Valery V. Kozlov

Traffic and Granular Flow '13

The Intelligent Driver Model (IDM) is studied and several dr awbacks with respect to driving simulators are defined. We present two mod ifications of the IDM. The first one gives any predefined distance to the leading vehi cle in a steady state. The second modification is a combination of the first one and th e optimal velocity model. It takes into account driver’s reaction time expl icitly and is described by delay differential equation. This model always results in r ealistic vehicles accelerations what allows simulating real traffic collisions. Necessary and sufficient conditions are obtained, that guar antee a non-oscillating solution near the equilibrium for the vehicle platoon. We su ggest the calibrating framework based on a numerical solution of the constrained o ptimization problem. Nonlinear constraints are generated by the numerical integ ra ion scheme. The suggested procedure incorporates the local stability conditi ons obtained and takes into account vehicle dynamics, drivers’ behavior and weather co nditions.Part I: Pedestrian Dynamics and Evacuation Dynamics.- Part II: Highway and Urban Vehicular Traffic.- Part III: Biological Systems and Granular Flow.

Asymptotic stability and associated problems of dynamics of falling rigid body

We consider two problems from the rigid body dynamics and use new methods of stability and asymptotic behavior analysis for their solution. The first problem deals with motion of a rigid body in an unbounded volume of ideal fluid with zero vorticity. The second problem, having similar asymptotic behavior, is concerned with motion of a sleigh on an inclined plane. The equations of motion for the second problem are non-holonomic and exhibit some new features not typical for Hamiltonian systems. A comprehensive survey of references is given and new problems connected with falling motion of heavy bodies in fluid are proposed.

The Euler-Jacobi-Lie integrability theorem

This paper addresses a class of problems associated with the conditions for exact integrability of systems of ordinary differential equations expressed in terms of the properties of tensor invariants. The general theorem of integrability of the system of n differential equations is proved, which admits n − 2 independent symmetry fields and an invariant volume n-form (integral invariant). General results are applied to the study of steady motions of a continuum with infinite conductivity.

Gibbs Ensembles, Equidistribution of the Energy of Sympathetic Oscillators and Statistical Models of Thermostat

The paper develops an approach to the proof of the “zeroth” law of thermodynamics. The approach is based on the analysis of weak limits of solutions to the Liouville equation as time grows infinitely. A class of linear oscillating systems is indicated for which the average energy becomes eventually uniformly distributed among the degrees of freedom for any initial probability density functions. An example of such systems are sympathetic pendulums. Conditions are found for nonlinear Hamiltonian systems with finite number of degrees of freedom to converge in a weak sense to the state where the mean energies of the interacting subsystems are the same. Some issues related to statistical models of the thermostat are discussed.

On rational integrals of geodesic flows

This paper is concerned with the problem of first integrals of the equations of geodesics on two-dimensional surfaces that are rational in the velocities (or momenta). The existence of nontrivial rational integrals with given values of the degrees of the numerator and the denominator is proved using the Cauchy-Kovalevskaya theorem.

Asymptotic solutions of strongly nonlinear systems of differential equations

Preface -- Semi-quasihomogeneous systems of ordinary differential equations -- 2. The critical case of purely imaginary kernels -- 3. Singular problems -- 4. The inverse problem for the Lagrange theorem on the stability of equilibrium and other related problems -- Appendix A. Nonexponential asymptotic solutions of systems of functional-differential equations -- Appendix B. Arithmetic properties of the eigenvalues of the Kovalevsky matrix and conditions for the nonintegrability of semi-quasihomogeneous systems of ordinary dierential equations -- Bibliography.

A dynamical communication system on a network

A dynamical system is introduced and investigated. The system contains N vertices. The vertices send messages at discrete time instants according to a given rule. A conflict of two vertices takes place if the vertices try to send messages to each other at the same instant. Each vertex sends a message to another vertex at every step if no conflict takes place. In case of a conflict, only one of the two competing vertices sends a message. Deterministic and stochastic conflict resolution rules are considered. We investigate the average number of messages sent by a vertex per a time unit, called the productivity of this vertex, the total productivity of the system and other characteristics. The productivity of vertices depends on the initial state of the system, and the criterion of efficiency is the expected average productivity of vertices provided all possible initial states of the system are equiprobable. An ergodic version of the system is also considered in which any particle moves with approximately equal to 1 probability provided there is no conflict.

The Vlasov kinetic equation, dynamics of continuum and turbulence

We consider a continuum of interacting particles whose evolution is governed by the Vlasov kinetic equation. An infinite sequence of equations of motion for this medium (in the Eulerian description) is derived and its general properties are explored. An important example is a collisionless gas, which exhibits irreversible behavior. Though individual particles interact via a potential, the dynamics of the continuum bears dissipative features. Applicability of the Vlasov equations to the modeling of small-scale turbulence is discussed.

Monotonic walks on a necklace and a coloured dynamic vector

Stochastic and deterministic versions of a discrete dynamical system on a necklace are investigated. This network consists of a sequence of contours NSWE with nodes, i.e. the nodes are common points at W and E. There are two cells and a particle on each contour. Each time instance, the particle occupies a cell and, at every time unit, comes to the next cell in the same direction. The particles of the neighbouring contours move in accordance with rules of stochastic or deterministic type. The behaviour of the model with the rule of the first type is stochastic only at the beginning and after a time interval becomes a pure deterministic system. The system with the second rule comes to a steady mode, which depends on the initial state. The average velocity of particles and characteristics of the system are studied.

The dynamics of systems with servoconstraints. II

This paper addresses the dynamics of systems with servoconstraints where the constraints are realized by controlling the inertial properties of the system. Vakonomic systems are a particular case. Special attention is given to the motion on Lie groups with left-invariant kinetic energy and a left-invariant constraint. The presence of symmetries allows the dynamical equations to be reduced to a closed system of differential equations with quadratic right-hand sides. As the main example, we consider the rotation of a rigid body with a left-invariant servo-constraint, which implies that the projection of the body’s angular velocity on some body-fixed direction is zero.