Yu. S. Kolesov

The attractor problem for nonlinear wave equations in plane domains

We consider a model example of a quasilinear wave equation in the unit square with zero boundary conditions and use the method of quasinormal forms to prove that there are quite a few dichotomic cycles and tori bifurcating from zero equilibrium. A conjecture concerning the attractor structure is presented.

Optimal method for swinging the swing

AbstractWe compute the greatest lower bound of multipliers of the generalized Mathieu equations % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqipC0xg9qqqrpepC0xbbL8F4rqqrFfpeea0xe9Wqpe0x % c9q8qqaqFn0dXdir-xcvk9pIe9q8qqaq-dir-f0-yqaqVeLsFr0-vr % 0-vr0db8meaabaqaciGacaGaaeqabaWaaeaaeaaakeaatCvAUfKttL % earyqr1ngBPrgaiuGacuWF4baEgaWaaaaa!3FD9!

Dynamics of a self-excited oscillator in a distortion-free line

\ddot x

Stability properties of cycles and tori of a simplest nonresonant wave-type equation

+(1+p(t))x = 0 with continuous π-periodic functions p(·) satisfying some additional constraints.

Asymptotic Methods in Singularly Perturbed Systems

An analysis of the dynamics of three-dimensional nonlinear waves on a torus has shown that their attractors are the so-called self-organization regimes, which are created from trajectories crowding together and have several remarkable features. Specifically, they are well ordered with respect to spatial and time variables, and their energy is fairly high and decreases gradually with decreasing elasticity coefficient, which itself evolves into a diffusion chaos. The role of this paper is twofold. First, the features of self-organization regimes are analyzed in the case of Neumann boundary conditions. Second, the stages leading to the detection of this phenomenon are described in detail.

BIFURCATION OF SELF-OSCILLATIONS OF NONLINEAR PARABOLIC EQUATIONS WITH SMALL DIFFUSION

The dynamics considered in the paper is related to some continuous and pointwise mapping. We show that the attractors of such mappings are much richer than the actual dynamics of the considered self-excited oscillator.

Asymptotics and stability of non-linear parametric oscillations of a?singularly perturbed telegraph equation

We state a theorem on the instability of self-similar cycles and tori of a certain type in a system which is a quasinormal form of a boundary-value problem for a nonlinear wave equation in the square.

BIFURCATION OF INVARIANT TORI OF PARABOLIC SYSTEMS WITH SMALL DIFFUSION

For a number of meaningful examples, nonresonant wave-type equations are shown to be characterized by periodic dynamics.