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Рассматривается цепочка последовательно связанных $RCL$-контуров, к одному из концов которой подсоединен источник постоянного напряжения $E$, а к другому - туннельный диод и постоянная емкость $C_0$. Показано, что математической моделью данного генератора служит некоторая нелинейная система обыкновенных дифференциальных уравнений, в которой, в свою очередь, реализуется известное явление буферности. А именно, с помощью сочетания аналитических и численных методов устанавливается неограниченный рост числа сосуществующих аттракторов этой системы при увеличении количества $RCL$-контуров и при надлежащем изменении остальных параметров. Проводится также сравнительный анализ локальной динамики в дискретной и непрерывной $RCL$-линиях.

Рассматриваются специальные системы обыкновенных дифференциальных уравнений - так называемые полносвязные сети нелинейных осцилляторов. Для данного класса систем предлагаются методы, позволяющие разобраться с вопросами о существовании и устойчивости периодических режимов двухкластерной синхронизации. В случае любого из таких режимов множество осцилляторов распадается на два непересекающихся класса. В пределах этих классов наблюдается полная синхронизация колебаний, а каждые два осциллятора из разных классов колеблются асинхронно.

На бесконечномерном торе $\mathbb{T}^{\infty} {=} E/2\pi\mathbb{Z}^{\infty}$, где $E$ - бесконечномерное вещественное банахово пространство, $\mathbb{Z}^{\infty}$ - абстрактная целочисленная решетка, рассматривается специальный класс диффеоморфизмов $\operatorname{Diff}(\mathbb{T}^{\infty})$. Упомянутый класс состоит из отображений $G\colon \mathbb{T}^{\infty}\to\mathbb{T}^{\infty}$, представляющих собой суммы линейных обратимых ограниченных операторов, сохраняющих решетку $\mathbb{Z}^{\infty}$, и $C^1$-гладких периодических добавок. Устанавливаются необходимые и достаточные условия, гарантирующие гиперболичность таких отображений (т.е. принадлежность их к диффеоморфизмам Аносова). Библиография: 15 названий.

На бесконечномерном торе $\mathbb{T}^{\infty}=E/2\pi\mathbb{Z}^{\infty}$, где $E$ - бесконечномерное вещественное банахово пространство, $\mathbb{Z}^{\infty}$ - абстрактная целочисленная ре-\linebreakшетка, рассматривается специальный класс диффеоморфизмов $\operatorname{Diff}(\mathbb{T}^{\infty})$. Упомянутый класс состоит из отображений $G\colon\mathbb{T}^{\infty}\to\mathbb{T}^{\infty}$, для которых дифференциалы $DG$ и $D(G^{-1})$ равномерно ограничены и равномерно непрерывны на $\mathbb{T}^{\infty}$. Для диффеоморфизмов из $\operatorname{Diff}(\mathbb{T}^{\infty})$ дается систематическое изложение элементов гиперболической теории, начиная с основных определений и ряда вспомогательных утверждений и заканчивая более продвинутыми результатами. К последним относятся критерий гиперболичности, теорема о $C^1$-грубости свойства гиперболичности для диффеоморфизмов из класса $\operatorname{Diff}(\mathbb{T}^{\infty})$, теорема Адамара-Перрона, а также один из базовых результатов гиперболической теории: существование у любого диффеоморфизма Аносова $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ устойчивого и неустойчивого инвариантных слоений. Библиография: 34 названия.

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 $$m$$ .

Рассматривается кольцевое множество вида $K=B\times\mathbb{T}^{\infty}$, где $B$ - замкнутый шар банахова пространства $E$, $\mathbb{T}^{\infty}$ - бесконечномерный тор (прямое произведение счетного числа окружностей с топологией равномерной покоординатной сходимости). Для некоторого класса гладких отображений $\Pi\colon K\to K$ устанавливаются достаточные условия существования и устойчивости инвариантного тороидального многообразия вида $$ A=\{(v,\varphi)\in K: v=h(\varphi)\in E, \varphi\in\mathbb{T}^{\infty}\}, $$ где $h(\varphi)$ - некоторая непрерывная функция аргумента $\varphi\in\mathbb{T}^{\infty}$. Исследуется также вопрос о $C^m$-гладкости этого многообразия при любом натуральном $m$. Библиография: 4 названия.

Рассматривается некоторый вполне естественный класс диффеоморфизмов $G$, действующих из $\mathbb{T}^{\infty}$ в $\mathbb{T}^{\infty}$, где $\mathbb{T}^{\infty}$ - бесконечномерный тор (прямое произведение счетного числа окружностей с топологией равномерной покоординатной сходимости). Интересующие нас диффеоморфизмы допускают представление в виде суммы линейного гиперболического отображения и периодической добавки. Предлагается набор конструктивных достаточных условий, при которых любое отображение $G$ из нашего класса является гиперболическим, т. е. диффеоморфизмом Аносова на торе $\mathbb{T}^{\infty}$. Кроме этого, при выполнении упомянутых условий устанавливаются следующие стандартные факты из гиперболической теории: наличие устойчивого и неустойчивого инвариантных слоений, топологическая сопряженность с линейным гиперболическим автоморфизмом тора, структурная устойчивость $G$. Библиография: 21 наименование.

This paper discusses mathematical modeling of artificial gene networks. We consider a phenomenological model of the simplest gene network, called a repressilator. This network contains three elements unidirectionally coupled into a ring. More specifically, the first element inhibits the synthesis of the second, the second inhibits the synthesis of the third, and the third, which closes the cycle, inhibits the synthesis of the first one. The interaction of protein concentrations and mRNA (messenger RNA) concentration is surprisingly similar to the interaction of six ecological populations, three predators and three preys. This makes it possible to propose a new phenomenological model, which is represented by a system of unidirectionally coupled ordinary differential equations. We investigate the problem of existence and stability of a relaxation periodic solution that is invariant with respect to cyclic permutations of coordinates. To find the asymptotics of this solution, a special relay system is constructed. We prove that the periodic solution of the relay system gives the asymptotic approximation of the orbitally asymptotically stable relaxation cycle of the problem under consideration.

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.

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.

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.

Для распределенных эволюционных динамических систем типа реакция-диффузия и реакция-диффузия-адвекция проведен анализ поведения инвариантных числовых характеристик аттрактора при уменьшении коэффициентов диффузии. Рассмотрен феномен многомодового диффузионного хаоса, одним из признаков которого является увеличение ляпуновской размерности аттрактора. Для ряда примеров выполнен обширный численный эксперимент, в котором проиллюстрирован этот эффект.

Рассматривается произвольный диффеоморфизм $f$ класса $C^1$, действующий из некоторого открытого подмножества $U$ риманова многообразия $M$ размерности $m$, $m\ge 2$, в $f(U)\subset M$. Пусть $A$ - компактное подмножество из $U$, инвариантное для $f$, т.е. $f(A)=A$. Предлагаются различные достаточные условия, при выполнении которых $A$ является гиперболическим множеством диффеоморфизма $f$.

—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.

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.

Рассматривается произвольный диффеоморфизм $\Pi$, преобразующий в себя некоторое кольцевое множество $K=B\times \mathbb{T}$, где $B$ - шар банахова пространства, $\mathbb{T}$ - тор (конечномерный или бесконечномерный). Предлагается набор конструктивных достаточных условий, при которых глобальный аттрактор $A=\bigcap\limits_{n\geqslant 0}\Pi^n(K)$ диффеоморфизма $\Pi$ существует и допускает представление в виде обобщенного соленоида, т. е. предела обратного спектра $\mathbb{T}\xleftarrow{G}\mathbb{T}\xleftarrow{G}\cdots \xleftarrow{G}\mathbb{T}\xleftarrow{G}\cdots$, где $G$ - некоторый линейный растягивающий эндоморфизм тора $\mathbb{T}$. При этом сужение $\Pi|_{A}$ топологически сопряжено со сдвиговым отображением соленоида. Библиография: 25 названий.

Рассматривается некоторый естественный класс растягивающих эндоморфизмов $G\in C^1$, действующих из $\mathbb{T}^{\infty}$ в $\mathbb{T}^{\infty}$, где $\mathbb{T}^{\infty}$ - бесконечномерный тор (прямое произведение счетного числа окружностей с топологией равномерной покоординатной сходимости). Интересующие нас эндоморфизмы допускают представление в виде суммы линейного растягивающего отображения и периодической добавки. Устанавливаются следующие стандартные факты из гиперболической теории: топологическая сопряженность растягивающего эндоморфизма $G$ из нашего класса с линейным эндоморфизмом тора, структурная устойчивость отображения $G$, справедливость для $G$ на $\mathbb{T}^{\infty}$ свойства топологического перемешивания.

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.

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.

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.

We consider a system of two relay type differential-difference equations, with delay in the coupling between oscillators. The equations are coupled in a special way The system models an association of coupled neural oscillators. An important feature of the system is an additional delay in the connection chain. The delay allows obtaining new effects which is an essential complication of the system dynamics, and an appearance of coexisting special form attractors. We show that there coexist asymptotically orbitally stable solutions with summary 2n [n ϵ N) spikes on a period. Moreover, the first oscillator has m spikes, and the second one has 2n — m (m = 1, 2,..., 2n — 1) spikes on a period. We conclude that the additional delay leads to an accumulation of coexisting attractors with a given number of spikes on a period.

The system of phase differences for a chain of diffuse weakly coupled oscillators on a stable integral manifold is constructed and analyzed. It is shown (by means of numerical methods) that Lyapunov dimension growth is close to linear as the number of oscillators in the chain increases. Extensive computations performed for the difference model of the Ginsburg–Landau equation illustrate this result and determine the applicability limits for asymptotic methods.

A model of neural association of three pulsed neurons with a delayed broadcast connection is considered. It is assumed that the parameters of the problem are chosen near the critical point of stability loss by the homogeneous equilibrium state of the system. Because of the broadcast connection the equation corresponding to one of the oscillators can be detached in the system. Two remaining pulsed neurons interact with each other and, in addition, there is a periodic external action, determined by the broadcast neuron. Under these conditions, the normal form of this system is constructed for the values of parameters close to the critical ones on a stable invariant integral manifold. This normal form is reduced to a four-dimensional system with two variables responsible for the oscillation amplitudes, and the other two are defined as the difference between the phase variables of these oscillators with the phase variable of the broadcast oscillator. The obtained normal form has an invariant manifold on which the amplitude and phase variables of the oscillators coincide. Dynamics of the problem is described on this manifold. An important result was obtained on the basis of numerical analysis of the normal form. It turned out that periodic and chaotic oscillatory solutions can occur when the coupling link between the oscillators is weakened. Moreover, a cascade of bifurcations associated with the same type of phase transformations was discovered, where a self-symmetric stable cycle alternately loses symmetry with the appearance of two symmetrical to each other cycles. A cascade of bifurcations of period doubling occurs with each of these cycles with the appearance of symmetric chaotic regimes. With further decreasing of the coupling parameter, these symmetric chaotic regimes are combined into a self-symmetric one, which is then rebuilt into a self-symmetric cycle of a more complex form compared to the cycle obtained at the previous step. Then the whole process is repeated. Lyapunov exponents were computed to study chaotic attractors of the system.

The dynamics of a generalized Hutchinson’s equation with two delays is investigated. A local analysis of stability loss for the nonzero equilibrium state of the problem is presented. Main bifurcations are analyzed by means of numerical methods with the aid of obtained asymptotic relations. The bifurcation curves corresponding to principal bifurcations taking place in the system are constructed on the plane of parameters.

A model of neural association of three pulsed neurons with a delayed broadcast connection is considered. It is assumed that the parameters of the problem are chosen near the critical point of stability loss by the homogeneous equilibrium state of the system. Because of the broadcast connection the equation corresponding to one of the oscillators can be detached in the system. The two remaining impulse neurons interact with each other and, in addition, there is a periodic external action, determined by the broadcast neuron. Under these conditions, the normal form of this system is constructed for the values of parameters close to the critical ones on a stable invariant integral manifold. This normal form is reduced to a four-dimensional system with two variables responsible for the oscillation amplitudes, and the other two, defined as the difference between the phase variables of these oscillators with the phase variable of the broadcast oscillator. The obtained normal form has an invariant manifold on which the amplitude and phase variables of the oscillators coincide. The dynamics of the problem on this manifold is described. An important result was obtained on the basis of numerical analysis of the normal form. It turned out that periodic and chaotic oscillatory solutions can occur when the coupling between the oscillators is weakened. Moreover, a cascade of bifurcations associated with the same type of phase rearrangements was discovered, where a self-symmetric stable cycle alternately loses symmetry with the appearance of two symmetrical cycles. A cascade of bifurcations of doubling occurs with each of these cycles with the appearance of symmetric chaotic regimes. With further reduction of the coupling parameter, these symmetric chaotic regimes are combined into a self-symmetric one, which is then rebuilt into a self-symmetric cycle of a more complex form compared to the cycle obtained at the previous step. Then the whole process is repeated. Lyapunov exponents were calculated to study chaotic attractors of the system.

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.

The models of association of two coupled neural oscillators with synaptic coupling are considered. An important feature of this association is an additional delay in the chain of connections. The system of relay differential-difference equations was chosen as the mathematical model of this association. The synaptic connection was modeled based on the modified idea of fast threshold modulation (FTM). The delay allows obtaining new effects which is an essential complication of the system dynamics, and an appearance of coexisting special form attractors. We show that there coexist asymptotically orbitally stable solutions with summary \(N\in {\mathbb {N}}\) spikes on a period. Moreover, the first oscillator has m spikes, and the second one has \(N-m\) (\(m=1,2,\ldots ,\,N-1\)) spikes on a period. We conclude that the additional delay leads to an accumulation of coexisting attractors with a given number of spikes on a period.

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 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.

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.