A. Yu. Kolesov

A modification of Hutchinson’s equation

A new mathematical object is introduced, namely, a scalar nonlinear delay differential-difference equation is considered that is a modification of Hutchinson’s equation, which is well known in ecology. The existence and stability of its relaxation self-oscillations are analyzed.

Finite-dimensional models of diffusion chaos

Some parabolic systems of the reaction-diffusion type exhibit the phenomenon of diffusion chaos. Specifically, when the diffusivities decrease proportionally, while the other parameters of a system remain fixed, the system exhibits a chaotic attractor whose dimension increases indefinitely. Various finite-dimensional models of diffusion chaos are considered that represent chains of coupled ordinary differential equations and similar chains of discrete mappings. A numerical analysis suggests that these chains with suitably chosen parameters exhibit chaotic attractors of arbitrarily high dimensions.

Chaos Phenomena in a Circle of Three Unidirectionally Connected Oscillators

A method is proposed for designing chaotic oscillators. Mathematically, three so-called partial oscillators Sj (j = 1, 2, 3) are chosen, each of which is modeled by a nonlinear system of ordinary differential equations with a single attractor—an equilibrium or a cycle (the case S1 = S2 = S3 is not excluded). It is shown that, when unidirectionally connected in a circle of the form with suitably chosen parameters, these oscillators can exhibit a joint chaotic behavior.

New methods for proving the existence and stability of periodic solutions in singularly perturbed delay systems

We carry out a detailed analysis of the existence, asymptotics, and stability problems for periodic solutions that bifurcate from the zero equilibrium state in systems with large delay. The account is based on a specific meaningful example given by a certain scalar nonlinear second-order differential-difference equation that is a mathematical model of a single-circuit RCL oscillator with delay in a feedback loop.

Optical Buffering and Mechanisms for Its Occurrence

We investigate a mathematical nonlinear-optics model that is a scalar parabolic equation on a circle with a small diffusion coefficient and a deviating spatial argument. We establish that the problem under consideration is characterized by the so-called buffering phenomenon, i.e., under an appropriate choice of the parameters, the coexistence of an arbitrary fixed number of time-periodic stable solutions of the problem can be obtained. We reveal the mechanisms for the occurrence of this phenomenon.

Discrete autowaves in neural systems

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.

Extremal dynamics of the generalized Hutchinson equation

A scalar nonlinear differential-difference equation with two delays that generalizes Hutchinson’s equation is considered. The bifurcation of self-oscillations of this equation from the zero equilibrium is studied in the extremal situation when one delay is asymptotically large while the other parameters are on the order of unity. Analytical methods combined with numerical techniques are used to show that the well-known buffer phenomenon occurs in the equation in this case. This means that an arbitrary finite number of different attractors coexist in the phase space of the equation with suitably chosen parameters.

Resonance dynamics of nonlinear flutter systems

We consider a special class of nonlinear systems of ordinary differential equations, namely, the so-called flutter systems, which arise in Galerkin approximations of certain boundary value problems of nonlinear aeroelasticity and in a number of radiophysical applications. Under the assumption of small damping coefficient, we study the attractors of a flutter system that arise in a small neighborhood of the zero equilibrium state as a result of interaction between the 1: 1 and 1: 2 resonances. We find that, first, these attractorsmay be both regular and chaotic (in the latter case, we naturally deal with numerical results); and second, for certain parameter values, they coexist with the stable zero solution; i.e., the phenomenon of hard excitation of self-oscillations is observed.

Characteristic features of the dynamics of the Ginzburg-Landau equation in a plane domain

We study the boundary value problem wt=ℵ0Δw+ℵ1w-ℵ2w|w|2,w|∂Ω0=0 in the domain Ω0={(x,y):0 ≤ x ≤ l1,0 ≤ y ≤ l2}. Here, w is a complex-valued function, Δ is the laplace operator, and ℵj, j=0,1,2, are complex constants withRe ℵj > 0. We show that under a rather general choice of the parameters l1 and l2, the number of stable invariant tori in the problem, as well as their dimensions, grows infinitely asRe ℵ0 → 0 andRe ℵ0 → 0.

Relaxation oscillations and diffusion chaos in the Belousov reaction

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.