Regular and Chaotic Dynamics最新文献

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.

We consider the dynamics of a rod on the plane in a flow of non-interacting point particles moving at a fixed speed. When colliding with the rod, the particles are reflected elastically and then leave the plane of motion of the rod and do not interact with it. A thin unbending weightless “knitting needle” is fastened to themassive rod. The needle is attached to an anchor point and can rotate freely about it. The particles do not interact with the needle.

The equations of dynamics are obtained, which are piecewise analytic: the phase space is divided into four regions where the analytic formulas are different. There are two fixed points of the system, corresponding to the position of the rod parallel to the flow velocity, with the anchor point at the front and the back. It is found that the former point is topologically a stable focus, and the latter is topologically a saddle. A qualitative description of the phase portrait of the system is obtained.

We consider a homotopic to the identity family of maps, obtained as a discretization of the Lorenz system, such that the dynamics of the last is recovered as a limit dynamics when the discretization parameter tends to zero. We investigate the structure of the discrete Lorenz-like attractors that the map shows for different values of parameters. In particular, we check the pseudohyperbolicity of the observed discrete attractors and show how touse interpolating vector fields to compute kneading diagrams for near-identity maps. For larger discretization parameter values, the map exhibits what appears to be genuinely-discrete Lorenz-like attractors, that is, discrete chaotic pseudohyperbolic attractors with a negative second Lyapunov exponent. The numerical methods used are general enough to be adapted for arbitrary near-identity discrete systems with similar phase space structure.

We examine smooth four-dimensional vector fields reversible under somesmooth involution (L) that has a smooth two-dimensional submanifold of fixedpoints. Our main interest here is in the orbit structure of such a systemnear two types of heteroclinic connections involving saddle-foci andheteroclinic orbits connecting them. In both cases we found families ofsymmetric periodic orbits, multi-round heteroclinic connections andcountable families of homoclinic orbits of saddle-foci. All this suggests that the orbitstructure near such connections is very complicated. A non-variational version of the stationary Swift – Hohenberg equation is considered, as an example, where such structure has been found numerically.

The present paper is devoted to a study of orientation-preserving homeomorphisms on three-dimensional manifolds with a non-wandering set consisting of a finite number of surface attractors and repellers. The main results of the paper relate to a class of homeomorphisms for which the restriction of the map to a connected component of the non-wandering set is topologically conjugate to an orientation-preserving pseudo-Anosov homeomorphism. The ambient (Omega)-conjugacy of a homeomorphism from the class with a locally direct product of a pseudo-Anosov homeomorphism and a rough transformation of the circle is proved. In addition, we prove that the centralizer of a pseudo-Anosov homeomorphisms consists of only pseudo-Anosov and periodic maps.

We show that the stable invariant foliation of codimension 1 near a zero-dimensional hyperbolic set of a (C^{beta}) map with (beta>1) is (C^{1+varepsilon}) with some (varepsilon>0). The result is applied to the restriction of higher regularitymaps to normally hyperbolic manifolds. An application to the theory of the Newhouse phenomenon is discussed.