The Quantum Cellular Automaton as a Markov Process

The dynamical behavior of the Quantum Cellular Automaton (QCA) is described here as a Markov Process. Ergodicity and recurrence, emergent properties of the discrete dynamical QCA system, are defined in the context of the characteristic polynomial of the Markov transition matrix. Except for a few anomalous cases, the transition matrix can be used to predict recurrence times. Finally, a correspondence between recurrence and elementary particle mass is proposed as an example of an emergent property of the QCA system. © 1999 Elsevier Science Ltd. All rights reserved.     

Non-smooth dynamics of a magnetic track brake

During the first stages of braking with a magnetic track brake in low suspension impacts occur between the force transmitting components, giving rise to a non-smooth system behaviour. Initially, when the electric current is switched on, the magnet moves down until it impinges on the rail. Thereafter, it is decelerated by the friction force, and subsequently there occurs a first impact of the transmission link of the magnet on the transmission link of the bogie frame. Due to the elasticity of the components there follows a high-frequency series of impacts with decreasing intensity until the velocity of the magnet relative to the bogie frame vanishes. Of course, the occurring forces are multiples of the steady-state ones, and this must be taken into account at the design of the force transmitting components.     

The complexity of social groups and social systems described by graph structures

This paper presents methodology which permits the complete ranking of nondirected graphs (NDG's) on an attribute labelled ‘complexity.’ The technique applies to both small and large systems as might arise in studies of group or organization behavior. The methodology extends to cover the complexity of directed graphs (DG's) and permits the detailed specification of individual and group behavior.For the NDG an abstract automaton representing the participants' interaction or communications function is sited at each node. Each automaton is constructed so its internal complexity is sufficient to realize the minimal social action (e.g. transmission of a rumor and the path followed by the rumor) within the framework of the NDG. It is shown that the complexity of each node automaton depends upon the order of the graph, the degree of the node and the longest path parameter of the graph. The combined complexity of node automata constitutes the complexity of the NDG. The complexity of a DG is specified as a composition of complexities computed for the associated NDG and logical devices which produce the observed behavior. Illustrative examples pertaining to the committee-subcommittee problem and to organizational structures are presented.     

Fractional dynamics of systems with long-range interaction

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction defined by a term proportional to 1/∣n − m∣^α+1. Continuous medium equation for this system can be obtained in the so-called infrared limit when the wave number tends to zero. We construct a transform operator that maps the system of large number of ordinary differential equations of motion of the particles into a partial differential equation with the Riesz fractional derivative of order α, when 0 < α < 2. Few models of coupled oscillators are considered and their synchronized states and localized structures are discussed in details. Particularly, we discuss some solutions of time-dependent fractional Ginzburg–Landau (or nonlinear Schrodinger) equation.     

The dynamics of a rigid body with a sharp edge in contact with an inclined surface in the presence of dry friction

In this paper we consider the dynamics of a rigid body with a sharp edge in contact with a rough plane. The body can move so that its contact point is fixed or slips or loses contact with the support. In this paper, the dynamics of the system is considered within three mechanical models which describe different regimes of motion. The boundaries of the domain of definition of each model are given, the possibility of transitions from one regime to another and their consistency with different coefficients of friction on the horizontal and inclined surfaces are discussed.     

Evolution of predator-prey systems described by a Lotka-Volterra equation under random environment

In this paper, we consider the evolution of a system composed of two predator-prey deterministic systems described by Lotka-Volterra equations in random environment. It is proved that under the influence of telegraph noise, all positive trajectories of such a system always go out from any compact set of with probability one if two rest points of the two systems do not coincide. In case where they have the rest point in common, the trajectory either leaves from any compact set of or converges to the rest point. The escape of the trajectories from any compact set means that the system is neither permanent nor dissipative.     

On the variational formulation of the dynamics of systems with friction

We discuss the basic problem of the dynamics of mechanical systems with constraints, namely, the problem of finding accelerations as a function of the phase variables. It is shown that in the case of Coulomb friction, this problem is equivalent to solving a variational inequality. The general conditions for the existence and uniqueness of solutions are obtained. A number of examples are considered. For systems with ideal constraints the problem under discussion was solved by Lagrange in his “Analytical Dynamics” (1788), which became a turning point in the mathematization of mechanics. In 1829, Gauss gave his principle, which allows one to obtain the solution as the minimum of a quadratic function of acceleration, called the constraint. In 1872 Jellett gave examples of non-uniqueness of solutions in systems with static friction, and in 1895 Painlevé showed that in the presence of friction, the absence of solutions is possible along with the nonuniqueness. Such situations were a serious obstacle to the development of theories, mathematical models and the practical use of systems with dry friction. An elegant, and unexpected, advance can be found in the work [1] by Pozharitskii, where the author extended the Gauss principle to the special case where the normal reaction can be determined from the dynamic equations regardless of the values of the coefficients of friction. However, for systems with Coulomb friction, where the normal reaction is a priori unknown, there are still only partial results on the existence and uniqueness of solutions [2–4]. The approach proposed here is based on a combination of the Gauss principle in the form of reactions with the representation of the nonlinear algebraic system of equations for the normal reactions in the form of a variational inequality. The theory of such inequalities [5] includes results on the existence and uniqueness, as well as the developed methods of solution.     

The longitudinal deformation of a rod in friction interaction with a compressing body

The deformation of a rod, confined in a fixed external housing, is considered. The friction forces in the contact surface are related to the deformation of the rod by a power relation. A wide range of variation of the friction parameter and the preliminary clearance parameter with which the rod is inserted into the housing is investigated and the characteristic features of the stress and strain distributions are revealed. The dissipation of energy due to friction and the formation of a hysteresis loop in the dependence of the stresses in the loaded end face on its displacement are considered. The problem is solved in a quasistatic formulation. Analytical relations are found for a number of important cases. Other results are obtained by numerical integration of the initial differential problem.     

The dynamics of a disc on an inclined plane with friction in the dynamically consistent model of friction

The problem of the motion of a uniform circular disc of non-zero thickness on a fixed rough inclined plane is discussed. It is proved that, if the ratio of the coefficient of sliding friction to the slope of the plane is greater than unity, the disc comes to rest after a finite time. If this ratio is equal to unity, the limit motion of the disc is uniform slip along the line of greatest slope, and if this ratio is less than unity, the slip velocity of the disc increases without limit with time.     

The natural oscillations of distributed inhomogeneous systems described by generalized boundary value problems

The problem of the constructive determination of the natural frequencies and modes of oscillations of distributed systems with substantially varying parameters is investigated. Unlike the classical case, the self-adjoint boundary-value problem allows of an arbitrary non-linear dependence of the coefficients of the equation on a numerical parameter, the eigenvalues of which are required to be obtained. An original numerical-analytic method is developed for a highly accurate construction of the desired solution. The computational efficiency of the algorithm, which possesses the property of accelerated (quadratic) convergence, is illustrated by the calculation of model examples. The approach can be extended to other classes of generalized problems of determining the critical values of the parameters and the forms corresponding to them, in particular, to the problem of the loss of stability of elastic systems with variable stiffnesses and inertial and force characteristics. A highly accurate solution of the classical Prandtl problem of determining the critical force which leads to lateral buckling of a long homogeneous cantilever beam is constructed, taking its weight into account.     

The conditions for the unique solvability of the equations of the dynamics of systems with friction

The problem of determining the generalized accelerations and reactions of constraints in systems with dry friction is investigated. The necessary and sufficient conditions for the unique solvability of the problem are obtained, applicable for cases of sliding and static friction. A geometrical approach is used, based on the introduction of a certain auxiliary parameter space divided into non-overlapping regions in terms of the number of possible types of motion. In each of these regions there are explicit expressions for the accelerations and reactions, which enable us, using piecewise-smooth mapping, to express, from the equations of motion, the generalized forces in terms of the parameters. The solution of the problem is equivalent to inverting the given mapping. A number of examples are given.     

The stability of equilibrium in systems with friction

Mechanical systems with non-ideal geometrical constraints are considered. The possible lack of uniqueness of the solution of the problem of determining the generalized accelerations and reactions with respect to specified coordinates and velocities is taken into account in solving the problem of the stability of an equilibrium state. A number of necessary and sufficient conditions of stability are obtained. It is shown that the results are also applicable in the case of unilateral constraints subject to the condition that a specific hypothesis concerning the character of the impacts on the constraints is adopted. A problem on the stability of a rigid body on a rough plane in the two-dimensional case is solved as an example.     

Interplay between dynamic systems described by the Klein-Gordon and Dirac equations

It is shown that in the two-dimensional space-time the dynamic system, described by the free Klein-Gordon equation, turns to the dynamic system, described by the free Dirac equation, provided the current and the energy-momentum tensor are redefined in a proper way.     

The dynamics of a homogeneous disc on an inclined plane with friction

The problem of the motion of a homogeneous circular cylinder along a fixed rough inclined plane is discussed. It is assumed that the cylinder is supported on the plane by its base and executes continuous motion. The frictional forces and moment are calculated within the limits of the dynamically consistent model proposed by Ivanov, for which the pressure distribution over the contact area is non-uniform. A qualitative analysis of the dynamics of the cylinder is given in the case when the slope of the plane is less then the Coulomb coefficient of friction.