First integrals and families of symmetric periodic motions of a reversible mechanical system

A reversible mechanical system which allows of first integrals is studied. It is established that, for symmetric motions, the constants of the asymmetric integrals are equal to zero. The form of the integrals of a reversible linear periodic system corresponding to zero characteristic exponents and the structure of the corresponding Jordan Boxes are investigated. A theorem on the non-existence of an additional first integral and a theorem on the structural stabilities of having a symmetric periodic motion (SPM) are proved for a system with m symmetric and k asymmetric integrals. The dependence of the period of a SPM on the constants of the integrals is investigated. Results of the oscillations of a quasilinear system in degenerate cases are presented. Degeneracy and the principal resonance: bifurcation with the disappearance of the SPM and the birth of two asymmetric cycles, are investigated. A heavy rigid body with a single fixed point is studied as the application of the results obtained. The Euler-Poisson equations are used. In the general case, the energy integral and the geometric integral are symmetric while the angular momentum integral turns out to be asymmetric. In the special case, when the centre of gravity of the body lies in the principal plane of the ellipsoid of inertia, all three classical integrals become symmetric. It is ascertained here that any SPM of a body contains four zero characteristic exponents, of which two are simple and two form a Jordan Box. In typical situation, the remaining two characteristic exponents are not equal to zero. All of the above enables one to speak of an SPM belonging to a two-parameter family and the absence of an additional first integral. It is established that a body also executes a pendulum motion in the case when the centre of gravity is close to the principal plane of the ellipsoid of inertia.     

Controllability and stabilization of programmed motions of reversible mechanical and electromechanical systems

Problems of controllability and methods of stabilizing programmed motions of a large class of mechanical and electromechanical systems which are reversible with respect to the control are considered. Criteria of the controllability and stabilizability of reversible systems are obtained. Programmed motions and algorithms of programmed control are designed in analytical form and algorithms of programmed motions for non-linear reversible systems are synthesized.     

Reverse motions of mechanical systems

The possibility of the occurrence of sections of reverse motions in natural mechanical systems, when, in the second half of a time interval, the motion in the first half of the interval is repeated in the reverse order and the opposite velocity with a specified accuracy, is investigated. It is shown that such motions are characteristic of natural mechanical systems in the neighbourhood of a non-degenerate equilibrium position if the natural frequencies are independent. Systems with gyroscopic and dissipative forces are also considered. It is shown that, in these systems, sections of reverse motion can be observed in a special system of coordinates. Examples are presented.     

Gabor systems on discrete periodic sets

Due to its good potential for digital signal processing, discrete Gabor analysis has interested some mathematicians. This paper addresses Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. Complete Gabor systems and Gabor frames on discrete periodic sets are characterized; a sufficient and necessary condition on what periodic sets admit complete Gabor systems is obtained; this condition is also proved to be sufficient and necessary for the existence of sets E such that the Gabor systems generated by χ _E are tight frames on these periodic sets; our proof is constructive, and all tight frames of the above form with a special frame bound can be obtained by our method; periodic sets admitting Gabor Riesz bases are characterized; some examples are also provided to illustrate the general theory. This work was supported by National Natural Science Foundation of China (Grant No. 10671008), Beijing Natural Science Foundation (Grant No. 1092001), PHR (IHLB) and the project sponsored by SRF for ROCS, SEM of China     

Stability of the periodic motions of the vibro-impact systems

In many other papers a new direct method is developed for the study of vibro-impact system stability, a case in which the stability conditions are directly and rapidly obtained. This new method is stated and applied for systems with one or several degrees of freedom whose motion is analysed by Lagrange's equations. Thus, the equations in variations determined from the differential equations are considered, which by applying the boundary conditions lead by integration to the linearized equation in perturbations necessary to write the stability conditions. As example, is analysed in detail a particular case with one degree of freedom realised by the rod-crank mechanism. Also, a general system with two degrees of freedom is completely studied.     

Families of periodic orbits in resonant reversible systems

We study the dynamics near an equilibrium point p ₀ of a Z ₂(ℝ)-reversible vector field in ℝ²ⁿ with reversing symmetry R satisfying R ² = I and dimFix(R) = n. We deal with one-parameter families of such systems X _λ such that X ₀ presents at p ₀ a degenerate resonance of type 0: p: q. We are assuming that the linearized system of X ₀ (at p ₀) has as eigenvalues: λ₁ = 0 and λ _j = ±iα _j , j = 2, … n. Our main concern is to find conditions for the existence of one-parameter families of periodic orbits near the equilibrium.     

Dissipative periodic systems and symmetric interpolation in Schur classes

Matrix-valued functions analytic and contractive in the open unit disk (Schur functions) play an important role in system theory. They represent transfer functions of causal time-invariant dissipative systems. In this paper we show how a Schur function can still be associated to a k-periodic system. This function satisfies a certain symmetry condition (conditions (1.7) for k=2 and (4.4) for general k). We study a general bitangential interpolation problem for the Schur functions satisfying this symmetry condition.     

Invariant tori for a reversible oscillator with a nonlinear damping and periodic forcing term

In this paper, the boundedness of all solutions of the oscillator

x^″+f(x,x^′)+ω²x+?(x)=p(t)$x^{″}+f(x,x^{\prime })+{\omega }^{2}x+?(x)=p(t)$

Invariant curves and boundedness of solutions of a class of reversible systems

In this paper, the boundedness of all solutions for the following planar reversible system Ju ′ = ?H (u) + G (u) + h (t) (1) is discussed, where the function H (u) ∈ C²(?², ?⁺) is positive for u ≠ 0 and positively (q, p)‐quasihomogeneous of quasi‐degree pq, G ∈ C⁵ is bounded, h ∈ C⁶ is 2π ‐periodic and J is the standard symplectic matrix. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stationary and nonstationary stability of periodic solutions of reversible systems

M. V. Lomonosov Moscow State Univeristy. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 23, No. 2, pp. 40–48, April–June, 1989.     

Stabilization of the motions of non-autonomous mechanical systems

The problem of stabilizing the motions of mechanical systems that can be described by non-autonomous systems of ordinary differential equations is considered. The sufficient conditions for stabilizing of the motions of mechanical systems with assigned forces due to forces of another structure are obtained by constructing a vector Lyapunov function and a reference system. Examples of the solution of the problems of stabilizing the rotational motion of an axisymmetric satellite in an elliptic orbit, a non-tumbling gyro horizon, etc. are considered ©2009     

Invariant manifolds of periodic orbits for piecewise linear three-dimensional systems

This paper analyses the existence of invariant manifolds ofperiodic orbits for a specific piecewise linear three-dimensionalsystem with two zones, whose linear parts share a pair of imaginaryeigenvalues. This degenerate situation is obtained from thelack of controllability. The analysis proceeds by its reductionto a periodic one-dimensional equation for which some resultsof the Ambrosetti–Prodi type are given.     

The determination of the periodic motions of systems with delay

Continuing the development of results previously obtained for systems with delay described by first-order differential equations with delay [1], a system without delay is constructed which enables the periodic motions of systems with delay to be found.