Regular and Chaotic Dynamics最新文献

In this paper, we consider the minimum time problem for a space rocket whose dynamics is given by a control-affine system with drift. The admissible control set is a disc.We study extremals in the neighbourhood of singular points of the second order.Our approach is based on applying the method of a descending system of Poissonbrackets and the Zelikin – Borisov method for resolution of singularities to the Hamiltonian system of Pontryagin’s maximum principle. We show that in the neighbourhoodof any singular point there is a family of spiral-like solutions of the Hamiltonian system that enter the singular point in a finite time, while the control performs an infinite number of rotations around the circle.

We consider a skew product (F_{A}=(sigma_{omega},A)) over irrational rotation (sigma_{omega}(x)=x+omega) of a circle (mathbb{T}^{1}). It is supposed that the transformation (A:mathbb{T}^{1}to SL(2,mathbb{R}))which is a (C^{1})-map has the form (A(x)=Rbig{(}varphi(x)big{)}Zbig{(}lambda(x)big{)}), where (R(varphi)) is a rotation in (mathbb{R}^{2}) through the angle (varphi) and (Z(lambda)=text{diag}{lambda,lambda^{-1}}) is a diagonal matrix. Assuming that (lambda(x)geqslantlambda_{0}>1) with a sufficiently large constant (lambda_{0}) and the function (varphi)is such that (cosvarphi(x)) possesses only simple zeroes, we study hyperbolic properties ofthe cocycle generated by (F_{A}). We apply the critical set method to show that, under someadditional requirements on the derivative of the function (varphi), the secondary collisions compensate weakening of the hyperbolicity due to primary collisions and the cocycle generated by (F_{A}) becomes uniformly hyperbolicin contrast to the case where secondary collisions can be partially eliminated.

We consider the nonholonomic systems of (n) homogeneous balls (mathbf{B}_{1},dots,mathbf{B}_{n}) with the same radius (r) that are rolling without slipping about a fixed sphere (mathbf{S}_{0}) with center (O) and radius (R).In addition, it is assumed that a dynamically nonsymmetric sphere (mathbf{S}) with the center that coincides with the center (O) of the fixed sphere (mathbf{S}_{0}) rolls withoutslipping in contact with the moving balls (mathbf{B}_{1},dots,mathbf{B}_{n}). The problem is considered in four different configurations, three of which are new.We derive the equations of motion and find an invariant measure for these systems.As the main result, for (n=1) we find two cases that are integrable by quadratures according to the Euler – Jacobi theorem.The obtained integrable nonholonomic models are natural extensions of the well-known Chaplygin ball integrable problems.Further, we explicitly integratethe planar problem consisting of (n) homogeneous balls of the same radius, but with differentmasses, which roll without slippingover a fixed plane (Sigma_{0}) with a plane (Sigma) that moves without slipping over these balls.

In this paper we investigate a nonholonomic system with parametric excitation, a Roller Racer with variable gyrostatic momentum. We examine in detail the problem of the existence of regimes with unbounded growth of energy (nonconservative Fermi acceleration). We find a criterion for the existence of trajectories for which one of the velocity components increases withound bound and has asymptotics (t^{1/3}). In addition, we show that the problem under consideration reduces to analysis of a three-dimensional Poincaré map. This map exhibits both regular attractors (a fixed point, a limit cycle and a torus) and strange attractors.

The problem of the rolling of a disk on a plane is considered under the assumption that there is no slipping in the direction parallel to the horizontal diameter of the disk and that the center of mass does not move in the horizontal direction. This problem is reduced to investigating a system of three first-order differential equations. It is shown that the reduced system is reversible relative to involution of codimension one and admits a two-parameter family of fixed points. The linear stability of these fixed points is analyzed. Using numerical simulation, the nonintegrability of the problem is shown. It is proved that the reduced system admits, even in the nonintegrable case, a two-parameter family of periodic solutions. A number of dynamical effects due to the existence of involution of codimension one and to the degeneracy of the fixed points of the reduced system are found.

We study billiard systems within an ellipsoid in the (4)-dimensional pseudo-Euclidean spaces. We provide an analysis and description of periodic and weak periodic trajectories in algebro-geometric and functional-polynomial terms.

In 1983 Bogoyavlenski conjectured that, if the Euler equations on a Lie algebra (mathfrak{g}_{0}) are integrable, then their certain extensions to semisimple lie algebras (mathfrak{g}) related to the filtrations of Lie algebras(mathfrak{g}_{0}subsetmathfrak{g}_{1}subsetmathfrak{g}_{2}dotssubsetmathfrak{g}_{n-1}subsetmathfrak{g}_{n}=mathfrak{g}) are integrable as well.In particular, by taking (mathfrak{g}_{0}={0}) and natural filtrations of ({mathfrak{so}}(n)) and (mathfrak{u}(n)), we haveGel’fand – Cetlin integrable systems. We prove the conjecturefor filtrations of compact Lie algebras (mathfrak{g}): the system is integrable in a noncommutative sense by means of polynomial integrals.Various constructions of complete commutative polynomial integrals for the system are also given.

We consider Hamiltonian systems possessing families of nonresonant invariant tori whose frequencies are all collinear.Then under certain conditions the frequencies depend on energy only.This is a generalization of the well-known Gordon’s theorem about periodic solutions of Hamiltonian systems.While the proof of Gordon’s theorem uses Hamilton’s principle, our result is based on Percival’s variational principle. This work was motivated by the problem of isochronicity in Hamiltonian systems.