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An infinite-dimensional torus where , is a space of sequences and is a natural integer lattice in is considered. We study a classical question in the theory of dynamical systems concerning the behavior of trajectories of a shift mapping on More precisely, sufficient conditions are proposed under which the -limit and -limit sets of any trajectory of the shift mapping on are empty. [ABSTRACT FROM AUTHOR]

We study the question, classical in the theory of dynamical systems, about the minimality of the shift on an infinite-dimensional torus; more precisely, the problem of finding sufficient conditions guaranteeing the absence of the minimality property is solved. [ABSTRACT FROM AUTHOR]

We consider a chain of in-series circuits; a constant-voltage source is connected to one of its ends, and a tunnel diode and a constant capacitor to another one. We show that a nonlinear system of ordinary differential equations in which the known buffer capacity phenomenon is realized serves as a mathematical model of the given generator. Namely, using the combination of analytic and numerical methods, we establish an unlimited growth of the number of coexisting attractors of this system as the number of circuits increases and the other parameters change appropriately. We also perform a comparative analysis of the local dynamics in the discrete and continuous circuits. [ABSTRACT FROM AUTHOR]

We consider special systems of ordinary differential equations, the so-called fully coupled networks of nonlinear oscillators. For a given class of systems, we propose methods that allow examining problems of the existence and stability of periodic two-cluster synchronization modes. For any of these modes, the set of oscillators falls into two disjoint classes. Within these classes, complete synchronization of oscillations is observed, and every two oscillators from different classes oscillate asynchronously. [ABSTRACT FROM AUTHOR]

We introduce a new mathematical model of a neural network with electrical connections. The model is a singularly perturbed system of delay differential–difference equations. Some methods for studying the existence and stability of periodic relaxation motions in this system are presented. In the case of diffusion and symmetric fully connected neural networks, we establish that the well-known buffering phenomenon can be realized in the corresponding mathematical models. [ABSTRACT FROM AUTHOR]

Special systems of ordinary differential equations—the so called fully coupled networks of nonlinear oscillators—are considered. For this class of systems, methods for analyzing the existence and stability of solutions of the traveling wave type are proposed. A feature of the proposed methods is that they use auxiliary delay systems for finding periodic solutions and for analyzing their stability properties. [ABSTRACT FROM AUTHOR]

An effective method is proposed for constructing specific examples of Anosov diffeomorphisms on the torus that are different from linear hyperbolic automorphisms. We introduce a special class of diffeomorphisms that are compositions of the well-known linear Arnold's cat map and some diffeomorphisms homotopic to the identity. Constructively verified sufficient hyperbolicity conditions are established for this class of mappings. [ABSTRACT FROM AUTHOR]

We consider an annular set of the form , where is a closed ball of the Banach space , is the infinite-dimensional torus (the direct product of a countable number of circles with the topology of coordinatewise uniform convergence). For a certain class of smooth maps , we establish sufficient conditions for the existence and stability of an invariant toroidal manifold of the form where is a continuous function of the argument . We also study the question of the -smoothness of this manifold for any natural . [ABSTRACT FROM AUTHOR]

A natural class of expansive endomorphisms of the infinite-dimensional torus (the Cartesian product of countably many circles with the product topology) is considered. The endomorphisms in this class can be represented in the form of the sum of a linear expansion and a periodic addition. The following standard facts of hyperbolic theory are proved: the topological conjugacy of any expansive endomorphism from the class under consideration to a linear endomorphism of the torus, the structural stability of , and the topological mixing property of on . [ABSTRACT FROM AUTHOR]

For an arbitrary expanding endomorphism of the class acting from to , where is an infinite-dimensional torus (the quotient space of some Banach space by an integer lattice), we establish the following standard assertions from the hyperbolic theory: the topological conjugacy of this endomorphism with the linear endomorphism of the torus and its structural stability, as well as the presence of the property of topological mixing for this endomorphism if the fundamental set of the torus is bounded. [ABSTRACT FROM AUTHOR]

We consider a system of two specially coupled differential-difference equations with delay in the coupling link. We establish that the system has a set of coexisting orbitally asymptotically stable solutions with the total number 2n, n ∈ ℕ, of bursts in the period; moreover, one of the oscillators has m bursts and the other has 2n − m bursts, m = 1,... , 2n−1. From the results obtained it follows that an additional delay leads to the appearance of coexisting attractors in the system with a given number of bursts in the period. Bibliography: 20 titles. Illustrations: 2 figures. [ABSTRACT FROM AUTHOR]

We consider the well-known Bazykin–Svirezhev model describing the predator–prey interaction. This model is a system of two nonlinear ordinary differential equations with a small parameter multiplying one of the derivatives. The existence and stability of a so-called relaxation cycle in such a system are studied. A peculiar feature of such a cycle is that as the small parameter tends to zero, its fast component changes in a -like manner, while the slow component tends to some discontinuous periodic function. [ABSTRACT FROM AUTHOR]

For distributed evolutionary dynamical systems of the "reaction-diffusion" and "reaction-diffusion-advection" types, we analyze the behavior of invariant numerical characteristics of the attractor as the diffusion coefficients decrease. We consider the phenomenon of multimode diffusion chaos, one of whose signatures is an increase in the Lyapunov dimensions of the attractor. For several examples, we perform broad numerical experiments illustrating this phenomenon. [ABSTRACT FROM AUTHOR]

A diffusion neuron model representing a system of , identical nonlinear delay differential equations coupled by linear diffusion terms is considered. It is shown that, with a suitable choice of the diffusion coefficient, the system has a set of stable relaxation cycles. [ABSTRACT FROM AUTHOR]

A family of mappings used in the numerical simulation of a delay logistic equation is considered. This equation finds wide applications in problems of mathematical ecology. The properties of solutions of these mappings and the original delay equation are compared. It is shown that the behavior of solutions of difference equations can be rather complex, while the delay logistic equation has only a stable equilibrium state or cycle. The constructed mappings can be used as models of population dynamics, so their study is of interest. [ABSTRACT FROM AUTHOR]

An arbitrary diffeomorphism f of class C1 acting from an open set U ⊂ ℝ m , m ≥ 2, into f (U) ⊂ ℝ m is considered. Sufficient conditions for such a diffeomorphism to admit a hyperbolic mixing attractor are obtained. [ABSTRACT FROM AUTHOR]

We consider the so-called bilocal neuron model, which is a special system of two nonlinear delay differential equations coupled by linear diffusion terms. The system is invariant under the interchange of phase variables. We prove that, under an appropriate choice of parameters, the system under study has a stable relaxation cycle whose components turn into each other under a certain phase shift. [ABSTRACT FROM AUTHOR]

A sufficiently general class of diffeomorphisms of the annulus (the direct product of a ball in Rk, k ≥ 2, by an m-dimensional torus) is studied. The so-called annulus principle, i.e., a set of sufficient conditions under which the diffeomorphisms of the class under study have a mixing hyperbolic attractor, is obtained. [ABSTRACT FROM AUTHOR]

A new mathematical model is proposed for a circular gene network representing a system of unidirectionally coupled ordinary differential equations. The existence and stability of special periodic motions (traveling waves) for this system is studied. It is shown that, with a suitable choice of parameters and an increasing number m of equations in the system, the number of coexisting traveling waves increases indefinitely, but all of them (except for a single stable periodic solution for odd m) are quasistable. The quasi-stability of a cycle means that some of its multipliers are asymptotically close to the unit circle, while the other multipliers (except for a simple unit one) are less than unity in absolute value. [ABSTRACT FROM AUTHOR]

We propose a new mathematical model of a repressilator, i.e., the simplest gene ring network consisting of three elements. The studied model is a three-dimensional system of ordinary differential equations depending on a single parameter. We study the existence and stability problems for relaxation periodic motion in this system. [ABSTRACT FROM AUTHOR]

The problem of density wave propagation governed by a logistic equation with delay and diffusion (Fisher-Kolmogorov-Petrovskii-Piskunov equation with delay) was studied. To analyze the qualitative behavior of solutions to this equation with periodic boundary conditions in the case of the diffusion parameter tending to zero, the normal form of the problem, i.e., the Ginzburg-Landau equation was constructed near the unit equilibrium. A numerical analysis of wave propagation showed that, for sufficiently small delays, this equation has solutions close to those of the standard Kolmogorov-Petrovskii-Piskunov equation. As the delay parameter increases, a decaying oscillatory component appears in the spatial distribution of the solution and, then, undamped (in time) and slowly propagating (in space) oscillations close to solutions of the corresponding boundary value problem with periodic boundary conditions are observed near the initial perturbation segment. [ABSTRACT FROM AUTHOR]

We study a special class of diffeomorphisms of an annulus (the direct product of a ball in ℝ, k ≥ 2, by an m-dimensional torus). We prove the so-called annulus principle; i.e., we suggest a set of sufficient conditions under which each diffeomorphism in a given class has an m-dimensional expanding hyperbolic attractor. [ABSTRACT FROM AUTHOR]

The three-dimensional nonlinear system of ordinary differential equations modeling the functioning of the simplest oscillatory genetic network, the so-called repressilator, is considered. The existence, asymptotics, and stability of the relaxation periodicmotion in this system are studied. [ABSTRACT FROM AUTHOR]

A new mathematical model of a one-dimensional array of FitzHugh-Nagumo neurons with resistive-inductive coupling between neighboring elements is proposed. The model relies on a chain of diffusively coupled three-dimensional systems of ordinary differential equations. It is shown that any finite number of coexisting stable invariant two-dimensional tori can be obtained in this chain by suitably increasing the number of its elements. [ABSTRACT FROM AUTHOR]

We consider cyclic chains of unidirectionally coupled delay differential-difference equations that are mathematical models of artificial oscillating gene networks. We establish that the buffering phenomenon is realized in these system for an appropriate choice of the parameters: any given finite number of stable periodic motions of a special type, the so-called traveling waves, coexist. [ABSTRACT FROM AUTHOR]

We consider nonlinear systems of differential-difference equations with delays that provide mathematical models for artificial complete genetic networks. We study problems of the existence and stability of special periodic motions referred to as two-cluster synchronization modes in these systems. [ABSTRACT FROM AUTHOR]

A new mathematical model for the electrical activity of the heart is proposed. The model represents a special singularly perturbed three-dimensional system of ordinary differential equations with one fast and two slow variables. A characteristic feature of the system is that its solution performs nonclassical relaxation oscillations and simultaneously undergoes a blue sky catastrophe bifurcation. Both these factors make it possible to achieve a phenomenological proximity between the time dependence of the fast component in the model and an ECG of the human heart. [ABSTRACT FROM AUTHOR]

We propose a mathematical model of a continuous annular chain of unidirectionally coupled generators given by some nonlinear advection-type hyperbolic boundary value problem. Such problems are constructed by a limit transition from annular chains of unidirectionally coupled ordinary differential equations with an unbounded increase in the number of links. We find that a certain buffering phenomenon is realized in our boundary value problem. Namely, we show that any preassigned finite number of stable periodic motions of the traveling-wave type can coexist in the model. [ABSTRACT FROM AUTHOR]

The article discusses the research on circular systems. Topics include unidirectionally coupled oscillators, attractors of circular systems, and Lyapunov dimensions. Euclidean norms, Hopf–Andronov bifurcation, and eigenvectors are also mentioned. The mathematical theorems used in the study are discussed.

A singularly perturbed system of ordinary differential equations with a fast and a slow variable is proposed, which is a modification of the well-known FitzHugh-Nagumo model from neuroscience. The existence and stability of a nonclassical relaxation cycle in this system are studied. The slow component of the cycle is asymptotically close to a discontinuous function, while the fast component is a δ-like function. [ABSTRACT FROM AUTHOR]

We introduce a new mathematical model of a circular neural network with unidirectional chemical bonds. The model is a singularly perturbed system of delay differential-difference equations. We study the existence and stability of relaxation periodic motions in the system. It is proved that the well-known buffer phenomenon can occur in the model. [ABSTRACT FROM AUTHOR]

We propose a new method for modeling the well-known phenomenon of 'bursting behavior' in neuron systems by invoking delay equations. Namely, we consider a singularly perturbed nonlinear difference-differential equation with two delays describing the functioning of an isolated neuron. Under a suitable choice of parameters, we establish the existence of a stable periodic motion with any prescribed number of spikes on a closed time interval equal to the period length. [ABSTRACT FROM AUTHOR]

We consider special systems of ordinary differential equations, so-called circular unidirectional chains. For this class of systems, we develop a new method for studying the existence and stability problems for periodic solutions. A feature of the proposed approach is that some auxiliary systems with delay are used in both seeking cycles and analyzing their stability properties. We illustrate the relevance of this approach with a concrete example of a circular Hopfield neural network [ABSTRACT FROM AUTHOR]

A singularly perturbed scalar nonlinear differential-difference equation with two delays is considered that is a mathematical model of an isolated neuron. It is shown that a one-dimensional chain of diffusively coupled oscillators of this type exhibits the well-known buffer phenomenon. Specifically, as the number of chain links increases consistently with decreasing diffusivity, the number of coexisting stable periodic motions in the chain grows indefinitely. [ABSTRACT FROM AUTHOR]

The article presents a study regarding the use of differential-difference equation to examine the electrical activity of neural systems. It mentions that differential-difference equation is used as a mathematical model to formulate the calculation of the solution. It states that a buffer phenomenon which is natural in neural systems can be observed with the proper choice of parameters in the equation. Furthermore, mathematical solutions showing the illustration of differential-difference equation were also discussed.

The mathematical model considered here of a neuron system is a chain of an arbitrary number m ≥ 2 of diffusion-coupled singularly perturbed nonlinear delay differential equations with Neumann-type conditions at the endpoints. We study the existence, asymptotic behavior, and stability of relaxation periodic solutions of this system. [ABSTRACT FROM AUTHOR]

A singularly perturbed system of nonlinear delay differential equations that models the diffusion interaction of two neurons is considered. We study the existence and stability of relaxation periodic motions in this system. [ABSTRACT FROM AUTHOR]

Asymptotic and numerical analysis of relaxation self-oscillations in a three-dimensional system of Volterra ordinary differential equations that models the well-known Belousov reaction is carried out. A numerical study of the corresponding distributed model-the parabolic system obtained from the original system of ordinary differential equations with the diffusive terms taken into account subject to the zero Neumann boundary conditions at the endpoints of a finite interval is attempted. It is shown that, when the diffusion coefficients are proportionally decreased while the other parameters remain intact, the distributed model exhibits the diffusion chaos phenomenon; that is, chaotic attractors of arbitrarily high dimension emerge. [ABSTRACT FROM AUTHOR]

We consider a scalar singularly perturbed nonlinear delay differential-difference equation modeling an individual neuron. We study the existence, asymptotics, and stability of its relaxation cycle. [ABSTRACT FROM AUTHOR]

We suggest a phenomenological model for the development of turbulence in the form of the nonlinear Klein-Gordon equation perturbed by nonconservative disturbances. Combining analytic and numerical methods, we establish that the transition to turbulence in this equation can follow both the Landau and the Landau-Sell scenarios. As is known, the first scenario is related to a cascade of bifurcations of stable invariant tori of increasing dimensions. The other scenario is related to a chaotic attractor whose Lyapunov dimension increases indefinitely under variation of a certain control parameter. [ABSTRACT FROM AUTHOR]

We establish conditions under which three-dimensional relaxational systems of the form where 0 ≤ ε ≪ 1, |µ| ≪ 1, and f, g ∈ C ∞, exhibit the so-called blue sky catastrophe [the appearance of a stable relaxational cycle whose period and length tend to infinity as µ tends to some critical value µ*( ε), µ*(0) = 0]. [ABSTRACT FROM AUTHOR]

The article discusses the solution of the separation of motions in a semistable cycle equation. An arbitrary system F of ordinary differential equations has been considered. The two-dimensional system F0 that is the restriction of the original system F to the center manifold Wc (L0) of the cycle L0 were introduced.

We study the problem of attractors of the two-dimensional mapping where 0 < µ ≪ 1 and q 1, q 2 > 0. This mapping is the mathematical model of a self-excited oscillator with relay amplifier and a part of the long transmission line without distortions in the feedback circuit. We prove that, in the system under study, there coexist stable cycles with arbitrarily large periods as the parameter µ decreases properly. We also show that the total number of these cycles increases without bound as µ → 0, i.e., the buffer phenomenon is realized. [ABSTRACT FROM AUTHOR]

The article studies the limit values of Mel’nikov Functions on periodic orbits. The statement of the problem considers the two-dimensional non-autonomous system and restrictions under which the system will be studied are stated for the limit Hamiltonian system. The article discusses various theorems of differential equation satisfying the conditions and their proof.

In the present paper, we suggest a modification of the well-known Benettin algorithm [1] for the computation of the higher Lyapunov exponent of a chaotic attractor. Recall that the abovementioned algorithm involves the renormalization of the vector of initial conditions at equidistant points in time (the time interval T is given) for the integration of the variational system along a trajectory belonging to the attractor. The main idea of our version of the method is to make the system itself to choose the renormaliza-tion times according to a certain rule, i.e., dynamically. [ABSTRACT FROM AUTHOR]

Focuses on chaotic buffering property in chains of coupled oscillators. Coexisting factors; Construction of various chains of coupled oscillators; Schrodinger equations.