Regular and Chaotic Dynamics最新文献

In this paper, we provide a complete answer to the question posed in its title. For all closed surfaces for which this question has not yet been resolved, namely, closed nonorientable surfaces of odd genus greater than 5, a series of examples of pseudo-Anosov homeomorphisms are constructed. Each of these homeomorphisms is defined bymeans of the so-called code, which implies using the construction of a band surface. Combined with previously obtained results, this allows us to fill in the gap and to formulate a general result: pseudo-Anosov homeomorphisms exist on a closed nonorientable surface if and only if its genus is at least 4.

We show that bifurcations of four-dimensional symplectic diffeomorphisms with a quadratic homoclinic tangency to a saddle periodic orbit with real multipliers produce 2-elliptic periodic orbits if the tangency is not partially hyperbolic. We show that a normal form for the rescaled first-return maps near such tangency is given by a four-dimensional symplectic Hénon-like map and study bifurcations of the first-return maps in generic two-parameter families.

We present some new Poisson bivectors that are invariants by the flow of the nonholonomic Suslov problem.Two rank-four invariant Poisson bivectors have globally defined Casimir functions and, therefore, define cubic Poisson brackets of the five-dimensional state space with standard symplectic leaves. For the Suslov gyrostat in the potential field we found rank-two Poisson bivectors having only two globally defined Casimir functions and, therefore, we say about formal Hamiltonian description in these cases.

This paper is concerned with abnormal geodesics on the Carnot group with growth vector ((2,3,5,8,14)). Because of a large number of symmetries, this problem reduces to an analysis of the five-dimensional flow. Using the Kovalevskaya method, integrable cases of the resulting system are identified. For these cases, first integrals and explicit solutions are found. It is shown that in the general case the system admits no additional meromorphic first integrals. The paper concludes by discussing some problems regarding the abnormal geodesics on Lie groups.

We propose and study a model for the mechanical system constituted bya chain of (ngeqslant 1) identical pendula hanging from a viscoelasticstring with fixed extrema. The novelty of our approach is todescribe the string as a continuous system, specifically, as aone-dimensional viscoelastic Kelvin – Voigt string. The resultingsystem is a hybrid nonlinear system of coupled PDEs and ODEs. Welinearize the system around the attractive equilibrium with pendulaand string pointing downwards. The (infinite-dimensional)linearization decouples into a “vertical” and a “horizontal”subsystems. The former is a viscoelastic version of the well knownRayleigh loaded string, and its point spectrum is known. We thusconsider the latter, which describes, at the linear level, thehorizontal oscillations of string and pendula. We obtain closed formexpressions for the eigenvalue equations and for the eigenfunctionsfor any value of (n). Next, we study the point spectrum with acombination of analytical and numerical techniques, adopting acontinuation approach from the limiting cases of massless pendula, whichinvolves the well known spectrum of the Kelvin – Voigt string. Finally,we focus on the identification, particularly when (n=2) and as afunction of the parameters, of the eigenvalues closest to theimaginary axis, whose eigenfunction(s) dominate the asymptoticdynamics of the (horizontal) linearized systems and can explain theappearance of synchronization patterns in the chain of pendula.

In this article, we present a classification of toric (b^{m})-symplectic manifolds and outline a program for the classification of semitoric singular symplectic manifolds. As a testing ground, we begin with the case of (b)-symplectic manifolds, where a Delzant-type theorem for toric actions has already been established [13]. Building on this foundation, we extend the framework to the broader setting of (b^{m})-symplectic manifolds, thereby establishing the first classification result for (b^{m})-toric manifolds. This achievement lays the groundwork for a conjectural classification theory of semitoric systems in dimension four on (b^{m})-symplectic manifolds. Furthermore, by employing the reduction theory developed in [21], we propose a conjectural classification scheme for higher-dimensional semitoric manifolds, aiming to generalize the Pelayo – Vũ Ngọc classification program [32, 33] to the singular (b^{m})-setting.

Equations of motion for a body moving through an ideal fluid when the flow is irrotational and incompressible are obtained taking account of embedded dipoles on the boundary and the Kutta – Chaplygin condition. We develop the embedded dipole model from the complex potential of a dipole on the boundary of a body, oriented so as to preserve no-penetration through the body, using a conformal mapping approach. The resulting hydrodynamic force and moment on the body depend on the dipoles’ strength and position along the body. Using the flat plate as a model geometry, we examine the evolution of the resulting system under the conditions of fixed and time-varying circulation with and without embedded dipoles. We assume two embedded dipoles symmetrically positioned about the center point of the plate, finding that the presence of the dipoles reduces the fluctuations of the angle of attack of the plate. We explore conserved quantities for the system and perform a linear stability analysis, which leads to a constraint on the dipole strength for stability of a plate moving at zero angle of attack with either circulation equal to zero or the Kutta – Chaplygin condition applied.

Nonholonomic systems are mechanical systems with ideal linear velocity constraints that are not derivable from position constraints and with dynamics identified by the Lagrange – d’Alembert principle.This paper surveys discrete-time nonholonomic mechanics with applications to numerical integration of nonholonomic systems.This includes an exposition of key elements of discrete mechanics, discrete Lagrange – d’Alembert principle, and exact nonholonomic integrators on vector spaces.Exact variational integrators were introduced and exposed in the context of Lagrangian mechanics by Marsden and West.These integrators sample the trajectories of mechanical systems and are useful for developing practical mechanical integrators.

We consider the Novikov problem, namely, the problem of describingthe global geometry of level lines of quasiperiodic functions on a plane,for a special class of two-dimensional potentials. Potentials of thisclass play an important role in the physics of two-dimensional systemsand are defined by superpositions of two periodic potentials withthe same rotational symmetry. For different orientations of the periodsof the original potentials, the resulting potential can have 4 quasiperiodesor be periodic. The main result of the paper is a proof that quasiperiodicpotentials of this class can have open level lines at only one energy level.This property brings these potentials closer to random potentials on a plane,as well as to potentials with 3 quasiperiodes possessing “chaotic” level lines.The paper also presents an estimate for the energy interval containing openlevel lines of periodic potentials arising at “magic” angles of rotationof the original potentials relative to each other.