Regular and Chaotic Dynamics最新文献

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.

The classical result of Eisenhart states that, if a Riemannian metric (g) admits a Riemannian metric that is not constantly proportional to (g) and has the same (parameterized) geodesics as (g) in a neighborhood of a given point, then (g) is a direct product of two Riemannian metrics in this neighborhood. We introduce a new generic class of step (2) graded nilpotent Lie algebras, called (mathrm{ad})-surjective, and extend the Eisenhart theorem to sub-Riemannian metrics on step (2) distributions with (mathrm{ad})-surjective Tanaka symbols. The class of ad-surjective step (2) nilpotent Lie algebras contains a well-known class of algebras of H-type as a very particular case.

We study bifurcations of symmetric elliptic fixed points in the case of p:q resonances with odd (qgeqslant 3). We consider the case where the initial area-preserving map (bar{z}=lambda z+Q(z,z^{*})) possesses the central symmetry, i. e., is invariant under the change of variables (zto-z), (z^{*}to-z^{*}). We construct normal forms for such maps in the case (lambda=e^{i2pifrac{p}{q}}), where (p) and (q) are mutually prime integer numbers, (pleqslant q) and (q) is odd, and study local bifurcations of the fixed point (z=0) in various settings. We prove the appearance of garlands consisting of four (q)-periodic orbits, two orbits are elliptic and two orbits are saddles, and describe the corresponding bifurcation diagrams for one- and two-parameter families. We also consider the case where the initial map is reversible and find conditions where nonsymmetric periodic orbits of the garlands are nonconservative (contain symmetric pairs of stable and unstable orbits as well as area-contracting and area-expanding saddles).

Heteroclinic cycles are widely used in neuroscience in order to mathematically describe different mechanisms of functioning of the brain and nervous system. Heteroclinic cycles and interactions between them can be a source of different types of nontrivial dynamics. For instance, as it was shown earlier, chaotic dynamics can appear as a result of interaction via diffusive couplings between two stable heteroclinic cycles between saddle equilibria. We go beyond these findings by considering two coupled stable heteroclinic cycles rotating in oppositedirections between weak chimeras. Such an ensemble can be mathematically described by a system of six phase equations. Using two-parameter bifurcation analysis, we investigate the scenarios ofemergence and destruction of chaotic dynamics in the system under study.