Study of a discrete dynamic system in a neighborhood of a quasi-periodic trajectory

We study a discrete system in a neighborhood of a quasi-periodic trajectory. We obtain conditions for reducing a system in this neighborhood to a system with quasi-periodic coefficients. We determine the behavior of this system under the action of small perturbations.This work was prepared with the financial support of the Ukrainian State Committee on Science and Technology.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 12, pp. 1702–1711, December, 1992.     

Investigation of a nonlinear difference equation in a Banach space in a neighborhood of a quasiperiodic solution

We investigate the behavior of a diserete dynamical system in a neighborhood of a quasiperiodic trajeetory for the case of an infinite-dimensional Banach space We find conditions sufficient for the system considered to reduce, in such a neighborhood, to a system with quasiperiodic coefficients. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1661–1676, December, 1997.     

On the nature of dynamic chaos in a neighborhood of a separatrix of a conservative system

In the present paper, we give a new treatment of the mechanism of generation of chaotic dynamics in a perturbed conservative system in a neighborhood of the separatrix contour of a hyperbolic singular point of the unperturbed system. We theoretically prove and justify by three numerical examples of classical Hamiltonian systems with one and a half degrees of freedom and by an example of a simply conservative three-dimensional system that the complication of the dynamics in a conservative system as the perturbation increases is caused by a nonlocal effect of multiplication of hyperbolic and elliptic cycles (and the tori surrounding them), which has nothing in common with the mechanism of separatrix splitting in classical Hamiltonian mechanics.     

Reliability analysis of a system in a randomly changing environment

In reliability analysis of a repairable system, it is usually assumed that the system operates in, a constant environment. In fact, it needn't be so. For example, environment may have different stress levels, and the lifelength of a repairable system under these stresses might differ from each other, too.In this paper, we present a reliability analysis of a repairable system whose life length is changing with environment.Institute of Applied Mathematics, Academia Sinica     

Spectral properties of a Hamiltonian of a four-particle system on a lattice

We consider the Hamiltonian of a system of four arbitrary quantum particles with two-particle contact (noncompact) interaction potentials on a three-dimensional lattice perturbed by three-particle contact potentials. We describe the location of the essential spectrum of the Schrödinger operator corresponding to a four-particle system.     

Evolution of a regular precession of a solid body carrying a visco-elastic disk

The motion of inertia is studied of a system consisting of an axisymmetric solid body with fixed point and a homogeneous visco-elastic disk lying in the equatorial plane of the ellipsoid of inertia of the solid body (the center of disk coincides with the fixed point). In the case of a solid disk immobilized relative to the solid body the system accomplishes a regular precession (the case of Euler motion of a symmetric solid body with a fixed point /1/). The deformation of the disk is taking place in the plane of the disk, and is accompanied by energy dissipation is the cause of the regular precession finishing by steady rotation about the vector of the moment of momentum of the system /2/.     

Rolling of a ball without spinning on a plane: the absence of an invariant measure in a system with a complete set of integrals

In the paper we consider a system of a ball that rolls without slipping on a plane. The ball is assumed to be inhomogeneous and its center of mass does not necessarily coincide with its geometric center. We have proved that the governing equations can be recast into a system of six ODEs that admits four integrals of motion. Thus, the phase space of the system is foliated by invariant 2-tori; moreover, this foliation is equivalent to the Liouville foliation encountered in the case of Euler of the rigid body dynamics. However, the system cannot be solved in terms of quadratures because there is no invariant measure which we proved by finding limit cycles.     

Optimal Motion of a Two-Body System in a Resistive Medium

Locomotion of a mechanical system consisting of two rigid bodies, a main body and a tail, is considered. The system moves in a resistive fluid and is controlled by angular oscillations of the tail relative to the main body. The resistance force acting upon each body is assumed to be a quadratic function of its velocity. Under certain assumptions, a nonlinear equation is derived that describes the progressive motion of the system as a whole.     

Null controllability of a system of viscoelasticity with a moving control

In this paper, we consider the wave equation with both viscous Kelvin–Voigt and frictional damping as a model of viscoelasticity in which we incorporate an internal control with a moving support. We prove the null controllability when the control region, driven by the flow of an ODE, covers all the domain. The proof is based upon the interpretation of the system as, roughly, the coupling of a heat equation with an ordinary differential equation (ODE). The presence of the ODE for which there is no propagation along the space variable makes the controllability of the system impossible when the control is confined into a subset in space that does not move. The null controllability of the system with a moving control is established in using the observability of the adjoint system and some Carleman estimates for a coupled system of a parabolic equation and an ODE with the same singular weight, adapted to the geometry of the moving support of the control. This extends to the multi-dimensional case the results by P. Martin et al. in the one-dimensional case, employing 1-d Fourier analysis techniques.     

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于一个具有历史相依临界状态的可维修n中取k:G系统, 论文给出了当系统平稳时它的可用度, 一个循环中的平均工作时间和平均失效时间. 并且和不具有年龄相依临界状态的可维修n中取k:G系统进行了比较.     

Vibrations of a beam-string complex system under a moving force

In this paper, the dynamic response of an Euler-Bernoulli beam and string system traversed by a constant moving force is considered. The force is moving with a constant velocity on the top beam. The complex system is finite, simply supported, parallel one upon the other and continuously coupled by a linear Winkler elastic element. The classical solution of the response of a beam-string system subjected to a force moving with a constant velocity has a form of an infinite series. The main goal of this paper is to show that in the considered case the aperiodic part of the solution can be presented in a closed, analytical form instead of an infinite series. The presented method of finding the solution in a closed, analytical form is based on the observation that the solution of the system of partial differential equations in the form of an infinite series is also a solution of an appropriate system of ordinary differential equations. The dynamic influence lines of complex systems may be used for the analysis the complex models of moving load. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the oil-whipping of a rotor-bearing system by a continuum model

The nonlinear dynamic behavior of a rotor-bearing system is analyzed based on a continuum model. The finite element method is adopted in the analysis. Emphasis is placed on the so-called “oil-whip phenomena” which might lead to the failure of the rotor system. The dynamic response of the system in unbalanced conditions is approached by a direct integration method. It is found that a typical “oil-whip phenomenon” is successfully simulated, and the effect of the refinement of the finite element mesh is also checked. Furthermore, the bifurcation behavior of the oil-whip phenomenon that is of much concern in recent nonlinear dynamics research is analyzed. The rotor-bearing system is also examined by a simple discrete model. Significant differences are found between these two models. It is suggested that a careful examination should be made in modeling the nonlinear dynamic behavior of a rotor system.     

Optimization of the motion in a resistant medium of a body with a movable internal mass

The motion, in a resistant medium, of a system consisting of a rigid body and movable internal mass is considered. The external medium acts on the body by a force that piecewise linearly depends on its speed. The class of periodic motions of the internal mass for which the speed of this mass relative to the body is piecewise constant is studied. It is shown that, under certain conditions, the forward movement of the whole system in the medium is possible. The average speed of this movement over a period is determined. Optimal parameters of the motion of the internal mass for which the average speed of the system movement is maximal are found.     

Asymptotic Behavior of Solutions of a Strongly Nonlinear Model of a Crystal Lattice

We consider a system of hyperbolic nonlinear equations describing the dynamics of interaction between optical and acoustic modes of a complex crystal lattice (without a symmetry center) consisting of two sublattices. This system can be considered a nonlinear generalization of the well-known Born-Huang Kun model to the case of arbitrarily large sublattice displacements. For a suitable choice of parameters, the system reduces to the sine-Gordon equation or to the classical equations of elasticity theory. If we introduce physically natural dissipative forces into the system, then we can prove that a compact attractor exists and that trajectories converge to equilibrium solutions. In the one-dimensional case, we describe the structure of equilibrium solutions completely and obtain asymptotic solutions for the wave propagation. In the presence of inhomogeneous perturbations, this system is reducible to the well-known Hopfield model describing the attractor neural network and having complex behavior regimes.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 357–367, June, 2005.     

Realization of a polylinear controller as a second-order differential system in a Hilbert space

We study the solvability of the inverse nonlinear system analysis problem viewed as qualitative solvability (necessary and sufficient conditions) of a realization of a time-varying polylinear controller as a second-order differential system whose admissible solutions include a given nonlinear pencil of arbitrary (finite, countable, or continual) cardinality of dynamic processes in a separable Hilbert space.     

一类具有备用部件的可修人机系统解的渐近稳定性

运用泛函分析的方法 ,通过分析系统主算子的谱特征 ,给出一类具有备用部件的可修人机系统解的渐近稳定性证明 .     

Control of a novel fractional hyperchaotic system using a located control method

Fractional order dynamics and chaotics systems have been recently combined, yielding interesting behaviours. In this paper, a novel integer order hyperchaotic system is considered. Then, a fractional order hyperchaotic representation of said system is proposed using a natural fractionalization. Two different linear control methodologies to deal with the complexity which introduce such systems are proposed. Those methods are able to modify the hyperchaotic behaviour of the system and force it to move towards a fixed point; i.e. steady state. These approaches give a general framework for taming such complex systems using simple linear controllers. The main tools for analysing the controlled system are Matignon stability criterion and RouthHurwitz test. Using a reliable numerical simulation, the designed system is simulated to verify the theoretical analysis.     

Lifetime properties of a cumulative shock model with a cluster structure

This paper studies a generalized cumulative shock model with a cluster shock structure. The system considered is subject to two types of shocks, called primary shocks and secondary shocks, where each primary shock causes a series of secondary shocks. The lifetime behavior of such a system becomes more complicated than that of a classical model with only one class of shocks. Under a non-homogeneous Poisson process of primary shocks, we analyze the lifetime behavior of the system with light-tailed and heavy-tailed distributed secondary shocks. We show some important characteristics of lifetime of this type of system. Our model, as an extension of the classical shock models, has wide applications in maintenance engineering, operations management, and insurance risk assessment.     

Rolling of a homogeneous ball over a dynamically asymmetric sphere

We consider a novel mechanical system consisting of two spherical bodies rolling over each other, which is a natural extension of the famous Chaplygin problem of rolling motion of a ball on a plane. In contrast to the previously explored non-holonomic systems, this one has a higher dimension and is considerably more complicated. One remarkable property of our system is the existence of “clandestine” linear in momenta first integrals. For a more trivial integrable system, their counterparts were discovered by Chaplygin. We have also found a few cases of integrability.     

Analysis of a single-server queue interacting with a fluid reservoir

We consider a single-server queueing system with Poisson arrivals in which the speed of the server depends on whether an associated fluid reservoir is empty or not. Conversely, the rate of change of the content of the reservoir is determined by the state of the queueing system, since the reservoir fills during idle periods and depletes during busy periods of the server. Our interest focuses on the stationary joint distribution of the number of customers in the system and the content of the fluid reservoir, from which various performance measures such as the steady-state sojourn time distribution of a customer may be obtained. We study two variants of the system. For the first, in which the fluid reservoir is infinitely large, we present an exact analysis. The variant in which the fluid reservoir is finite is analysed approximatively through a discretization technique. The system may serve as a mathematical model for a traffic regulation mechanism - a two-level traffic shaper - at the edge of an ATM network, regulating a very bursty source. We present some numerical results showing the effect of the mechanism. This revised version was published online in June 2006 with corrections to the Cover Date.