Comments on “Modeling of chaotic motion of gyrostats in resistant environment on the base of dynamical systems with strange attractors” (Commun Nonlinear Sci Numer Simul 2011;16:3188-3202)

Extensive prior literature (not referenced in the paper) on links between mechanical gyrostats and nonlinear dynamical systems with strange attractors is called to the readers’ attention.     

Bifurcations of relative equilibria of an oblate gyrostat with a discrete damper

We investigate relative equilibria of an oblate gyrostat with a discrete damper. Linear and nonlinear methods yield stability conditions for simple spins about the nominal principal axes. We use analytical and numerical methods to explore other equilibria, including bifurcations that occur for varying rotor momentum and damper parameters. These bifurcations are complex structures that are perturbations of the zero rotor momentum case. We use Lyapunov–Schmidt reduction to determine an analytic relationship between parameters to determine conditions for which a jump phenomenon occurs. This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States     

Chaotic motion in a class of asymmetrical Kelvin type gyrostat satellite

This work is an investigation on the roots of chaotic attitudinal motion in a class of asymmetrical gyrostat satellites. The result shows that for a class of Kelvin type gyrostat satellite, there is an equivalent rigid spinning satellite with the same attitude dynamics. Finding some constants of motion and eliminating the cyclic coordinates, the rotational kinetic energy is changed to a quadratic form and using Jordan canonical form of the associated inertia tensor and transforming the coordinate system, the Hamiltonian has been changed to those of a rigid satellite. The Hamiltonian has been split into integrable and non-integrable parts. Using Deprit canonical transformation and Andoyer variables the integrable part has been reduced to a one-dimensional form. The reduced Hamiltonian shows that the regular dynamics of the satellite can be chaotic, under the influence of gravitational effects. To demonstrate various attitudinal dynamics of the satellite, a second-order Poincaré map is employed. This research shows firstly, that the attitudinal dynamics of Kelvin type gyrostat satellites and rigid satellites follow the same dynamical patterns, secondly, for non-linear analysis of dynamics of gyrostat satellite based on the perturbation methods, there is a preferable form for Hamiltonian of the system in the near-integrable fashion and thirdly the chaotic motion is originated from the gravitational field effects that can be suppressed by increasing the attitudinal energy of the satellite in comparison with the translational energy.     

Analytical solutions for dynamics of dual-spin spacecraft and gyrostat-satellites under magnetic attitude control in omega-regimes

The attitude dynamics of a dual-spin spacecraft (a gyrostat with one rotor) with magnetic actuators attitude control is considered in the constant external magnetic field at the presence of the spacecraft’s own magnetic dipole moment, which is created proportionally to the angular velocity components (this motion regime can be called as “the omega-regime” or “the omega-maneuver”). The research of the dual-spin spacecraft angular motion under the action of the magnetic restoring torque is fulfilled in the generalized formulation close to the classical mechanics’ task of the heavy body/gyrostat motion in the Lagrange top. Analytical exact solutions of differential equations of the motion are obtained for all parameters in terms of elliptic integrals and the Jacobi functions. New obtained analytical solutions can be classified as results developing the classical fundamental problem of the rigid body and gyrostat motion around the fixed point. The technical application of the omega-regime to the angular reorientation of the spacecraft longitudinal axis along the angular momentum vector is considered.     

Analysis of attitude motion evolutions of variable mass gyrostats and coaxial rigid bodies system

This work involves the research into angular motion of variable mass gyrostats, coaxial bodies systems and dual-spin spacecraft in a translating coordinate frame. The variability of mass-inertia parameters of coaxial bodies causes non-trivial changes of system angular motion. The article describes qualitative method for phase space analysis, based on the evaluation of a phase trajectory curvature. The method can be used to investigate the phase trajectory shape and to synthesize conditions for special motion modes realization (for example, monotone decreasing/increasing of nutation angle). The paper results can be used to describe the motion of a variable mass dual-spin spacecraft, performing active maneuvers.     

Homoclinic solutions and motion chaotization in attitude dynamics of a multi-spin spacecraft

The attitude dynamics of the multi-spin spacecraft (MSSC) and the torque-free angular motion of the multi-rotor system are considered. Some types of homoclinic and general solutions are obtained in hyperbolic and elliptic functions. The local homoclinic chaos in the MSSC angular motion is investigated under the influence of polyharmonic perturbations. Some possible applications of the multi-rotor system are indicated, including gyrostat-satellites, dual-spin spacecraft, roll-walking robots, and also the inertialess method of the spacecraft attitude (angular) reorientation/control.     

带偏心转子陀螺体的混沌运动

本文讨论了无力矩条件下带有质量偏心轴对称转子的非对称陀螺体的运动。利用动量变量列写动力学方程,并将系统化作受周期微扰作用下的Ｅｕｌｅｒ－Ｐｏｉｎｓｏｔ运动。应用Ｍｅｌｎｉｋｏｖ方法预测系统存在Ｓｍａｌｄ马蹄意义下的混沌运动,此结论与Ｐｏｉｎｃａｒｅ截面的数值计算相符。从Ｐｏｉｎｃａｒｅ截面的相图也可明显看出转子对于双自旋卫星的姿态稳定作用。     

Global synchronization criteria for a class of third-order non-autonomous chaotic systems via linear state error feedback control

This paper investigates the global synchronization of a class of third-order non-autonomous chaotic systems via the master–slave linear state error feedback control. A sufficient global synchronization criterion of linear matrix inequality (LMI) and several algebraic synchronization criteria for single-variable coupling are proven. These LMI and algebraic synchronization criteria are then applied to two classes of well-known third-order chaotic systems, the generalized Lorenz systems and the gyrostat systems, proving that the local synchronization criteria for the chaotic generalized Lorenz systems developed in the existing literature can actually be extended to describe global synchronization and obtaining some easily implemented synchronization criteria for the gyrostat systems.     

Umbilical torus bifurcations in Hamiltonian systems

We consider perturbations of integrable Hamiltonian systems in the neighbourhood of normally umbilic invariant tori. These lower dimensional tori do not satisfy the usual non-degeneracy conditions that would yield persistence by an adaption of KAM theory, and there are indeed regions in parameter space with no surviving torus. We assume appropriate transversality conditions to hold so that the tori in the unperturbed system bifurcate according to a (generalised) umbilical catastrophe. Combining techniques of KAM theory and singularity theory we show that such bifurcation scenarios of invariant tori survive the perturbation on large Cantor sets. Applications to gyrostat dynamics are pointed out.