Search

We study a quite natural class of diffeomorphisms on , where is the infinite-dimensional torus (the direct product of countably many circles endowed with the topology of uniform coordinatewise convergence). The diffeomorphisms under consideration can be represented as the sums of a linear hyperbolic map and a periodic additional term. We find some constructive sufficient conditions, which imply that any in our class is hyperbolic, that is, an Anosov diffeomorphism on . Moreover, under these conditions we prove the following properties standard in the hyperbolic theory: the existence of stable and unstable invariant foliations, the topological conjugacy to a linear hyperbolic automorphism of the torus and the structural stability of .

We consider an annular set of the form $$K=B\times \mathbb{T}^{\infty}$$ , where $$B$$ is a closed ball of the Banach space $$E$$ , $$\mathbb{T}^{\infty}$$ 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 $$\Pi\colon K\to K$$ , we establish sufficient conditions for the existence and stability of an invariant toroidal manifold of the form $$A=\{(v, \varphi)\in K: v=h(\varphi)\in E,\,\varphi\in\mathbb{T}^{\infty}\},$$ where $$h(\varphi)$$ is a continuous function of the argument $$\varphi\in\mathbb{T}^{\infty}$$ . We also study the question of the $$C^m$$ -smoothness of this manifold for any natural $$m$$ .

A natural class of expansive endomorphisms $$G\in C^1$$ of the infinite-dimensional torus $$\mathbb{T}^{\infty}$$ (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 $$G$$ from the class under consideration to a linear endomorphism of the torus, the structural stability of $$G$$ , and the topological mixing property of $$G$$ on $$\mathbb{T}^{\infty}$$ .

For an arbitrary expanding endomorphism of the class $$C^1 $$ acting from $$\mathbb {T}^{\infty } $$ to $$\mathbb {T}^{\infty } $$ , where $$\mathbb {T}^{\infty } $$ 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.

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 $$\delta $$ -like manner, while the slow component tends to some discontinuous periodic function.

An arbitrary diffeomorphism of an annular set of the form is considered, where is a ball in a Banach space and is a (finite- or infinite-dimensional) torus. A system of effective sufficient conditions is proposed which ensure that has a global attractor that can be represented as a generalized solenoid, that is, the inverse limit , where is an expanding linear endomorphism of the torus . Furthermore, the restriction is topologically conjugate to a shift map of the solenoid. Bibliography: 25 titles.

—We consider an arbitrary C1 diffeomorphism f that acts from an open subset U of a Riemannian manifold M of dimension m, m ≥ 2, to f(U) ⊂ M. Let A be a compact f-invariant (i.e., f(A) = A) subset in U. We propose various sufficient conditions under which A is a hyperbolic set of f.

A diffusion neuron model representing a system of $$m,$$ $$m \geqslant 2$$ , 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 $$m$$ stable relaxation cycles.

An arbitrary diffeomorphism f of class C1 acting from an open set $$\mathcal{U}\subset \mathbb{R}^{m}$$ , m ≥ 2, into $$f(\mathcal{U})\subset \mathbb{R}^{m}$$ is considered. Sufficient conditions for such a diffeomorphism to admit a hyperbolic mixing attractor are obtained.

A system of two generalized Hutchinson’s equations coupled by linear diffusion terms is considered. It is established that for an appropriate choice of parameters, the system has a stable relaxation cycle whose components turn into each other under a certain phase shift. A number of additional properties of this cycle are presented that allow one to interpret it as a self-organization mode.

Nonsingular endomorphisms of the m-torus $$\mathbb{T}$$ m, m ≥ 2, which are C1 perturbations of linear hyperbolic endomorphisms are considered. Sufficient conditions for such maps to be hyperbolic (i.e., belong to the class of Anosov endomorphisms) are found.

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.

A sufficiently general class of diffeomorphisms of the annulus (the direct product of a ball in $$\mathbb{R}^{k}$$ , 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.

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.

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.

We consider mappings of the m-dimensional torus Tm (m ≥ 2) that are C 1-perturbations of linear hyperbolic automorphisms. We obtain sufficient conditions for such mappings to be one-to-one hyperbolic mappings (i.e., Anosov diffeomorphisms). These results are used to study the blue-sky catastrophe related to the vanishing of a saddle-node invariant torus with a quasiperiodic winding in a system of ordinary differential equations.

We study a special class of diffeomorphisms of an annulus (the direct product of a ball in ℝ k , 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.

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.

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.

A modification of the well-known FitzHugh–Nagumo model from neuroscience has been proposed. This model is a singularly perturbed system of ordinary differential equations with a fast variable and a slow variable. The existence and stability of a nonclassical relaxation cycle in this system have been studied. The slow component of the cycle is asymptotically close to a discontinuous function, while the fast component is a δ-like function. A one-dimensional circle of unidirectionally coupled neurons has been considered. The existence of an arbitrarily large number of traveling waves for this chain has been shown. In order to illustrate the increase in the number of stable traveling waves, numerical methods were involved.

We consider circular chains of unidirectionally coupled ordinary differential equations which are mathematical models of artificial gene networks. We study the problems of the existence and stability of special periodic solutions, the so-called travelling waves, in these chains. We establish that the number of such periodic solutions grows unboundedly as the number of links in the chain grows. However, at most one of these travelling waves can be stable. We give an explicit algorithm for choosing the stable cycle.

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.

A certain special class of diffeomorphisms of an 'annulus' (equal to the Cartesian product of a ball in , , and a circle) is investigated. The so-called annulus principle is established, that is, a list of sufficient conditions ensuring that each diffeomorphism in this class has a strange hyperbolic attractor of Smale-Williams solenoid type is given.Bibliography: 20 titles.

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.

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.

In this paper, we consider so-called finite-dimensional flutter systems, i.e., systems of ordinary differential equations that come about, first, from the Galerkin approximation of certain boundary value problems of the aeroelasticity theory and, second, in a number of radiophysics applications. Small parametric oscillations of these equations in the case of 1: 3 resonance are studied. By combining analytical and numerical methods, it is found that this resonance can cause the hard excitation of oscillations. Namely, for flutter systems, there is shown the possibility of emerging, in parallel with the stable equilibrium zero state, both stable invariant tori of arbitrary finite dimension and chaotic attractors.

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.

We introduce a new object, namely, a system of ordinary dif- ferential equations which is a mathematical model of a ring-like chain of unidirectionally connected RCL-generators. To study periodic solutions of travelling wave type of this system, some special tricks are used which reduce the existence and stability problems for cycles to the investigation of auxiliary delay equations. Using this approach, we establish that the num- ber of stable travelling waves simultaneously existing in the chain increases unboundedly as the number of links of the chain increases, that is, the well-known buer phenomenon occurs. phenomenon.

We consider a special class of abstract nonlinear equations of hyperbolic type that contains the main boundary value problems of panel flutter theory. For these equations, we study the existence and stability of parametric oscillations bifurcating from the zero equilibrium in the case of small damping.

We introduce a mathematical model of a continual circular chain of unidirectionally coupled oscillators. It is a nonlinear hyperbolic boundary value problem obtained from a circular chain of unidirectionally coupled ordinary differential equations in the limit as the number of equations indefinitely increases. We study the attractors of this boundary value problem. Combining analytic and numerical methods, we establish that one of the following two alternatives takes place in this problem: either the buffer phenomenon (unbounded accumulation of stable periodic motions) or chaotic attractors of arbitrarily high Lyapunov dimensions.

Some special classes of relaxation systems are introduced, with one slow and one fast variable, in which the evolution of the slow compo- nent x(t) in time is described by an ordinary differential equation, while the evolution of the fast component y(t) is described by a Volterra-type differential equation with delay y(t − h), h = const > 0, and with a small parameter e > 0 multiplying the time derivative. Questions relating to the existence and stability of impulse-type periodic solutions, in which the x-component converges pointwise to a discontinuous function as e → 0 and the y-component is shaped like a δ-function, are investigated. The results obtained are illustrated by several examples from ecology and laser theory. Bibliography: 11 titles.

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.

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.

The paper looks at quite a wide class of nonlinear evolution equations in a Banach space, including the typical boundary value problems for the main wave equations in mathematical physics (the telegraph equation, the equation of a vibrating beam, various equations from the elastic stability and so on). For this class of equations a unified approach to the bifurcation of invariant tori of arbitrary finite dimension is put forward. Namely, the problem of the birth of such tori from the zero equilibrium is investigated under the assumption that in the stability problem for this equilibrium the situation arises close to an infinite-dimensional degeneracy. Bibliography: 28 titles.

We consider special systems of ordinary differential equations, namely, the so-called ring chains of unidirectionally coupled oscillators. A new method is developed for the investigation of the problems of existence and stability of periodic solutions for this class of systems. As a specific feature of this approach, we can mention the use of certain auxiliary delay systems for the determination of cycles and the analysis of their properties. The proposed method is illustrated by a specific example.

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.

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

The article analyzes the current state, problems, and possibilities of the development of upbringing work in an institution of higher learning. It examines the importance of upbringing for professi...

We propose a theory of relaxation oscillations for a nonlinear scalar delay differential-difference equation that represents a modification of the well-known Hutchinson equation in ecology. In particular, we establish that a one-dimensional chain of diffusively coupled equations of this type exhibits the well-known buffer phenomenon. Namely, under an increase in the number of links in the chain and a consistent decrease in the coupling constant, the number of coexisting stable periodic motions indefinitely increases.