Isochronicity conditions for some planar polynomial systems II

We study the isochronicity of centers at $O\in \mathbb{R}^2$ for systems $$\dot x=-y+A(x,y),\;\dot y=x+B(x,y),$$ where $A,\;B\in \mathbb{R}[x,y]$, which can be reduced to the Li\'enard type equation. When $deg(A)\leq 4$ and $deg(B) \leq 4$, using the so-called C-algorithm we found $36$ new families of isochronous centers. When the Urabe function $h=0$ we provide an explicit general formula for linearization. This paper is a direct continuation of \cite{BoussaadaChouikhaStrelcyn2010} but can be read independantly

On estimates of the balanced gyro drift and the accuracy of the magnus formula

We construct efficient estimates of the balanced gyro drift due to nutation oscillations. We show that, for oscillation amplitudes not exceeding one degree and for amajority of gyromotions, the relative error in the drift rate calculations on the basis of the obtained estimates does not exceed tenths of one percent. The numerical results show that the accuracy of the Magnus formula is not worse than predicted by these estimates

On Steady Motions of a Rigid Body Bearing Three-Degree-of-Freedom Control Momentum Gyroscopes and Their Stability

Equations of motion are obtained for a rigid body bearing N three-degree-of-freedom control momentum gyroscopes in gimbals and the entire set of steady motions in a homogeneous external field is determined. The steady motion dependence on the magnitude of the system angular momentum is studied and a detailed analysis of the secular stability is performed. In the case of dissipative forces acting in the gyroscope gimbal axes, the Barbashin-Krasovskii theorem is used to study stability in the sense of Lyapunov. It is shown that, depending on the angular momentum magnitude, either static states of the system or two motions corresponding to rotations of the bearing body about the axis of the greatest moment of inertia are asymptotically stable, while all the other stationary motions are unstable in the sense of Lyapunov

On the dynamical symmetry points and the orientations of the principal axes of inertia of a rigid body

For an arbitrary rigid body, all dynamical symmetry points are found, and the directions of the axes of dynamical symmetry are determined for these points. We obtain conditions on the principal central moments of inertia under which the Lagrange and Kovalevskaya cases can be realized for the rigid body. We also analyze the set of orientations of the bases formed by the principal axes of inertia for various points of the rigid body

On steady rotations of a rigid body bearing a single-axis powered gyroscope whose precession axis is parallel to a principal plane of inertia

The set of steady motions of the system named in the title is represented parametrically via the gyro gimbal rotation angle for an arbitrary position of the gimbal axis. We study the set of steady motions for a system in which the gyro gimbal axis is parallel to a principal plane of inertia as well as for a system with a dynamic symmetry. We determine all motions satisfying sufficient stability conditions. In the presence of dissipation in the gimbal axis, we use the Barbashin-Krasovskii theorem to identify each steady motion as either conditionally asymptotically stable or unstable

On Steady Motions of a Rigid Body Bearing a Two-Degree-of-Freedom Control Momentum Gyroscope with Precession Axis Arbitrarily Oriented in the Carrying Body

Steady motions of a rigid body with a control momentum gyroscope are studied versus the gimbal axis direction relative to the body and the magnitude of the system angular momentum. The study is based on a formula that gives a parametric representation of the set of the system steady motions in terms of the rotation angle of the gimbal. It is shown that, depending on the values of the parameters, the system has S, 12, or 16 steady motions and the number of stable motions is 2 or 4