The stability of some classes of systems in standard Bogolyubov form

The stability of the equilibrium of quasilinear systems in standard Bogolyubov form is investigated. Classes of systems are distinguished out for which it is possible to determine the threshold value ?₀ of the small parameter ? which ensures qualitative agreement between the solutions of the initial systems of equations and the solutions of the averaged system corresponding it in an infinite time interval when ? < ?₀.     

GENERALIZED STOCHASTIC PERTURBATION-DEPENDING DIFFERENTIAL EQUATIONS

The paper is devoted to the generalized stochastic differential equations of the Ito? type whose coefficients are additionally perturbed and dependent on a small parameter. Their solutions are compared with the solutions of the corresponding unperturbed equations. We give conditions under which the solutions of these equations are close in the (2m)-th moment sense on finite intervals or on intervals whose length tends to infinity as the small parameter tends to zero. We also give the degree of the closeness of these solutions.     

Periodic solutions of the N-body problem

This paper proves the existence of six new classes of periodic solutions to the N-body problem by small parameter methods. Three different methods of introducing a small parameter are considered and an appropriate method of scaling the Hamiltonian is given for each method. The small parameter is either one of the masses, the distance between a pair of particles or the reciprocal of the distances between one particle and the center of mass of the remaining particles. For each case symmetric and non-symmetric periodic solutions are established. For every relative equilibrium solution of the (N ? 1)-body problem each of the six results gives periodic solutions of the N-body problem. Under additional mild non-resonance conditions the results are roughly as follows. Any non-degenerate periodic solutions of the restricted N-body problem can be continued into the full N-body problem. There exist periodic solutions of the N-body problem, where N ? 2 particles and the center of mass of the remaining pair move approximately on a solution of relative equilibrium and the pair move approximately on a small circular orbit of the two-body problems around their center of mass. There exist periodic solutions of the N-body problem, where one small particle and the center of mass of the remaining N ? 1 particles move approximately on a large circular orbit of the two body problems and the remaining N ? 1 bodies move approximately on a solution of relative equilibrium about their center of mass. There are three similar results on the existence of symmetric periodic solutions.     

The coexistence problem for Heun's differential equation

For certain parameter sets all solutions of Heun's equation are analytic in ? \ [0, 1], where ? is a (small) neighborhood of [0, 1]. These parameters sets are investigated. The results generalize those on coexistence of periodic solutions of the Lamé and Ince equations due to Ince, Magnus and Winkler. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Difference singular perturbations—I. A priori estimates

Difference approximations for differential singular perturbations with small parameter ? are considered. We point out ellipticity and coerciveness conditions which are necessary and sufficient for a two-sided a priori estimate to hold for the solutions of difference singular perturbation uniformly with respect to the ratio of both small parameters: the original one ? and the meshsize h.     

Asymptotic representation of solutions of regularly perturbed systems of differential equations with nonclassical right-hand side

The problem of the B-asymptotic representation, with respect to a small parameter, of solutions of a regularly perturbed system of differential equations with impulse action on surfaces and of differential equations with discontinuous righthand side is considered. Theorems concerning the B-analytic dependence of solutions on the small parameter are proved. Algorithms for calculating the coefficients of the expansion are developed.Translated from Ukrainskii Maternaticheskii Zhurnal, Vol. 43, No. 10, pp. 1298–1304, October, 1991.     

On small perturbations of a nonlinear periodic system with degenerate equilibrium

This paper considers the reducibility and existence of periodic solutions for a class of nonlinear periodic system with a degenerate equilibrium point under small perturbations. By introducing some parameter, we consider an equivalent periodic system. Then we prove that by an affine linear periodic transformation the parameterized periodic system is reducible to one with zero as an equilibrium. Topological degree theorem ensures that for some parameter the result can go back to the original system. Then, we obtain a small periodic solution.     

Existence and nonexistence results for a singular boundary value problem arising in the theory of epitaxial growth

The existence of stationary radial solutions to a partial differential equation arising in the theory of epitaxial growth is studied. It turns out that the existence or not of such solutions depends on the size of a parameter that plays the role of the velocity at which mass is introduced into the system. For small values of this parameter, we prove the existence of solutions to this boundary value problem. For large values of the same parameter, we prove the nonexistence of solutions. We also provide rigorous bounds for the values of this parameter, which separate existence from nonexistence. The proofs come as a combination of several differential inequalities and the method of upper and lower functions applied to an associated two‐point boundary value problem. Copyright © 2013 John Wiley & Sons, Ltd.     

Following homology in singularly perturbed systems

The fast systems associated to a system of first-order equations where a small parameter multiplies the derivatives (hereinafter, an SPMD system) are obtained by stretching time by a factor of one over the small parameter and then reducing the small parameter to zero. Curves of isolated invariant sets associated to the fast system serve as approximate solutions to the SPMD system over finite intervals in the following sense: A manifold of initial points representing non-zero homology α in the Morse-Conley index of the invariant set at the left endpoint has a submanifold which is carried by the flow of the SPMD system to a manifold representing that non-zero homology class in the index of the invariant set at the right endpint which is the continuation of α along the curve of invariants sets. This is a type of linearization of the approximation problem since the continuation of homology classes along the curve of invariant sets is given by a linear transformation which varies functorially with the curve of invariant sets. Examples are given for systems in R² and R³. The results are used elsewhere to prove existence theorems for two point boundary value problems to obtain solutions with boundary and interior transition layers.     

Multiplicity of solutions for a fourth-order quasilinear nonhomogeneous equation

We consider a fourth-order quasilinear nonhomogeneous equation which is equivalent to a nonhomogeneous Hamiltonian system. The purpose of this work is to prove the existence of at least two solutions for such equation when a certain parameter is small enough. Furthermore, under an additional hypothesis on positiveness of the nonhomogeneous part we prove that our solutions are positive.     

Representation of Solutions to the Equations of Statics for Transversely Isotropic Mildly Sloping Shells

A representation of solutions to the system of differential equations for the statics of elastic, transversely isotropic, mildly sloping shells is found in the form of a series in terms of a parameter that characterizes the difference of the median surface from spherical. Recurrence relations are obtained that can be used for solving boundary value problems by expansion in terms of a small parameter to arbitrary approximations.     

小展弦比波的Green-Naghdi渐进模型的行波解

本文研究了小展弦比波的Green-Naghdi渐进模型. 利用平面自治系统的稳定性分析方法, 在不同的参数条件下, 讨论了它的行波系统的分岔并且给出了对应的相图, 得到了光滑周期波解, 广义扭波解, 广义反扭波解, 广义紧波解, 周期尖波解, 孤波解和孤立尖波解的精确表达式. 进一步, 通过数学软件Maple模拟了这些解.     

Hybrid competitive Lotka–Volterra ecosystems: countable switching states and two-time-scale models

This work is concerned with competitive Lotka–Volterra model with Markov switching. A novelty of the contribution is that the Markov chain has a countable state space. Our main objective of the paper is to reduce the computational complexity by using the two-time-scale systems. Because existence and uniqueness as well as continuity of solutions for Lotka–Volterra ecosystems with Markovian switching in which the switching takes place in a countable set are not available, such properties are studied first. The two-time scale feature is highlighted by introducing a small parameter into the generator of the Markov chain. When the small parameter goes to 0, there is a limit system or reduced system. It is established in this paper that if the reduced system possesses certain properties such as permanence and extinction, etc., then the complex system also has the same properties when the parameter is sufficiently small. These results are obtained by using the perturbed Lyapunov function methods.     

Asymptotic behaviour of the energy for electromagnetic systems in the presence of small inhomogeneities

In this article we consider solutions to the time-harmonic and time-dependent Maxwell's systems with piecewise constant coefficients with a finite number of small inhomogeneities in ?³. In time-harmonic case and for such solutions, we derive the asymptotic expansions due to the presence of small inhomogeneities embedded in the entire space. Further, we analyse the behaviour of the electromagnetic energy caused by the presence of these inhomogeneities. For a general time-dependent case, we show that the local electromagnetic energy, trapped in the total collection of these well-separated inhomogeneities, decays towards zero as the shape parameter decreases to zero or as time increases.     

Quasistationary Approximation in the Rotating Ring Problem

We consider the planar rotation-symmetric motion by inertia of a viscous incompressible fluid in a ring with free boundary. We reduce the corresponding initial-boundary value problem for the Navier–Stokes equations to some problem for a coupled system of one parabolic equation and two ordinary differential equations. We suppose that the coefficient of the derivatives of the sought functions with respect to time (the quasistationary parameter) is small; so the system is singularly perturbed. In this article we construct an asymptotic expansion for a solution to the rotating ring problem in a small quasistationary parameter and obtain a smallness estimate for the difference between the exact and approximate solutions.     

A linear system of differential equations with small parameter at a part of derivatives,a deviation of the argument,and a turning point

For a system of differential equations with small parameter at a part of derivatives, a linear deviation of the argument, and a turning point, we obtained conditions, under which its solutions are solutions of a system of differential equations with small parameter at a part of derivatives such that its matrices possess the asymptotic expansions at |ε| ≤ ε₀ with the coefficients holomorphic at |x| ≤ x ₀ . The existence and the infinite differentiability of a solution of the system of differential equations with small parameter at a part of derivatives and with a linear deviation of the argument in the presence of a turning point are proved.     

On the holomorphic regularization of singularly perturbed systems of differential equations

A method for constructing pseudo-holomorphic solutions to strongly nonlinear singularly perturbed systems of differential equations, which a logical continuation of the Lomov regularization method, is proposed. The existence of integrals of such systems, holomorphic in the small parameter, is proven, and sufficient conditions for the convergence of expansion of solutions to these systems in powers of the small parameter in the usual sense are obtained.     

Asymptotic expansions of solutions to inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative

Two inverse problems for a hyperbolic equation with a small parameter multiplying the highest derivative are considered. The existence and uniqueness of their solutions are proved. As the small parameter tends to zero, the solutions of the inverse problems are proved to converge to solutions of inverse problems for a parabolic equation.     

Multiple solutions and their limiting behavior of coupled nonlinear Schrödinger systems

We study the multiplicity of positive solutions and their limiting behavior as ? tends to zero for a class of coupled nonlinear Schrödinger system in R^N. We relate the number of positive solutions to the topology of the set of minimum points of the least energy function for ? sufficiently small. Also, we verify that these solutions concentrate at a global minimum point of the least energy function.     

Continuation of periodic motions of a reversible system in non-structurally stable cases. Application to the N-planet problem

The problem of continuing symmetric periodic solutions of an autonomous or periodic system with respect to a parameter is solved. Non-structurally stable cases, when the generating system does not guarantee that the solution can be continued, are considered. Three approaches are proposed to solving the problem: (a) particular consideration of terms that depend on the small parameter and the selection of generating solutions; (b) the selection of a generating system depending on the small parameter; (c) reduction to a quasi-linear system which is then analysed using the first approach. Within the framework of the third approach the existence of a periodic motion is also established that differs from the generating one by a quantity whose order is a fractional power of the small parameter. The theoretical results are used to prove the existence of two families of periodic three-dimensional orbits in the N-planet problem. The orbit of each planet is nearly elliptical and situated in the neighbourhood of its fixed plane; the angle between the planes is arbitrary. The average motions of the planets in these orbits relate to one another as natural numbers (the resonance property), and at instants of time that are multiples of the half-period the planes are either aligned in a straight line—the line of nodes (the first family), or cross the same fixed plane (the second family). The phenomenon of a parade of planets is observed. The planets' directions of motion in their orbit are independent.