On the orbital stability of pendulum-like oscillations and rotations of a symmetric rigid body with a fixed point

We deal with the problem of orbital stability of planar periodic motions of a dynamically symmetric heavy rigid body with a fixed point. We suppose that the center of mass of the body lies in the equatorial plane of the ellipsoid of inertia. Unperturbed periodic motions are planar pendulum-like oscillations or rotations of the body around a principal axis keeping a fixed horizontal position. Local coordinates are introduced in a neighborhood of the unperturbed periodic motion and equations of the perturbed motion are obtained in Hamiltonian form. Regions of orbital instability are established by means of linear analysis. Outside the above-mentioned regions, nonlinear analysis is performed taking into account terms up to degree 4 in the expansion of the Hamiltonian in a neighborhood of unperturbed motion. The nonlinear problem of orbital stability is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients. The orbital stability is investigated analytically in two limiting cases: small amplitude oscillations and rotations with large angular velocities when a small parameter can be introduced.     

On the hydrodynamic motion in a domain with mixed boundary conditions: Existence,uniqueness, stability and linearization principle

Summary Existence, uniqueness, stability and instability theorems of hydrodynamic motion in a bounded domain with mixed (free boundary type) conditions are proved. Moreover a linearization principle is proved for an unperturbed periodic motion.Work performed under auspices of G.N.F.M.-C.N.R., also M.P.I. (40%) 20120201/81.     

Logarithmic matrix norms in motion stability problems

The problem of the stability of the motions of mechanical systems, described by non-linear non-autonomous systems of ordinary differential equations, is considered. Using the logarithmic matrix norm method, and constructing a reference system, the sufficient conditions for the asymptotic and exponential stability of unperturbed motion and for the stabilization of progammed motions of such systems are obtained. The problem of the asymptotic stability of a non-conservative system with two degrees of freedom is solved, taking for parametric disturbances into account. Examples of the solution of the problem of stabilizing programmed motions – for an inverted double pendulum and for a two-link manipulator on a stationary base – are considered.     

On the orbital stability of planar periodic motions of a rigid body in the Bobylev-Steklov case

We deal with the problem of orbital stability of pendulum-like periodic motions of a heavy rigid body with a fixed point. We suppose that a mass geometry corresponds to the Bobylev-Steklov case. The stability problem is solved in nonlinear setting. In the case of small amplitude oscillations and rotations with large angular velocities the small parameter can be introduced and the problem can be investigated analytically. In the case of unspecified oscillation amplitude or rotational angular velocity the problem is reduced to analysis of stability of a fixed point of the symplectic map generated by the equations of the perturbed motion. The coefficients of the symplectic map are determined numerically. Rigorous results on the orbital stability or instability of unperturbed motion are obtained by analyzing these coefficients.     

The oscillations of a satellite about a direction fixed in absolute space

The stability of the plane oscillations of a satellite about the centre of mass in a central Newtonian gravitational field is investigated. The orbit of the centre of mass is circular and the principal central moments of inertia of the satellite are different. In unperturbed motion, one of the axes of inertia is perpendicular to the plane of the orbit, while the satellite performs periodic oscillations about a direction fixed in absolute space. The problem of the stability of these oscillations with respect to plane and spatial perturbations is investigated.     

Stability by the linear approximation for discrete equations

This paper is concerned with exponential stability of solutions of perturbed discrete equations. For a given m>1 we will provide necessary and sufficient conditions for exponential stability of all perturbed systems with perturbation of order m under the assumption that the unperturbed linear system is exponentially stable. Basing on this result we obtained necessary and sufficient conditions for exponential stability of the perturbed system for all perturbations of order m>1 for regular systems. Our results are expressed in terms of regular coefficients of the unperturbed system.     

On the stability in terms of two measures for perturbed impulsive integro-differential equations

This paper establishes several stability criteria for perturbed impulsive integro-differential equations with fixed moments of impulsive effect. By using a new comparison theorem, which connects the solutions of perturbed system and the unperturbed one, some sufficient conditions for the stability in terms of two measures are obtained for the perturbed system while unperturbed one dissatisfied which because of the effect of the perturbed terms.     

On the instability of a periodic system in the case of internal resonance

The stability of periodic motion is studied in the critical case of n pairs of purely imaginary characteristic indices. It is shown that in the case of resonance, when the ratio of the modulus of one of the characteristic indices to the frequency of the unperturbed motion is an integer, instability usually occurs. The results obtained are used to study the free oscillations of an autonomous quasilinear system when the Andronov-Witt criterion /1/ cannot be used. The instability of free oscillations of the Froude pendulum at the bifurcation point is proved.     

Stabilization of a quasi-conservative system subjected to high frequency excitation

The existence and stability conditions for the steady motions and equilibrium positions of non-linear quasi-conservative systems with fast external perturbations having quasi-periodic and random components are investigated. A change of variables is proposed which reduces Lagrange's equations of the system to standard form. It is shown the averaged system of the first approximation has a canonical form and the effect of fast perturbations (not necessarily potential) is equivalent to a change in the system's potential. This leads to stabilization of unstable equilibrium positions and to the appearance of additional stationary points different from the equilibrium positions of the unperturbed system. The approach used is illustrated by examples; the stability of a pendulum on an elastic suspension when there is suspension point, and the steady motion of a sphere subjected to a high-frequency load. The critical loading of a double pendulum loaded by a pulsating tracking force is estimated. A form of wide-band random perturbations capable of stabilizing the system is considered.     

Rates of decay and growth of solutions to linear stochastic differential equations with state-independent perturbations

The paper studies the almost sure asymptotic convergence to zero of solutions of perturbed linear stochastic differential equations, where the unperturbed equation has an equilibrium at zero, and all solutions of the unperturbed equation tend to zero, almost surely. The perturbation is present in the drift term, and both drift and diffusion coefficients are state‐dependent. We determine necessary and sufficient conditions for the almost sure convergence of solutions to the equilibrium of the unperturbed equation. In particular, a critical polynomial rate of decay of the perturbation is identified, such that solutions of equations in which the perturbation tends to zero more quickly that this rate are almost surely asymptotically stable, while solutions of equations with perturbations decaying more slowly that this critical rate are not asymptotically stable. As a result, the integrability or convergence to zero of the perturbation is not by itself sufficient to guarantee the asymptotic stability of solutions when the stochastic equation with the perturbing term is asymptotically stable. Rates of decay when the perturbation is subexponential are also studied, as well as necessary and sufficient conditions for exponential stability.     

The exponential stability and stabilization of non-autonomous mechanical systems with non-conservative forces

The exponential stability of the unperturbed motion of a non-autonomous mechanical system with a complete set of forces, that is, dissipative, gyroscopic, potential and non-conservative positional forces, is investigated. The problem of stabilizing a non-autonomous system with specified non-conservative forces is considered with and without the use of potential forces. The problem of stabilizing a non-autonomous system with specified potential forces by the action of the forces of another structure is studied. The domain of stabilizability of the relative equilibrium position of a satellite in a circular orbit is found.     

Stability of a pit lifting vessel on a periodic system of vertical beams

Stability is considered for the unperturbed motion of a lifting vessel in two-sided guides in a vertical pit shaft, where allowance is made for the inertial behavior of the guides. A Hill equation is derived from the equations for the perturbed motion of the center of mass of the vessel and the elastic vibrations in the guides after certain simplifications. Numerical studies have been made on the stability of the solutions, and a region has been constructed for the main parametric resonance in this system.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 20, pp. 109–112, 1989.     

The Poincaré Map of Randomly Perturbed Periodic Motion

A system of autonomous differential equations with a stable limit cycle and perturbed by small white noise is analyzed in this work. In the vicinity of the limit cycle of the unperturbed deterministic system, we define, construct, and analyze the Poincaré map of the randomly perturbed periodic motion. We show that the time of the first exit from a small neighborhood of the fixed point of the map, which corresponds to the unperturbed periodic orbit, is well approximated by the geometric distribution. The parameter of the geometric distribution tends to zero together with the noise intensity. Therefore, our result can be interpreted as an estimate of the stability of periodic motion to random perturbations. In addition, we show that the geometric distribution of the first exit times translates into statistical properties of solutions of important differential equation models in applications. To this end, we demonstrate three distinct examples from mathematical neuroscience featuring complex oscillatory patterns characterized by the geometric distribution. We show that in each of these models the statistical properties of emerging oscillations are fully explained by the general properties of randomly perturbed periodic motions identified in this paper.     

On the stability of the vertical rotation of a solid suspended on a rod

The problem of the motion of a dynamically symmetric solid suspended from a fixed point by a weightless rod and two ball and socket joints one of which is fixed at the fixed point O', and the other is on the body axis of symmetry at the point O is considered. The question of the stability of the uniform body rotation when points O' and O, and the body centre of inertia C lie on the same vertical, and at the same time point O may be either above or below point O', and point C either above or below point O, is discussed. An analysis of the necessary and sufficient conditions for stability is given. The set of all the system's parameters is reduced to three independent dimensionless parameters L, Ω and β, and in the plane (L, Ω), for fixed values of β, the regions for which the unperturbed rotation is steady, or steady to a first approximation, or non-steady are indicated. The regions for which the body rotation is steady to a first approximation when the point O is situated higher than the point O', and the point C lies higher or lower than the point O are determined.

The sufficient conditions for the vertical rotation of a dynamically symmetric body suspended on a filament were obtained in /1/ and investigated for the cases where in non-perturbed motion the point C is below point O, when points C and O coincide, and when the length of the filament is zero (Lagrange gyroscope). In /2/ an analysis is given of the sufficient conditions for stability obtained in /1/, and also the necessary conditions for the cases where in a non-perturbed motion point C is located above point O.     

The use of first integrals in the problem of optimal control of a linear system with a quadratic functional

We propose a method for constructing an optimal control of a linear system in a variational problem with fixed time for the control process, fixed endpoints of the phase trajectory, and a quadratic functional. The method is based on the use of first integrals of the equations of unperturbed motion. We obtain sufficient conditions for complete controllability of the linear nonstationary system.     

Stabilization of Cowell's method

This paper offers a modification of the Cowell method for the integration of orbits. The modification is characterized by the property that it will integrate unperturbed Kepler motion exactly (excluding truncation errors), thus the slight instability of the Cowell Method is avoided. Furthermore, the modification takes into account the most important secular effects of orbit motion. As an example of the applicability of the modified method to perturbed motion, the equations of motion of an artificial earth satellite are integrated. In the case of elliptic initial conditions regularization by a Levi-Civita transformation was used.Visiting Professor from Institut für Angewandte Mathematik, Eidgenössische Technische Hochschule, Zurich, Switzerland.     

On the solution of the equations of perturbed motion of an object-parachute system

We propose a method of finding the solutions of the equations of perturbed motion of an object-parachute system in analytic form. We perform an analysis of the roots of the characteristic equation of the linearized system. On the basis of the analysis we determine all the solutions of the equations of perturbed motion of the object-parachute system. We exhibit conditions under which the unperturbed motion (descent with slipping) is asymptotically stable.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 41–46.     

Lower semicontinuity of the solution map to a parametric vector variational inequality

This paper is concerned with the study of solution stability of a parametric vector variational inequality, where mappings may not be strongly monotone. Under some requirements that the operator of a unperturbed problem is monotone or it satisfies degree conditions then we show that the solution map of a parametric vector variational inequality is lower semicontinuous.     

Nonlinear stability of a parabolic velocity profile in a plane periodic channel

An inviscid or viscous incompressible flow with a general parabolic velocity profile in an infinite plane periodic channel with parallel walls that can move is considered with the impermeability conditions (for the Euler equations) or the no-slip conditions (for the Navier-Stokes equations). The nonlinear (for the original equations) and nonlocal (for all Reynolds numbers) stability of the unperturbed flow with respect to arbitrary two-dimensional smooth perturbations of the initial velocity field is established.     

On complete controllability of linear nonstationary systems

By use of the first integrals of the equations of unperturbed motion we obtain necessary and sufficient conditions for complete controllability of linear nonstationary dynamical systems.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 35, 1992, pp. 160–163.