We demonstrate that bifurcations of periodic orbits underlie the dynamics of the Hindmarsh–Rose model and other square-wave bursting models of neurons of the Hodgkin–Huxley type. Such global bifurcations explain in-depth the transitions between the tonic spiking and bursting oscillations in a model.We show that a modified Hindmarsh-Rose model can exhibit the blue sky bifurcation, and a bistability of the coexisting tonic spiking and bursting activities.

For a nonlinear Reyleigh-like system with a periodic perturbation, we give some fragments of the study to be connected with the problems of the existence of both chaotic dynamics and invariant tori of certain types. We construct bifurcation diagrams explaining a character of boundaries for regions corresponding to the existence of chaotic dynamics and the invariant tori. Besides, we construct bifurcation curves (for a series of periodic motions) which play the principal role at scenarios of creation of the boundaries pointed out.

We study bifurcations of of three-dimensional diffeomorphisms with non-transversal heteroclinic cycles which lead to the birth of wild hyperbolic Lorenz-like attractors. As known, such attractors can be appeared under small periodic perturbations of the classical Lorenz attractor and they allow homoclinic tangencies, however, do not contain stable periodic orbits.

In this paper we consider time-periodic perturbations of self-oscillating pendulum equation which arises from analysis of one system with two degrees of freedom. We derive averaged systems which describe the behavior of solutions of original equation in resonant areas and we find existence condition of Poincare homoclinic structure. In the case when autonomous equation has 5 limit cycles in oscillating region we give results of numerical computation. Under variation of perturbation frequency we investigate bifurcations of phase portraits of Poincare map.

In this paper diffeomorphisms on orientable surfaces are considered, whose non-wandering set consists of a finite number of hyperbolic fixed points and the wandering set contains a finite number of heteroclinic orbits of transversal and non-transversal intersections. We investigate substantial class of diffeomorphisms for which it is found complete topological invariant — a scheme consisting of a set of geometrical objects equipped by numerical parametres (moduli of topological conjugacy).

In the present paper autonomous and nonautonomous oscillations of dynamical and stochastic systems are analyzed in the framework of common concepts. The definition of an attractor is introduced for a nonautonomous system. The definitions of self-sustained oscillations and a self-sustained oscillatory system is proposed, that generalize A.A.Andronov’s concept introduced for autonomous systems with one degree of freedom.

We consider the problems of Hamiltonian representation and integrability of the nonholonomic Suslov system and its generalization suggested by S. A. Chaplygin. These aspects are very important for understanding the dynamics and qualitative analysis of the system. In particular, they are related to the nontrivial asymptotic behaviour (i. e. to some scattering problem). The paper presents a general approach based on the study of the hierarchy of dynamical behaviour of nonholonomic systems.

For nonautonomous systems of differential equations of second order which represent the family of control dynamical systems with given constraints on the control, we propose a method for constructing the borders of controllability and attainability. For this, we introduce the notions of singular points and singular trajectories, and study the structure of punctured neighborhood of a singular point. Some concrete examples of self interest are considered.

We prove the existence of solution in the problem of time averaged optimization of cyclic processes with both profit and effort discounts and find the respective necessary optimality condition. It is shown that optimal strategy could be selected piecewise continuous if a differentiable profit density has a finite number of critical points. In such a case the optimal motion uses only maximum and minimum velocities as in Arnold’s case without any discount.

Chaotic oscillator based on CMOS structure is proposed, fabricated and investigated. Monolithic IC сhip of the oscillator is fabricated in 0.18 um process technology. As is shown, the transition to chaos in this system occurs through destruction of 2D torus. In experiments with the IC, stable generation of chaotic oscillations is observed, with spectral density maximum in the range 2.8–3.8GHz.

This work deals with local dynamics of difference-differential equation with two delays. Supposed that both delays are asymptotically large and relatively close to each other. In critical cases of equlibrium state stability problem, which all have infinite dimention, special equations — normal forms — were built. Shown that normal forms are Ginzburg–Landau equations.

General scheme of asymptotic investigation of the questions of existence and stability of the contrast structures is proposed. This scheme is based on the development of the asymptotic method of differential inequalities, which was developed by author for different classes of singularly perturbed problems.

In 1998 in paper of D.V. Turaev and L.P. Shilnikov there was introduced the definition of the pseudohyperbolic flow. The pseudohyperbolic flow is the flow such that in every point of the phase space there exists decomposition of the tangent bandle to sum of two spaces such that in one of these spaces there is expanding of volume. Independently in paper of C. Morales, M.J. Pacifico and E. Pujals was introduced the definition of the singular hyperbolic flow. Singular hyperbolic attractors satisfy more strong conditions then pseudohyperbolic ones. This paper is devoted to the theory of Sinai–Bowen–Ruelle measures for singular hyperbolic attractors. There are established such properties as ergodicity, mixing, continuous dependence of the invariant measures on flow.

The coexistence of different types of homoclinic and periodic trajectories for dynamical systems generated by continuous maps of interval into itself is investigated.

A foliation that admits a Weil geometry as its transverse structure is called by us a Weil foliation. We proved that there exists an attractor for any Weil foliation that is not Riemannian foliation. If such foliation is proper, there exists an attractor coincided with a closed leaf. The above assertions are proved without assumptions of compactness of foliated manifolds and completeness of the foliations.

We proved also that an arbitrary complete Weil foliation either is a Riemannian foliation, with the closure of each leaf forms a minimal set, or it is a trasversally similar foliation and there exists a global attractor. Any proper complete Weil foliation either is a Riemannian foliation, with all their leaves are closed and the leaf space is a smooth orbifold, or it is a trasversally similar foliation, and it has a unique closed leaf which is a global attractor of this foliation.