Regular and Chaotic Dynamics最新文献

We construct (C^{infty}) time-periodic fluctuating surfaces in (mathbb{R}^{3}) such that the corresponding non-autonomous geodesic flow has orbits along which the energy, and thus the speed goes to infinity. We begin with a static surface (M) in (mathbb{R}^{3}) on which the geodesic flow (with respect to the induced metric from (mathbb{R}^{3})) has a hyperbolic periodic orbit with a transverse homoclinic orbit. Taking this hyperbolic periodic orbit in an interval of energy levels gives us a normally hyperbolic invariant manifold (Lambda), the stable and unstable manifolds of which have a transverse homoclinic intersection. The surface (M) is embedded into (mathbb{R}^{3}) via a near-identity time-periodic embedding (G:Mtomathbb{R}^{3}). Then the pullback under (G) of the induced metric on (G(M)) is a time-periodic metric on (M), and the corresponding geodesic flow has a normally hyperbolic invariant manifold close to (Lambda), with stable and unstable manifolds intersecting transversely along a homoclinic channel. Perturbative techniques are used to calculate the scattering map and construct pseudo-orbits that move up along the cylinder. The energy tends to infinity along such pseudo-orbits. Finally, existing shadowing methods are applied to establish the existence of actual orbits of the non-autonomous geodesic flow shadowing these pseudo-orbits. In the same way we prove the existence of oscillatory trajectories, along which the limit inferior of the energy is finite, but the limit superior is infinite.

In this paper we consider the dynamics of a rollerbicycle on a horizontal plane. For this bicycle we derive anonlinear system of equations of motion in a form that allowsus to take into account the symmetry of the system in anatural form. We analyze in detail the stability of straight-linemotion depending on the parameters of the bicycle.We find numerical evidence that, in addition to stable straight-line motion,the roller bicycle can exhibit other, more complex,trajectories for which the bicycle does not fall.

We prove here the criterion of (C^{1})- (Omega)-stability of self-maps of a 3D-torus, whichare skew products of circle maps. The (C^{1})- (Omega)-stability property is studied with respect to homeomorphisms of skew products type. We give here an example of the (Omega)-stable map on a 3D-torus and investigate approximating properties of maps under consideration.

This paper discusses a range of questions concerning the application ofsolvable Lie algebras of vector fields to exact integration of systems of ordinarydifferential equations. The set of (n) independent vector fieldsgenerating a solvable Lie algebra in (n)-dimensional space is locallyreduced to some “canonical” form. This reduction is performed constructively (usingquadratures), which, in particular, allows a simultaneous integration of (n) systems ofdifferential equations that are generated by these fields.Generalized completely integrable systems are introduced and their properties are investigated.General ideas are applied to integration of the Hamiltonian systems of differential equations.

It is well known that the presence of horseshoes leads to positive entropy. If our goal is to construct a continuous map with infinite entropy, we can consider an infinite sequence of horseshoes, ensuring an unbounded number of legs.

Estimating the exact values of both the metric mean dimension and mean Hausdorff dimension for a homeomorphism is a challenging task. We need to establish a precise relationship between the sizes of the horseshoes and the number of appropriated legs to control both quantities.

Let (N) be an (n)-dimensional compact Riemannian manifold, where (ngeqslant 2), and (alphain[0,n]). In this paper, we construct a homeomorphism (phi:Nrightarrow N) with mean Hausdorff dimension equal to (alpha). Furthermore, we prove that the set of homeomorphisms on (N) with both lower and upper mean Hausdorff dimensions equal to (alpha) is dense in (text{Hom}(N)). Additionally, we establish that the set of homeomorphisms with upper mean Hausdorff dimension equal to (n) contains a residual subset of (text{Hom}(N).)

In this article, we consider mechanical billiard systems defined with Lagrange’s integrable extension of Euler’s two-center problems in the Euclidean space, the sphere, and the hyperbolic space of arbitrary dimension (ngeqslant 3). In the three-dimensional Euclidean space, we show that the billiard systems with any finite combinations of spheroids and circular hyperboloids of two sheets having two foci at the Kepler centers are integrable.The same holds for the projections of these systems on the three-dimensional sphere andin the three-dimensional hyperbolic space by means of central projection. Using the same approach, we also extend these results to the (n)-dimensional cases.

In this paper, we consider the equation (u_{xx}+Q(x)u+P(x)u^{3}=0) where (Q(x)) and (P(x)) are periodicfunctions. It is known that, if (P(x)) changes sign, a “great part” of the solutions for thisequation are singular, i. e., they tend to infinity at a finite point of the real axis. Our aim is to describe as completely as possible solutions, which are regular (i. e., not singular) on (mathbb{R}). For this purpose we consider the Poincaré map (mathcal{P}) (i. e., the map-over-period) for this equation and analyse the areas of the plane ((u,u_{x})) where (mathcal{P}) and (mathcal{P}^{-1}) are defined. We give sufficient conditions for hyperbolic dynamics generated by (mathcal{P}) in these areas and show that the regular solutions correspond to a Cantor set situated in these areas. We also present a numerical algorithm for verifying these sufficient conditions at the level of “numerical evidence”. This allows us to describe regular solutions of this equation, completely or within some class, by means of symbolic dynamics. We show that regular solutions can be coded by bi-infinite sequences of symbols of some alphabet, completely or within some class. Examples of the application of this technique are given.

We consider time-dependent perturbations of integrable and near-integrable Hamiltonians. Assuming the perturbation decays polynomially fast as time tends to infinity, we prove the existence of biasymptotically quasi-periodic solutions. That is, orbits converging to some quasi-periodic solutions in the future (as (tto+infty)) and the past (as (tto-infty)).

Concerning the proof, thanks to the implicit function theorem, we prove the existence of a family of orbits converging to some quasi-periodic solutions in the future and another family of motions converging to some quasi-periodic solutions in the past. Then, we look at the intersection between these two familieswhen (t=0). Under suitable hypotheses on the Hamiltonian’s regularity and the perturbation’s smallness, it is a large set, and each point gives rise to biasymptotically quasi-periodic solutions.

The recent detection of gravitational waves emanating from inspiralling black hole binaries has triggered a renewed interest in the dynamics of relativistic two-body systems. The conservative part of the latter are given by Hamiltonian systems obtained from so-called post-Newtonian expansions of the general relativistic description of black hole binaries. In this paper we study the general question of whether there exist relativistic binaries that display Kepler-like dynamics with elliptical orbits. We show that an orbital equivalence to the Kepler problem indeed exists for relativistic systems with a Hamiltonian of a Kepler-like form. This form is realised by extremal black holes with electric charge and scalar hair to at least first order in the post-Newtonian expansion for arbitrary mass ratios and to all orders in the post-Newtonian expansion in the test-mass limit of the binary. Moreover, to fifth post-Newtonian order, we show that Hamiltonians of the Kepler-like form can be related explicitly through a canonical transformation and time reparametrisation to the Kepler problem, and that all Hamiltonians conserving a Laplace – Runge – Lenz-like vector are related in this way to Kepler.

Let (Sigma) be a compact manifold without boundary whose first homology is nontrivial. The Hodge decomposition of the incompressible Euler equation in terms of 1-forms yields a coupled PDE-ODE system. The (L^{2})-orthogonal components are a “pure” vorticity flow and a potential flow (harmonic, with the dimension of the homology). In this paper we focus on (N) point vortices on a compact Riemann surface without boundary of genus (g), with a metric chosen in the conformal class. The phase space has finite dimension (2N+2g). We compute a surface of section for the motion of a single vortex ((N=1)) on a torus ((g=1)) with a nonflat metric that shows typical features of nonintegrable 2 degrees of freedom Hamiltonians. In contradistinction, for flat tori the harmonic part is constant. Next, we turn to hyperbolic surfaces ((ggeqslant 2)) having constant curvature (-1), with discrete symmetries. Fixed points of involutions yield vortex crystals in the Poincaré disk. Finally, we consider multiply connected planar domains. The image method due to Green and Thomson isviewed in the Schottky double. The Kirchhoff – Routh Hamiltoniangiven in C. C. Lin’s celebrated theorem is recovered byMarsden – Weinstein reduction from (2N+2g) to (2N).The relation between the electrostatic Green function and thehydrodynamic Green function is clarified.A number of questions are suggested.