E. F. Mishchenko

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.

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.

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.

Buffer phenomenon in systems close to two-dimensional Hamiltonian ones

Plane Hamiltonian systems perturbed by small time-periodic terms are considered. The conditions are established under which exponentially stable periodic solutions are accumulated infinitely in these systems as the perturbations tend to zero or, in other words, the buffer phenomenon occurs. It is shown that this phenomenon is typical for a wide range of classical mechanical problems described by equations of the pendulum type.

Parametric bufferness in systems of parabolic and hyperbolic equations with small diffusion

We investigate the problem of parametric excitation of oscillations in systems of parabolic and hyperbolic equations with small coefficient of diffusion. We establish the phenomenon of parametric bufferness, i.e., the existence of an arbitrary fixed number of stable periodic solutions for a proper choice of the parameters of equations.

Buffer phenomenon in the spatially one-dimensional Swift-Hohenberg equation

We consider a boundary value problem for the spatially one-dimensional Swift-Hohenberg equation with zero Neumann boundary conditions at the endpoints of a finite interval. We establish that as the length l of the interval increases while the supercriticality ɛ is fixed and sufficiently small, the number of coexisting stable equilibrium states in this problem indefinitely increases; i.e., the well-known buffer phenomenon is observed. A similar result is obtained in the 2l-periodic case.

Attractors of the sine-Gordon equation in the field of a quasiperiodic external force

The well-known sine-Gordon equation, supplemented with small damping and small quasiperiodic external force, is studied under the zero Dirichlet boundary conditions at the endpoints of a finite interval. The main assumption is that all frequencies of the external force are in 1:1 resonance with certain eigenfrequencies of the unperturbed equation; i.e., the socalled fundamental multifrequency resonance is observed. It is shown that in this case, by an appropriate choice of the parameters of the external force, one can make it so that the boundary value problem has a stable invariant torus of any finite dimension that bifurcates from zero on any preassigned finite set of spatial modes. It is also shown (by numerical analysis) that in a number of cases the above-mentioned torus coexists with a chaotic attractor.