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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 $$m$$ .

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.

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.

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.

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 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 study the problem of attractors of the two-dimensional mapping $$(u,v) \to (v, - (1 - \mu )u - F(v)),F(v) = \left\{ {\begin{array}{*{20}c} {q_1 forv > 0,} \\ {0forv > 0,} \\ { - q_2 forv > 0,} \\ \end{array} } \right.$$ where 0 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.