Optimal method for truncating a chain of equations for two-time Green’s functions

We use an example of a chain of equations describing a system of Bose particles with pairwise interaction to develop a method for decoupling the chain at its second element. We obtain an approximation of the interacting-modes type, which results in a system of nonlinear equations for one-, two-, and three-particle functions. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 3, pp. 503–510, June, 2006.     

The equations of the approximate theory of the motion of a rigid body with a vibrating suspension point

The motion of a rigid body in a uniform gravity field is investigated. One of the points of the body (the suspension point) performs specified small-amplitude high-frequency periodic or conditionally periodic oscillations (vibrations). The geometry of the body mass is arbitrary. An approximate system of differential equations is obtained, which does not contain the time explicitly and describes the rotational motion of the rigid body with respect to a system of coordinates moving translationally together with the suspension point. The error with which the solutions of the approximate system approximate to the solution of the exact system of equations of motion is indicated. The problem of the stability with respect to the equilibrium of the rigid body, when the suspension point performs vibrations along the vertical, is considered as an application.     

Two-time temperature Green’s functions in kinetic theory and molecular hydrodynamics: II. Equations for pair-interaction systems

We consider two approaches to the calculation of correlation functions for a system of particles with direct pair interaction. The first is based on a chain of equations that determines a Boltzmann-type kinetic equation; the second is based on a chain of molecular hydrodynamic equations. We demonstrate that the two approaches are equivalent in the sense that they completely describe the system under consideration. We discuss the advantages of the approach based on the molecular hydrodynamic equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 1, pp. 142–166, April, 1999.     

Approximate equations of rotational motion of a rigid body carrying a movable point mass

The dynamics of a compound system, consisting of a rigid body and a point mass, which moves in a specified way along a curve, rigidly attached to the body is investigated. The system performs free motion in a uniform gravity field. Differential equations are derived which describe the rotation of the body about its centre of mass. In two special cases, which allow of the introduction of a small parameter, an approximate system of equations of motion is obtained using asymptotic methods. The accuracy with which the solutions of the approximate system approach the solutions of the exact equations of motion is indicated. In one case, it is assumed that the point mass has a mass that is small compared with the mass of the body, and performs rapid motion with respect to the rigid body. It is shown that in this case the approximate system is integrable. A number of special motions of the body, described by the approximate system, are indicated, and their stability is investigated. In the second case, no limitations are imposed on the mass of the point mass, but it is assumed that the relative motion of the point is rapid and occurs near a specified point of the body. It is shown that, in the approximate system, the motion of the rigid body about its centre of mass is Euler–Poinsot motion.     

Computation of periodic orbits in three‐dimensional Lotka‐Volterra systems

This paper deals with an adaptation of the Poincaré‐Lindstedt method for the determination of periodic orbits in three‐dimensional nonlinear differential systems. We describe here a general symbolic algorithm to implement the method and apply it to compute periodic solutions in a three‐dimensional Lotka‐Volterra system modeling a chain food interaction. The sufficient conditions to make secular terms disappear from the approximate series solution are given in the paper.     

Solutions of the Vlasov equation in Lagrange coordinates

We show that the method of “finite-size” particles is a discrete model of the Vlasov equation but in a different (effective) interaction potential. We calculate the effective potential explicitly in the most interesting case of the Coulomb interaction. We find the equations of motion of particles of “finite size” for the Gaussian form factor. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 1, pp. 138–148, April, 2007.     

The solution of a problem in the theory of epidemics

In a 1971 paper, Hoppensteadt and Waltman consider a deterministic epidemic model that accounts for certain threshold phenomena occurring in the spread of infection. A system of nonlinear delay integral equations describe this model. We describe a method for constructing functions which approximate the solution of the system of integral equations. The approximating functions are shown to exist and to converge to the solution of the system.     

Application of the Averaging Method to the Investigation of Nonlinear Wave Processes in Elastic Systems with Circular Symmetry

We apply asymptotic methods of nonlinear mechanics (the Bogolyubov–Mitropol'skii averaging method) to the construction of approximate solutions of a system of nonlinear equations describing wave processes in elastic systems with circular symmetry. As an example, we study the dynamics of interaction of two flexural waves that propagate in a cylindrical shell under the conditions of free oscillations and periodic excitation.     

The equations of motion of a rigid body without parametrization of rotations

New dynamic equations are proposed for a rigid body, without using local parametrization of the rotation group to describe the rotational part of the motion. A simple system of differential-algebraic equations, well suited for constructing the equations of motion of articulated bodies, is obtained.     

Approximation of the perturbation equations of a quasi-linear hyperbolic system in the neighborhood of a bicharacteristic

In 1956 Whitham gave a nonlinear theory for computing the intensity of an acoustic pulse of an arbitrary shape. The theory has been used very successfully in computing the intensity of the sonic bang produced by a supersonic plane. [4.] derived an approximate quasi-linear equation for the propagation of a short wave in a compressible medium. These two methods are essentially nonlinear approximations of the perturbation equations of the system of gas-dynamic equations in the neighborhood of a bicharacteristic curve (or rays) for weak unsteady disturbances superimposed on a given steady solution. In this paper we have derived an approximate quasi-linear equation which is an approximation of perturbation equations in the neighborhood of a bicharacteristic curve for a weak pulse governed by a general system of first order quasi-linear partial differential equations in m + 1 independent variables (t, x₁,…, x_m) and derived Gubkin's result as a particular case when the system of equations consists of the equations of an unsteady motion of a compressible gas. We have also discussed the form of the approximate equation describing the waves propagating upsteam in an arbitrary multidimensional transonic flow.     

The Hamiltonian dynamics of self-gravitating liquid and gas ellipsoids

The dynamics of self-gravitating liquid and gas ellipsoids is considered. A literary survey and authors’ original results obtained using modern techniques of nonlinear dynamics are presented. Strict Lagrangian and Hamiltonian formulations of the equations of motion are given; in particular, a Hamiltonian formalism based on Lie algebras is described. Problems related to nonintegrability and chaos are formulated and analyzed. All the known integrability cases are classified, and the most natural hypotheses on the nonintegrability of the equations of motion in the general case are presented. The results of numerical simulations are described. They, on the one hand, demonstrate a chaotic behavior of the system and, on the other hand, can in many cases serve as a numerical proof of the nonintegrability (the method of transversally intersecting separatrices).      

A transient electromechanical process in a liquid crystal dipole

On the basis of the equations of the electromagnetic field of an orthotropic medium and the Lagrange equations for rotational motion, we obtain the equations that describe the electromechanical state of a liquid crystal dipole in an external electric field. The results of computer modeling of the transient process when an electric field is abruptly imposed are given. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 160–162.     

Independence principles in the equations of state of a deformable material

We consider the principles of coordinate, rotational, and initial independence of the equations of state for a deformable material and the theorem on the existence of elasticity potential connected with them. We show that the well-known axiomatic substantiation and mathematical representation of these principles in “rational continuum mechanics” as well as the proof of the theorem are erroneous. A correct proof of the principles and theorem is presented for the most general case (a stressed anisotropic body under the action of an arbitrary tensor field) without applying any axioms. On this basis, we eliminated the dependence on an arbitrary initial state and the corresponding accumulated strain from the system of equations of state of a deformable material. The obtained forms of equations are convenient for constructing and analyzing the equations of local influence of initial stresses on physical fields of different nature. Finally, these equations represent governing equations for the problems of nondestructive testing of inhomogeneous three-dimensional stress fields and for theoretical-and-experimental investigation of the nonlinear equations of state.     

Reduction of hugoniot-maslov chains for trajectories of solitary vortices of the “shallow water” equations to the hill equation

According to Maslov’s idea, many two-dimensional, quasilinear hyperbolic systems of partial differential equations admit only three types of singularities that are in general position and have the property of “structure self-similarity and stability.” Those are: shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is described by an infinite chain of ordinary differential equations (ODE) that generalize the well-known Hugoniot conditions for shock waves. After some reasonable closure of the chain for the case of solitary vortices in the “shallow water” equations, we obtain a nonlinear system of sixteen ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential. This means that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result can be used to predict the trajectory of the vortex center if we know its observable part. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 47–66.     

Renormalization group symmetries in problems of nonlinear geometrical optics

We construct renormalization group symmetries in the geometrical optics approximation for the boundary value problem of the system of equations describing the propagation of strong radiation in a nonlinear medium. Using the renormalization group symmetries, new exact and approximate analytic solutions to the equations of nonlinear geometrical optics are obtained. Explicit analytic expressions are presented that characterize the spatial evolution of a laser beam having an arbitrary dependence on intensity at the nonlinear medium boundary. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 111, No. 3, pp. 369–388, June, 1997.     

Hopf bifurcation in an hexagonal governor system with a spring

The complex dynamical behaviors of the hexagonal governor system with a spring are studied in this paper. We go deeper investigating the stability of the equilibrium points in the hexagonal governor system with a spring. These systems have a rich variety of nonlinear behaviors, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincaré maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. By studying numerical simulations, it is possible to provide reliable theory and effective numerical method for other systems.     

Global Complexity Bound Analysis of the Levenberg–Marquardt Method for Nonsmooth Equations and Its Application to the Nonlinear Complementarity Problem

We investigate a global complexity bound of the Levenberg–Marquardt Method (LMM) for nonsmooth equations. The global complexity bound is an upper bound to the number of iterations required to get an approximate solution that satisfies a certain condition. We give sufficient conditions under which the bound of the LMM for nonsmooth equations is the same as smooth cases. We also show that it can be reduced under some regularity assumption. Furthermore, by applying these results to nonsmooth equations equivalent to the nonlinear complementarity problem (NCP), we get global complexity bounds for the NCP. In particular, we give a reasonable bound when the mapping involved in the NCP is a uniformly P-function.     

Application of the principle of possible displacements to problems of the dynamics of a “shell-fluid” system

We state in general form the principle of possible displacements for a “shell-fluid” mechanical system, on the basis of which it is possible to solve dynamic problems taking account of a geometrically nonlinear process of deformation of the shell and nonpotential motions of a viscous fluid. It is shown that this principle yields the equations of motion of the shell and fluid as components of this system, confirming the reliability of the principle. The conditions of force contact are taken into account as a load term in the equations of motion of the shell. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 117–123.     

Approximate Noether-type symmetries and conservation laws via partial Lagrangians for PDEs with a small parameter

We show how one can construct approximate conservation laws of approximate Euler-type equations via approximate Noether-type symmetry operators associated with partial Lagrangians. The ideas of the procedure for a system of unperturbed partial differential equations are extended to a system of perturbed or approximate partial differential equations. These approximate Noether-type symmetry operators do not form a Lie algebra in general. The theory is applied to the perturbed linear and nonlinear (1+1) wave equations and the Maxwellian tails equation. We have also obtained new approximate conservation laws for these equations.     

Hopf bifurcations, Lyapunov exponents and control of chaos for a class of centrifugal flywheel governor system

In this paper, complex dynamical behavior of a class of centrifugal flywheel governor system is studied. These systems have a rich variety of nonlinear behavior, which are investigated here by numerically integrating the Lagrangian equations of motion. A tiny change in parameters can lead to an enormous difference in the long-term behavior of the system. Bubbles of periodic orbits may also occur within the bifurcation sequence. Hyperchaotic behavior is also observed in cases where two of the Lyapunov exponents are positive, one is zero, and one is negative. The routes to chaos are analyzed using Poincaré maps, which are found to be more complicated than those of nonlinear rotational machines. Periodic and chaotic motions can be clearly distinguished by all of the analytical tools applied here, namely Poincaré sections, bifurcation diagrams, Lyapunov exponents, and Lyapunov dimensions. This paper proposes a parametric open-plus-closed-loop approach to controlling chaos, which is capable of switching from chaotic motion to any desired periodic orbit. The theoretical work and numerical simulations of this paper can be extended to other systems. Finally, the results of this paper are of practical utility to designers of rotational machines.