Regular and Chaotic Dynamics最新文献

The aim of this note is to present a simple observation that a slight refinement of thecontraction mapping principle allows one to recover the precise convergence rate in thePicard – Lindelöf theorem.

We consider the problem of existence of forced oscillations on a Riemannian manifold, the metric on which is defined by the kinetic energy of a mechanical system. Under the assumption that the generalized forces are periodic functions of time, we find periodic solutions of the same period. We present sufficient conditions for the existence of such solutions, which essentially depend on the behavior of geodesics on the corresponding Riemannian manifold.

The celebrated Poncelet porism is usually studied for a pair of smooth conics that are in a general position. Here we discuss Poncelet porism in the real plane — affine or projective, when that is not the case, i. e., the conics have at least one point of tangency or at least one of the conics is not smooth.In all such cases, we find necessary and sufficient conditions for the existence of an (n)-gon inscribed in one of the conics and circumscribed about the other.

Nonholonomic systems are mechanical systems with ideal velocity constraints that are not derivable from position constraints and with dynamics identified by the Lagrange – d’Alembert principle.This paper reviews infinite-dimensional and field-theoretic nonholonomic systems as well as Hamel’s formalism for these settings.

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.