Regular and Chaotic Dynamics最新文献

The problem of the rolling of a disk on a plane is considered under the assumption that there is no slipping in the direction parallel to the horizontal diameter of the disk and that the center of mass does not move in the horizontal direction. This problem is reduced to investigating a system of three first-order differential equations. It is shown that the reduced system is reversible relative to involution of codimension one and admits a two-parameter family of fixed points. The linear stability of these fixed points is analyzed. Using numerical simulation, the nonintegrability of the problem is shown. It is proved that the reduced system admits, even in the nonintegrable case, a two-parameter family of periodic solutions. A number of dynamical effects due to the existence of involution of codimension one and to the degeneracy of the fixed points of the reduced system are found.

We study billiard systems within an ellipsoid in the (4)-dimensional pseudo-Euclidean spaces. We provide an analysis and description of periodic and weak periodic trajectories in algebro-geometric and functional-polynomial terms.

In 1983 Bogoyavlenski conjectured that, if the Euler equations on a Lie algebra (mathfrak{g}_{0}) are integrable, then their certain extensions to semisimple lie algebras (mathfrak{g}) related to the filtrations of Lie algebras(mathfrak{g}_{0}subsetmathfrak{g}_{1}subsetmathfrak{g}_{2}dotssubsetmathfrak{g}_{n-1}subsetmathfrak{g}_{n}=mathfrak{g}) are integrable as well.In particular, by taking (mathfrak{g}_{0}={0}) and natural filtrations of ({mathfrak{so}}(n)) and (mathfrak{u}(n)), we haveGel’fand – Cetlin integrable systems. We prove the conjecturefor filtrations of compact Lie algebras (mathfrak{g}): the system is integrable in a noncommutative sense by means of polynomial integrals.Various constructions of complete commutative polynomial integrals for the system are also given.

We consider Hamiltonian systems possessing families of nonresonant invariant tori whose frequencies are all collinear.Then under certain conditions the frequencies depend on energy only.This is a generalization of the well-known Gordon’s theorem about periodic solutions of Hamiltonian systems.While the proof of Gordon’s theorem uses Hamilton’s principle, our result is based on Percival’s variational principle. This work was motivated by the problem of isochronicity in Hamiltonian systems.

A spin-transfer oscillator is a nanoscale device demonstrating self-sustainedprecession of its magnetization vector whose length is preserved. Thus, thephase space of this dynamical system is limited by a three-dimensionalsphere. A generic oscillator is described by the Landau – Lifshitz – Gilbert – Slonczewskiequation, and we consider a particular case of uniaxial symmetry when theequation yet experimentally relevant is reduced to a dramatically simpleform. The established regime of a single oscillator is a purely sinusoidal limitcycle coinciding with a circle of sphere latitude (assuming that points wherethe symmetry axis passes through the sphere are the poles). On the limit cyclethe governing equations become linear in two oscillating magnetization vector componentsorthogonal to the axis, while the third one along the axis remains constant. In this paperwe analyze how this effective linearity manifests itself when two such oscillators aremutually coupled via their magnetic fields. Using the phase approximation approach, wereveal that the system can exhibit bistability betweensynchronized and nonsynchronized oscillations. For the synchronized one the Adler equation is derived, and theestimates for the boundaries of the bistability area are obtained. The two-dimensionalslices of the basins of attraction of the two coexisting solutions areconsidered. They are found to be embedded in each other, forming a series ofparallel stripes. Charts of regimes and charts of Lyapunov exponents are computednumerically. Due to the effective linearity the overall structure of thecharts is very simple; no higher-order synchronization tongues except the mainone are observed.

This paper is concerned with the classical Duffing equation whichdescribes the motion of a nonlinear oscillator with an elastic force that is odd withrespect to the value of deviation from itsequilibrium position, and in the presence of an external periodic force. The equationdepends on three dimensionless parameters. When they satisfy some relation, the equationadmits exact periodic solutions with a period that is a multiple of the period of externalforcing. These solutions can be written in explicit form without using series.The paper studies the nonlinear problem of the stability of these periodic solutions.The study is based on the classical Lyapunov methods, methods of KAM theory forHamiltonian systems and the computer algorithms for analysis ofarea-preserving maps. None of the parameters of the Duffing equation is assumed to be small.

In this paper, we are concerned with the shape of the attractor (mathcal{A}^{lambda}) of the scalar Chafee – Infante equation. We construct a Morse – Smale vector field in the disk (mathbb{D}^{k}) topologically equivalent toinfinite-dimensional dynamics of the Chafee – Infante equation. As a consequence,we obtain geometric properties of (mathcal{A}^{lambda}) using the Morse – Smale inequalities.

We propose a new method for constructing multidimensional reversible maps by only two input data: a diffeomorphism (T_{1}) and an involution (h), i. e., a map (diffeomorphism) such that (h^{2}=Id). We construct the desiredreversible map (T) in the form (T=T_{1}circ T_{2}), where (T_{2}=hcirc T_{1}^{-1}circ h). We also discuss how this method can be used to construct normal forms of Poincaré maps near mutually symmetric pairs of orbits of homoclinic or heteroclinic tangencies in reversible maps. One of such normal forms, as we show, is a two-dimensional double conservative Hénon map(H) of the form (bar{x}=M+cx-y^{2}; y=M+cbar{y}-bar{x}^{2}).We construct this map by the proposed method for the case when (T_{1}) is the standard Hénon map and the involution (h) is(h:(x,y)to(y,x)).For the map (H),we study bifurcations of fixed and period-2 points, among which there are both standard bifurcations (parabolic, period-doubling and pitchfork) and singular ones (during transition through (c=0)).

The family of generalized Schrödinger equations is considered with the Kerr nonlinearity. The partial differential equations are not integrable by the inverse scattering transform and new solutions of this family are sought taking into account the traveling wave reduction. The compatibility of the overdetermined system of equations is analyzed and constraints for parameters of equations are obtained.A modification of the simplest equation method for finding embedded solitons is presented.A block diagram for finding a solution to the nonlinear ordinary differential equation isgiven. The theorem on the existence of bright solitons for differential equations of any orderwith Kerr nonlinearity of the family considered is proved. Exact solutions of embedded solitonsdescribed by fourth-, sixth-, eighth and tenth-order equations are found using the modified algorithm of the simplest equation method. New solutions for embedded solitons of generalized nonlinear Schrödinger equations with several extremes are obtained.