Regular and Chaotic Dynamics最新文献

Recently it was proved that every billiard trajectory inside a (C^{3}) convex cone has a finite number of reflections. Here, by a (C^{3}) convex cone, we mean a cone whose section with some hyperplane is a strictly convex, closed (C^{3}) hypersurface of that hyperplane, with an everywhere nondegenerate second fundamental form. In this paper, we prove that there exist (C^{2}) convex cones with billiard trajectories that undergo infinitely many reflections in finite time. We also provide an estimation of the number of reflections for billiard trajectories inside elliptic cones in (mathbb{R}^{3}) using two first integrals.

We study two-dimensional Riemannian metrics which are superintegrable in the class ofintegrals polynomial in momenta.The study is based on our main technical result, Theorem 2, which states that thePoisson bracket of two integrals polynomial in momenta is an algebraic function ofthe integrals and of the Hamiltonian. We conjecture that two-dimensional superintegrable Riemannian metrics are necessarily real-analytic in isothermal coordinate systems, and give arguments supporting this conjecture. A small modification of the arguments, discussed in the paper, provides a method to construct new superintegrable systems. We prove a special case of the above conjecture which is sufficient to show thatthe metrics constructed by K. Kiyohara [9], which admit irreducibleintegrals polynomial in momenta, of arbitrary high degree (k), are not superintegrable andin particular do not admit nontrivial integrals polynomial in momenta, of degree lessthan (k). This result solves Conjectures (b) and (c) explicitly formulated in [4].

In this note, we briefly discuss how the singular KAM theory of [7] — which was worked out for the mechanical case (frac{1}{2}|y|^{2}+varepsilon f(x)) — can be extended to convex real-analyticnearly integrable Hamiltonian systemswith Hamiltonian in action-angle variables given by (h(y)+varepsilon f(x)) with (h) convex and(f) generic.

In this paper the two-dimensional Lorentzian problem on the anti-de Sitter plane is studied. Using methods of geometric control theory and differential geometry, we describe the reachable set, investigate the existence of Lorentzian length maximizers, compute extremal trajectories, construct an optimal synthesis, characterize Lorentzian distance and spheres, and describe the Lie algebra of Killing vector fields.

We present some new Poisson bivectors that are invariants by the Clebsch system flow. Symplectic integrators on their symplectic leaves exactly preserve the corresponding Casimir functions, which have different physical meanings. The Kahan discretization of the Clebsch system is discussed briefly.

In [7] an open set of structurally unstable families of vector fields on a sphere was constructed. More precisely, a vector field with a degeneracy of codimension three was discovered whose bifurcation in a generic three-parameter family has a numeric invariant. This vector field has a polycycle and two saddles, one inside and one outside this polycycle; one separatrix of the outside saddle winds towards the polycycle and one separatrix of the inside saddle winds from it. Families with functional invariants were constructed also. In [2] a hyperbolic polycycle with five vertices and no saddles outside it was constructed whose bifurcations in its arbitrary narrow neighborhood (semilocal bifurcations in other words) have a numeric invariant and thus are structurally unstable.This paper deals with semilocal bifurcations. A hyperbolic polycycle with nine edges is constructed whose semilocal bifurcation in an open set of nine-parameter families has a functional invariant.

In the present paper, we obtain real-analytic symplectic normal forms for integrable Hamiltoniansystems with (n) degrees of freedom in small neighborhoods of singular points having the type “universal unfolding of (A_{n}) singularity”, (ngeqslant 1) (local singularities), as well as in small neighborhoods of compact orbits containing such singular points (semilocal singularities).We also obtain a classification, up to real-analytic symplectic equivalence, of real-analytic Lagrangian foliations in saturated neighborhoods of such singular orbits (semiglobal classification).These corank-one singularities (local, semilocal and semiglobal ones) are structurally stable.It turns out that all integrable systems are symplectically equivalent near their singular points of this type, thus there are no local symplectic invariants.A complete semilocal (respectively, semiglobal) symplectic invariant of the singularityis given by a tuple of (n-1) (respectively (n-1+ell)) real-analytic function germs in (n) variables, where (ell) is the number of connected components of the complement of the singular orbit in the fiber.The case (n=1) corresponds to nondegenerate singularities (of elliptic and hyperbolic types)of one-degree-of-freedom Hamiltonians; their symplectic classifications were known.The case (n=2) corresponds to parabolic points, parabolic orbits and cuspidal tori.

We describe the Ozawa solution to the Davey – Stewartson II equation from the point of view of surface theory by presenting a soliton deformation of surfaces which is ruled by the Ozawa solution. The Ozawa solution blows up at a certain moment and we describe explicitly the corresponding singularity of the deformed surface.

In this paper we consider nonsingular Morse – Smale flows on closed orientable 3-manifolds, under the assumption that among the periodic orbits of the flow there is only one saddle orbit and it is twisted. It is found that any manifold admitting such flows is either a lens space, or a connected sum of a lens space with a projective space, or Seifert manifolds with base sphere and three special layers. A complete topological classification of the described flows is obtained and the number of their equivalence classes on each admissible manifold is calculated.