Oscillations of mechanical systems that do not become linear when the parameter vanishes : PMM vol. 42, no. 6, 1978, pp. 1039–1048

The results of investigations in [1] are extended to multidimensional systems that become nonlinear at μ = 0. Two-dimensional mechanical systems were investigated in [2,3]. The characteristic equations of systems considered here contain in the critical system either a pair of pure imaginary roots or two zero roots with one or two groups of solutions and n roots with negative real parts in the adjoint system. It is shown that the investigation of such systems necessitates the imposition on the system of some constraints that supplement those specified in [1], The auxilliary function u⁽¹⁾_k (θ) used in the determination of Liapunov's function is derived by a different method than in [1 – 3], In two of the three investigated cases the problem is reduced to the determination of roots of some integral real irrational function. An example is presented.     

On approximate methods of analysing certain singularly-perturbed systems

A certainclass of sigularly-perturbed systems which have a variety of m-dimensional stationary positions is considered. When a small parameter disappears, the system also has an m-dimensional manifold of stationary positions and, therefore, the corresponding characteristic equation has m zero roots. The conditions under which the solution of a stability problem reduces to the same problem for a degenerate system are defined. As an application in practice gyroscopic stabilizing systems (the critical case corresponds to such systems) with elastic elements of high stiffness are discussed. The conditions under which the solution of the problem of the stability of steady motion follows from the solution of this problem for an ideal system (with absolutely rigid elements) are obtained. The problem of the closeness of the corresponding solutions of the complete and a simplified system of differential equations over an infinite time interval is discussed.     

On instability in the presence of several resonances : PMM vol. 43, no. 6, 1979, pp. 970–974

The equilibrium position stability of an autonomous system of ordinary differential equations is considered in the case of n pairs of pure imaginary roots with the simultaneous presence of several resonances. It is shown using Chetaev's theorem [1] that when among the solutions of the model system there are increasing solutions of the invariant ray type, the complete system is Liapunov unstable.     

Method of homogeneous solutions in the mixed problem for a finite circular cylinder : PMM vol. 43, no. 6, 1979, pp. 1073–1081

The mixed axisymmetric problem of elasticity theory on the torsion of a finite circular cylinder by a stamp is considered. The stamp is fixed rigidly to one plane face of the cylinder, the other plane face is fixed, and conditions for no displacements or stresses are given on the cylinder surface. The problem is investigated by the method of homogeneous solutions [1], which permits obtaining its approximate solution for practically any values of the parameters. Such efficiency of the method is determined by the fact that the solution of the problem reduces to investigating an infinite algebraic system of the Poincaré — Koch normal systems type. When the ratio of the cylinder height to the radius of the stamp is sufficiently large, the system coefficients, the contact stresses, and the other characteristics of the problem are evaluated to any degree of accuracy, and effective asymptotic expressions are obtained for small values of this ratio. Results of numerical computations are presented.

A solution of the problem for the case of a large value of the ratio (R − a) /h and small values of the ratio λ = h / a is obtained in [2].     

On constructing periodic solutions of quasi-linear non-self-contained systems with one degree of freedom, when amplitude equations possess multiple roots

Construction of periodic solutions of quasilinear non-self-contained systems with one degree of freedom, was investigated in [1 and 2]. In [1] the case of simple roots of amplitude equations was considered together with the case of a double root when the solution could be expanded into a series in integral powers of μ. In [2] the case of a double root is investigated in more detail Including expansions of solutions into series in μ^1/2. In the present paper, the case of arbitrary multiple roots for non-self-contained systems is reduced to the corresponding case for self-contained systems, which simplifies computations.     

On the stability of space-periodic convective flows in a vertical layer with sinuous boundaries : PMM vol. 43, no. 6, 1979, pp. 998–1007

The problem of convection in a vertical layer with harmonically distorted boundaries is examined by perturbation theory methods for a small amplitude of sinuosity. The solutions obtained are applicable both in the stability region as well as in the supercritical region of the plane-parallel flow. The stability of the solutions found is investigated with respect to a certain class of space-bounded perturbations that are not necessarily space-periodic. The method of amplitude functions [1], generalized to the case of curved boundaries, is used. The Grashof critical number is found as a function of the period of sinuosity and the form of the neutral curve for the space-periodic motions and their stability region are obtained. It is established that if the deformation period of the boundaries is close to the wavelength of the critical perturbation for the plane-parallel flow or is twice as great, then as the Grashof number grows stability loss does not occur and the motion's amplitude changes continuously (cf. [2 — 4]). A comparison is made with the results of the numerical calculation in [5], An attempt was made in [6] to construct a stationary periodic motion in a layer with weakly-deformed boundaries, in the form of series in powers of a small sinuosity amplitude. However, the solution obtained diverges in a neighborhood of the neutral curve of the plane-parallel flow and approximates unstable motion in the supercritical region of the unperturbed problem. Flows under a finite sinuosity amplitude are calculated by the net method in [5] wherein the stability of the flows was investigated as well, but only with respect to perturbations with wave numbers that are multiples of 2π/l, where l is the length of the calculated region.     

Small-parameter method for constructing approximate strategies in a class of differential games : PMM vol. 39, n 6, 1975, pp. 1006–1016

We examine a class of problems in which the pay-off is some function of the terminal state of a conflict-controlled system. When the opportunities of one of the players are small in relation with the opportunities of the other, we propose methods for constructing approximate optimal strategies of the players, based on solving the Bellman equation containing a small parameter. We have shown that the players' approximate optimal strategies can be constructed if the solutions of the corresponding optimal control problems are known. The error bounds for the methods are proved and examples are considered. The arguments used rely on the results in [1–6] on the theory of differential games and on [7–11] devoted to optimal control synthesis methods for systems subject to random perturbations of small intensity.     

Periodic and almost periodic solutions of strongly non-linear impulsive systems

The problem of the existence and stability of periodic and almost periodic solutions of strongly non-linear impulsive systems is investigated. The Poincaré method [1] is justified for the case of an isolated generating solution. A dynamical system consisting of a bead on a vibrating surface is considered as an example.

The small parameter method for investigating systems with discontinuous solutions was previously applied [2, 3] to the case when the periodic solution is non-isolated.

A method is used below for reducing the investigation of a system of equations with impulsive actions on surfaces to equations with fixed moments of inpulsive action.     

Case of a generating family of quasi-periodic solutions in the theory of small parameter : PMM vol. 37, n6, 1973, pp. 990–998

We consider the problem of the existence and the stability in-the-small of periodic solutions of systems of ordinary differential equations with a small parameter μ, which in the generating approximation (μ = 0) admit of a family of quasi-periodic solutions (we are concerned only with the solutions belonging to the indicated family when μ = 0). The case to be investigated is in a specific sense a more general case of the unisolated generating solution in the small parameter theory and, therefore, includes everything previously treated by Malkin [1], Blekhman [2], and others. The main difficulty in the investigation is the presence of a multiple zero root in the characteristic determinant of the problem's generating system, to which both simple as well as quadratic elementary divisors [3] correspond. This fact predestines the presence of three groups of stability criteria for the solution being examined. The method for constructing these criteria, proposed here, assumes, in contrast to a previous one [1], the preliminary determination of not only the generating approximation but also the first one to the desired periodic solution. Particular aspects of the general “mixed” problem treated here were studied earlier in [4, 5].     

The problem of minimax mean square filtration in parabolic systems : PMM vol. 42, no. 6, 1978, pp, 1016–1025

The problem of estimating parameters of state of a distributed parabolic system by observation results is considered. The system is assumed to function under conditions of undefined perturbations in the measurement channel and specified initial distribution. The problem is considered in minimax formulation [1] in conformity with the scheme accepted for ordinary differential equations [2].(^*), Analytic definition of sets X (/gJ, y (·)) (/gJ > 0) of states of a parabolic system compatible at instant /gJ with the realizable signal y (t) (t ε [0, /gJ]) is obtained. An element of region X (/gJ, y (·)) which satisfies the specified minimax criterion is chosen as the optimal estimate of the true state at instant /gJ. Integradifferential equations in partial derivatives are derived for parameters that define the evolution of regions X (/gJ, y (·)) in time. One of the methods of approximating the input problem of observation by similar problems for systems of ordinary differential equations is discussed on a specific example. Problems of observation for distributed systems in different formulations appear in [3 – 6].     

The stability of a class of reversible systems

The problem of the stability of the point of rest of an autonomous system of ordinary differential equations from a class of reversible systems [1] characterized by the critical case of m zero roots and n pairs of pure imaginary roots is considered. When there are no internal resonances [2, 3], the point of rest always has Birkhoff complete stability [2]. Internal resonances may lead to Lyapunov instability. The conditions of stability and instability of the model system when there are third-order resonances may be obtained from a criterion previously developed [4] for the case of pure imaginary roots. The results are used to analyse the stability of the translational-rotational motion of an active artificial satellite in a non-Keplerian circular orbit, including a geostationary satellite in any latitude [4, 5]. The region of stability of relative equilibria and regular precession of the satellite is constructed assuming a central gravitational field and the resonance modes are analysed.     

On partial regularity of suitable weak solutions to the stationary fractional Navier–Stokes equations in dimension four and five

In this paper, we investigate the partial regularity of suitable weak solutions to the multidimensional stationary Navier Stokes equations with fractional power of the Laplacian (-△)~α 1 and α≠ 1/2). It is shown that the n + 2-6α(3 ≤ n ≤ 5) dimensional Hausdorff measure of the set of the possible singular points of suitable weak solutions to the system is zero, which extends a recent result of Tang and Yu [19] to four and five dimension. Moreover, the pressure in e-regularity criteria is an improvement of corresponding results in [1, 13, 18, 20].     

Some recent results on a free boundary problem for microelectromechanical systems

The mathematical modeling of idealized microelectromechanical systems leads to a nonlinear singular elliptic-parabolic free boundary problem involving non-smooth domains and depending on a parameter λ reflecting the applied voltage across the device. Recent results derived in [1,4] are presented which confirm physically important and expected features: For small λ there is (at least) one stable stationary solution and solutions to the evolution equation exist globally in time, while for large λ no stationary solution exists and the evolution equation has no global solution. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Approximate solution of nonlinear problems of optimal control of oscillatory processes using the method of canonical separation of motions : PMM vol.40, n 6, 1976, pp. 1024–1033

An asymptotic method of solving certain problems of optimal control of motion of the standard type systems with rotating phase is developed. It is assumed that the controls enter only the small perturbing terms, and that the fixed time interval over which the process is being considered is long enough to ensure that the slow variables change essentially. Assuming also that the system and the controls satisfy the necessary requirements of smoothness, the method of canonical averaging [1] is used to construct a scheme for deriving a simplified boundary value problem of the maximum principle. The structure of the set of solutions of the boundary value problem is investigated and a scheme for choosing the optimal solution with the given degree of accuracy in the small parameter is worked out. The validity of the approximate method of solving the boundary value problem is proved. The method suggested in [2] for constructing a solution in the first approximation for similar problems of optimal control is developed.     

Chaos, self-organized criticality, and SETAR nonlinearity: An analysis of purchasing power parity between Canada and the United States

This paper uses monthly observations for the real exchange rate between Canada and the United States over the recent flexible exchange rate period (from January 1, 1973 to August 1, 2004) to test purchasing power parity between Canada and the United States using unit root and stationarity tests. Moreover, given the apparent random walk behavior in the real exchange rate, various tests from dynamical systems theory, such as for example, the Nychka et al. [Nychka DW, Ellner S, Ronald GA, McCaffrey D. Finding chaos in noisy systems. J Roy Stat Soc B 1992;54:399–426] chaos test, the Li [Li W. Absence of 1/f spectra in Dow Jones average. Int J Bifurcat Chaos 1991;1:583–97] self-organized criticality test, and the Hansen [Hansen, B.E. Inference when a nuisance parameter is not identified under the null hypothesis. Econometrica 1996;64:413–30] threshold effects test are used to distinguish between stochastic and deterministic origin for the real exchange rate.     

Propagation failure in discrete periodic media

We study the periodic lattice dynamical systems with bistable nonlinearity. We use Moser's theorem to show that there exist infinitely many stationary solutions when one of the migration coefficients is sufficiently small. Moreover, we prove that the propagation failure occurs when both migration coefficients are sufficiently small.     

Application of the Bubnov-Galerkin procedure to the problem of searching for selfoscillations : PMM vol. 37, n6, 1973, pp. 1015–1019

We propose the use of the Bubnov-Galerkin procedure to the search for self-oscillations. We establish the existence and the convergence of the approximations. In the basic case we have obtained the asymptotics of the rate of convergence. In [1] it was shown, on the basis of the results in [2], how we can construct finite-dimensional approximations to the periodic solutions of autonomous systems. Below we have pointed out another approach to solving the approximation problem, based on the parameter functionalization method proposed in [3].     

Pseudo almost periodic solutions for the systems of differential equations with piecewise constant argument [<Emphasis Type="Italic">t</Emphasis> + 1/2]

In this paper we investigate the existence and uniqueness of pseudo almost periodic solutions and unboundedness of other solutions for the systems of differential equations with piecewise constant argument [t + 1/2] by means of new notion of pseudo almost periodic vector sequences. The case in which the characteristic equation has multiple roots is considered.     

Solitary Waves, periodic peakons, pseudo-peakons and compactons given by three ion-acoustic wave models in electron plasmas

The nonlinear ion-acoustic oscillations models are governed by three partial differential equation systems. Their travelling wave equations are three first class singular traveling wave systems depending on different parameter groups, respectively. By using the method of dynamical system and the theory of singular traveling wave systems, in this paper, it is shown that there exist parameter groups such that these singular systems have solitary wave solutions, pseudo-peakons, periodic peakons and compactons as well as kink and anti-kink wave solutions. The results of this paper complete the studies of three papers [5,13] and [14].     

Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential

Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross–Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the 1D stationary coupled-mode system for a relevant elliptic problem by employing the method of Lyapunov–Schmidt reductions in Fourier space. In particular, existence of periodic/anti-periodic and decaying solutions is proved and the error terms are controlled in suitable norms. The use of multi-dimensional stationary coupled-mode systems is justified for analysis of bifurcations of periodic/anti-periodic solutions in a small multi-dimensional periodic potential.