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2013
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Volume 21, Number 4

Bardin B. S.,  Chekina E. A.
On the Stability of Resonant Rotation of a Symmetric Satellite in an Elliptical Orbit
Abstract
We deal with the stability problem of resonant rotation of a symmetric rigid body about its center of mass in an elliptical orbit. The resonant rotation is a planar motion such that the body completes one rotation in absolute space during two orbital revolutions of its center of mass. In [1–3] the stability analysis of the above resonant rotation with respect to planar perturbations has been performed in detail.
In this paper we study the stability of the resonant rotation in an extended formulation taking into account both planar and spatial perturbations. By analyzing linearized equations of perturbed motion, we found eccentricity intervals, where the resonant rotation is unstable. Outside of these intervals a nonlinear stability study has been performed and subintervals of formal stability and stability for most initial data have been found. In addition, the instability of the resonant rotation was established at several eccentricity values corresponding to the third and fourth order resonances.
Our study has also shown that in linear approximation the spatial perturbations have no effect on the stability of the resonant rotation, whereas in a nonlinear system they can lead to its instability at some resonant values of the eccentricity.
Keywords: Hamiltonian system, symplectic map, normal form, resonance, satellite, stability
Citation: Bardin B. S.,  Chekina E. A., On the Stability of Resonant Rotation of a Symmetric Satellite in an Elliptical Orbit, Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 377-389
DOI:10.1134/S1560354716040018
Eldering J.
Realizing Nonholonomic Dynamics as Limit of Friction Forces
Abstract
The classical question whether nonholonomic dynamics is realized as limit of friction forces was first posed by Carath´eodory. It is known that, indeed, when friction forces are scaled to infinity, then nonholonomic dynamics is obtained as a singular limit.
Our results are twofold. First, we formulate the problem in a differential geometric context. Using modern geometric singular perturbation theory in our proof, we then obtain a sharp statement on the convergence of solutions on infinite time intervals. Secondly, we set up an explicit scheme to approximate systems with large friction by a perturbation of the nonholonomic dynamics. The theory is illustrated in detail by studying analytically and numerically the Chaplygin sleigh as an example. This approximation scheme offers a reduction in dimension and has potential use in applications.
Keywords: nonholonomic dynamics, friction, constraint realization, singular perturbation theory, Lagrange mechanics
Citation: Eldering J., Realizing Nonholonomic Dynamics as Limit of Friction Forces, Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 390-409
DOI:10.1134/S156035471604002X
Stolovitch L.,  Verstringe F.
Holomorphic Normal Form of Nonlinear Perturbations of Nilpotent Vector Fields
Abstract
We consider germs of holomorphic vector fields at a fixed point having a nilpotent linear part at that point, in dimension $n\geqslant 3$. Based on Belitskii's work, we know that such a vector field is formally conjugate to a (formal) normal form. We give a condition on that normal form which ensures that the normalizing transformation is holomorphic at the fixed point. We shall show that this sufficient condition is a nilpotent version of Bruno's condition ($A$). In dimension 2, no condition is required since, according to Stróżyna–Żoładek, each such germ is holomorphically conjugate to a Takens normal form. Our proof is based on Newton's method and $\mathfrak{sl}_2(\mathbb C)$-representations.
Keywords: local analytic dynamics, fixed point, normal form, Belitskii normal form, small divisors, Newton method, analytic invariant manifold, complete integrability
Citation: Stolovitch L.,  Verstringe F., Holomorphic Normal Form of Nonlinear Perturbations of Nilpotent Vector Fields, Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 410-436
DOI:10.1134/S1560354716040031
Norouzi H.,  Younesian D.
Chaos Control for the Plates Subjected to Subsonic Flow
Abstract
The suppression of chaotic motion in viscoelastic plates driven by external subsonic air flow is studied. Nonlinear oscillation of the plate is modeled by the von-Kármán plate theory. The fluid-solid interaction is taken into account. Galerkin’s approach is employed to transform the partial differential equations of the system into the time domain. The corresponding homoclinic orbits of the unperturbed Hamiltonian system are obtained. In order to study the chaotic behavior of the plate, Melnikov’s integral is analytically applied and the threshold of the excitation amplitude and frequency for the occurrence of chaos is presented. It is found that adding a parametric perturbation to the system in terms of an excitation with the same frequency of the external force can lead to eliminate chaos. Variations of the Lyapunov exponent and bifurcation diagrams are provided to analyze the chaotic and periodic responses. Two perturbation-based control strategies are proposed. In the first scenario, the amplitude of control forces reads a constant value that should be precisely determined. In the second strategy, this amplitude can be proportional to the deflection of the plate. The performance of each controller is investigated and it is found that the second scenario would be more efficient.
Keywords: chaos control, von-Kármán plate theory, Melnikov’s method, Lyapunov exponent, bifurcation
Citation: Norouzi H.,  Younesian D., Chaos Control for the Plates Subjected to Subsonic Flow, Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 437-454
DOI:10.1134/S1560354716040043
Borisov A. V.,  Mamaev I. S.,  Bizyaev I. A.
Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period
Abstract
In this historical review we describe in detail the main stages of the development of nonholonomic mechanics starting from the work of Earnshaw and Ferrers to the monograph of Yu.I. Neimark and N.A. Fufaev. In the appendix to this review we discuss the d’Alembert–Lagrange principle in nonholonomic mechanics and permutation relations.
Keywords: nonholonomic mechanics, nonholonomic constraint, d’Alembert–Lagrange principle, permutation relations
Citation: Borisov A. V.,  Mamaev I. S.,  Bizyaev I. A., Historical and Critical Review of the Development of Nonholonomic Mechanics: the Classical Period, Regular and Chaotic Dynamics, 2016, vol. 21, no. 4, pp. 455-476
DOI:10.1134/S1560354716040055