the model, i.e., the activation parameter, the autowave process inhibitors, and the characteristic scales of the activator and inhibitor. The model was tested on the example of Moscow expansion in the period of 1952&ndash, 1968 and showed good agreement with the map data. By means of the model, a prediction of Shanghai and surrounding territory development until 2030 was made.

This paper investigates a two-point boundary value problem for a second-order singularly perturbed ordinary differential equation in the case of multiple roots of the degenerate equation. This is a new class of problems, namely, problems with discontinuous nonlinear terms on the right-hand side of the equation, which leads to the formation of a multizonal interior transitional layer in a neighborhood of the discontinuity point. For sufficiently small parameter values, the existence of a smooth solution is proved, and its asymptotic expansion is constructed, showing that this solution qualitatively differs from the case when the degenerate equation has simple roots.

In this paper, we consider the Dirichlet boundary value problem for a singularly perturbed reaction-diffusion equation with discontinuous reactive term. We use the asymptotic analysis to construct the formal asymptotic approximation of the solution with internal and boundary layers. The internal layer is located in the vicinity of a curve of the discontinuous reactive term. By using sufficiently precise lower and upper solutions, we prove the existence of a periodic solution and estimate the accuracy of its asymptotic approximation.

We investigate the step-type contrast structure for high dimensional Tikhonov system with Neumann boundary conditions. We not only propose a key condition with the existence of the number of mutually independent first integrals under which there exists a step-type contrast structure, but also determine where an internal transition time is. Using the method of boundary function, we construct the formal asymptotic solution and give the analytical expression for the higher order terms. At the same time, the uniformly valid asymptotic expansion and the existence of such an available step-type contrast structure are obtained by sewing connection method.

Consideration was given to the problem of synchronizing the master and slave nonlinear discrete systems interrelated through a communication-constrained channel. The effect of quantization error was examined, and an efficient quantization scheme was proposed such that the error of transmission vanishes under constrained communications. The desired data transmission rate was calculated. A control law such that the error of quantized synchronization is input-to-state stable with respect to the transmission error was constructed in terms of the linear matrix inequalities. The proposed scheme supports full quantized synchronization of the master and slave systems. The result obtained was exemplified by a model of simply connected flexible joint robot.

We consider a singularly perturbed boundary value problem for a differential equation with a retarded and a deviating argument. By using the method of boundary functions and the sewing method, we find not only a continuous but also a smooth solution of the problem. We prove the existence of a solution with an internal transition layer. A graphical numerical example is presented.

This note addresses a controlled synchronization problem for chaotic systems involving a communication network with data packet dropout. A chaotic master system and its slave system are connected with a controller via a limited channel and data packet dropout is modeled as Bernoulli process. Then, a coder-decoder pair is designed with the controller such that the master system and the slave system are completely synchronized, not necessarily bounded for synchronization. Necessary data capacity of a channel is explicitly stated. Finally, a numerical example for the 3-double-scroll chaotic system is applied to illustrate the effectiveness of the obtained result.

We consider a nonlinear singularly perturbed boundary value problem with delay. By using the method of boundary functions and the theory of contrast structures, we prove the existence of a smooth solution with an internal transition layer and construct its uniform asymptotic expansion in a small parameter.

The quantized feedback stabilization problem for a class of linear system with nonlinear disturbances is addressed, in which the system and controller are connected via a communication channel. In this case, the effect of quantization errors is studied. A practical quantized scheme is designed such that the transmission error decays to zero exponentially. Meanwhile, a sufficient condition in terms of linear matrix inequality for input-to-state stability (ISS) of the system is presented with regard to the transmission error. Therefore, a quantized stabilization of the system is guaranteed based on the ISS property. A simulation example for the single-link flexible joint robot is presented to demonstrate the effectiveness of the result.

In this paper, we address the existence and asymptotic analysis of higher-dimensional contrast structure of singularly perturbed Dirichlet problem. Based on the existence, an asymptotical analysis of a steplike contrast structure (i.e., an internal transition layer solution) is studied by the boundary function method via a proposed smooth connection. In the framework of this paper, we propose a first integral condition, under which the existence of a heteroclinic orbit connecting two equilibrium points is ensured in a higher-dimensional fast phase space. Then, the step-like contrast structure is constructed, and the internal transition time is determined. Meanwhile, the uniformly valid asymptotical expansion of such an available step-like contrast structure is obtained. Finally, an example is presented to illustrate the result.

In this paper we investigate the step-type contrast structure of a second-order semilinear differential equation with integral boundary conditions. The asymptotic solution is constructed by the boundary function method, and the uniform validity of the formal solution is proved by the theory of differential equalities. 2000 Mathematics Subject Classification. 34B15; 34E05; 34E20.

A novel approach about iterative homotopy harmonic balancing is presented to determine the periodic solution for a strongly nonlinear oscillator. This approach does not depend upon the small/large parameter assumption and incorporates the salient features of both methods of the parameter-expansion and the harmonic balance. Importantly, in obtaining the higher-order analytical approximation, all the residual errors are considered in the process of every order approximation to improve the accuracy. With this procedure, the higher-order approximate frequency and corresponding periodic solution can be obtained easily. Comparison of the obtained results with those of the exact solutions shows the high accuracy, simplicity, and efficiency of the approach. The approach can be extended to other nonlinear oscillators in engineering and physics.

The existence of a step-like contrast structure for a class of high-dimensional singularly perturbed system is shown by a smooth connection method based on the existence of a first integral for an associated system. In the framework of this paper, we not only give the conditions under which there exists an internal transition layer but also determine where an internal transition time is. Meanwhile, the uniformly valid asymptotic expansion of a solution with a step-like contrast structure is presented.