Limit cycles for a perturbation of a quadratic center with symmetry

To estimate the number of limit cycles appearing under a perturbation of a quadratic system that has a center with symmetry, we use the method of generalized Dulac functions. To this end, we reduce the perturbed system to a Liénard system with a small parameter, for which we construct a Dulac function depending on the parameter. This permits one to estimate the number of limit cycles in the perturbed system for all sufficiently small parameter values. We find the Dulac function by solving a linear programming problem. The suggested method is used to analyze four specific perturbed systems that globally have exactly three limit cycles [i.e., the limit cycle distribution 3 or (3, 0)] and two systems that have the limit cycle distribution (3, 1) (i.e., one nest around each of the two foci).     

Asymptotics of the Solution of a Singularly Perturbed System of Equations with a Multizone Internal Layer

Differential Equations - We consider a boundary value problem for a singularly perturbed system of two second-order ordinary differential equations with distinct powers of the small parameter...     

Eliminating indeterminacy for localized solutions to reaction-diffusion equations in multidimensional domains

Exponentially ill-conditioned localized solutions are constructed asymptotically in the limit of small diffusion for two classes of singularly perturbed reaction-diffusion equations in a multidimensional domain.     

Distributional convergence in planar dynamics and singular perturbations

Motivated by applications to singular perturbations, the paper examines convergence rates of distributions induced by solutions of ordinary differential equations in the plane. The solutions may converge either to a limit cycle or to a heteroclinic cycle. The limit distributions form invariant measures on the limit set. The customary gauges of topological distances may not apply to such cases and do not suit the applications. The paper employs the Prohorov distance between probability measures. It is found that the rate of convergence to a limit cycle and to an equilibrium are different than the rate in the case of heteroclinic cycle; the latter may exhibit two paces, depending on a relation among the eigenvalues of the hyperbolic equilibria. The limit invariant measures are also exhibited. The motivation is stemmed from singularly perturbed systems with non-stationary fast dynamics and averaging. The resulting rates of convergence are displayed for a planar singularly perturbed system, and for a general system of a slow flow coupled with a planar fast dynamics.     

Asymptotics of the optimal time in a time-optimal control problem with a small parameter

A time-optimal control problem for a singularly perturbed linear autonomous system is considered. The main difference between this case and the case of systems with fast and slow variables studied earlier is that the eigenvalues of the matrix at the fast variables do not satisfy the standard requirement of negativity of the real part. We obtain and justify a complete power asymptotic expansion in the sense of Erdélyi of the optimal time and optimal control in a small parameter at the derivatives in the equations of the system.     

A SINGULARLY PERTURBED PROBLEM OF THIRD ORDER EQUATION WITH TWO PARAMETERS

A singularly perturbed problem of third order equation with two parameters is studied. Using singular perturbation method, the structure of asymptotic solutions to the problem is discussed under three possible cases of two related small parameters. The results obtained reveal the different structures and limit behaviors of the solutions in three different cases. And in comparison with the exact solutions of the autonomous equation they are relatively perfect.     

Singular limit cycle bifurcations to closed orbits and invariant tori

This paper investigates singular limit cycle bifurcations for a singularly perturbed system. Based on a series of transformations (the modified curvilinear coordinate, blow-up, and near-identity transformation) and bifurcation theory of periodic orbits and invariant tori, the bifurcations of closed orbits and invariant tori near singular limit cycles are discussed.     

Turning point in a system of differential equations with analytic operator

We construct uniform asymptotics for a solution of a system of singularly perturbed differential equations with turning point. We consider the case where the boundary operator analytically depends on a small parameter.     

The Poincaré Map of Randomly Perturbed Periodic Motion

A system of autonomous differential equations with a stable limit cycle and perturbed by small white noise is analyzed in this work. In the vicinity of the limit cycle of the unperturbed deterministic system, we define, construct, and analyze the Poincaré map of the randomly perturbed periodic motion. We show that the time of the first exit from a small neighborhood of the fixed point of the map, which corresponds to the unperturbed periodic orbit, is well approximated by the geometric distribution. The parameter of the geometric distribution tends to zero together with the noise intensity. Therefore, our result can be interpreted as an estimate of the stability of periodic motion to random perturbations. In addition, we show that the geometric distribution of the first exit times translates into statistical properties of solutions of important differential equation models in applications. To this end, we demonstrate three distinct examples from mathematical neuroscience featuring complex oscillatory patterns characterized by the geometric distribution. We show that in each of these models the statistical properties of emerging oscillations are fully explained by the general properties of randomly perturbed periodic motions identified in this paper.     

Approximation of derivative in a system of singularly perturbed convection-diffusion equations

In this paper, a numerical method for a weakly coupled system of two singularly perturbed convection-diffusion second order ordinary differential equations with a small parameter multiplying the highest derivative is presented. Parameter-uniform error bounds for the numerical solution and also to numerical derivative are established. Numerical results are provided to illustrate the theoretical results.     

System of differential equations with a strong turning point

The uniform asymptotics of a solution of a system of singularly perturbed differential equations with strong turning point is constructed. We study the case where the boundary operator is analytic with respect to a small parameter.     

Limit cycles under perturbations of a quadratic Hamiltonian system

The perturbed quadratic Hamiltonian system is reduced to a Lienard system with a small parameter for which a Dulac function depending on it is constructed. This permits one to estimate the number of limit cycles of the perturbed system for all sufficiently small parameter values. To find the Dulac function, we use the solution of a linear programming problem. The suggested method is used for studying three specific perturbed systems that have exactly two limit cycles, i.e., the distribution 2 or (0, 2), and one system with distribution (1, 1).     

具有两个转向点的大参数奇摄动方程的渐近解

利用匹配法研究了一类具有两个转向点的大参数奇摄动方程,通过Liouville-Green变换和Airy函数分别构造了方程在不同区域的外部解和内层解,得出了方程的渐近解,即解在不同范围内的5个渐近表达式及其5对常数之间的4个匹配条件.     

Factorization method for linear and quasilinear singularly perturbed boundary value problems for ordinary differential equations

For linear singularly perturbed boundary value problems, we come up with a method that reduces solving a differential problem to a discrete (difference) problem. Difference equations, which are an exact analog of differential equations, are constructed by the factorization method. Coefficients of difference equations are calculated by solving Cauchy problems for first-order differential equations. In this case nonlinear Ricatti equations with a small parameter are solved by asymptotic methods, and solving linear equations reduces to computing quadratures. A solution for quasilinear singularly perturbed equations is obtained by means of an implicit relaxation method. A solution to a linearized problem is calculated by analogy with a linear problem at each iterative step. The method is tested against solutions to the known Lagerstrom-Cole problem.     

On a system of singularly perturbed second-order quasilinear ordinary differential equations in the critical cases

A singularly perturbed system of second-order quasilinear ordinary differential equations with a small parameter multiplying the second derivatives is examined in the case where the coefficient matrix of the first derivatives is singular and does not depend on the unknown functions.     

Limiting behaviour of invariant manifolds for a system of singularly perturbed evolution equations

In this paper we consider a class of specific singularly perturbed abstract evolution equations. It is shown that, for small values of the singular parameter, the invariant manifold for the perturbed equation is C¹ close to that of the unpertubed equation. The results obtained are applied to the second-order evolution equations with strong damping arising in the mathematical théory of viscoelasticity.     

一类二阶拟线性微分方程边值问题的奇摄动(英文)

本文讨论了一类二阶拟线性微分方程的奇摄动问题 .在适当的条件下 ,本文用一种新的方法分析了原问题解的存在性、唯一性及渐近性态 .     

Homoclinic,heteroclinic and periodic orbits of singularly perturbed systems

The main aims of this paper are to study the persistence of homoclinic and heteroclinic orbits of the reduced systems on normally hyperbolic critical manifolds, and also the limit cycle bifurcations either from the homoclinic loop of the reduced systems or from a family of periodic orbits of the layer systems. For the persistence of homoclinic and heteroclinic orbits, and the limit cycles bifurcating from a homolinic loop of the reduced systems, we provide a new and readily detectable method to characterize them compared with the usual Melnikov method when the reduced system forms a generalized rotated vector field. To determine the limit cycles bifurcating from the families of periodic orbits of the layer systems, we apply the averaging methods.We also provide two four-dimensional singularly perturbed differential systems, which have either heteroclinic or homoclinic orbits located on the slow manifolds and also three limit cycles bifurcating from the periodic orbits of the layer system.     

Asymptotic analysis of singularly perturbed Hamilton-Jacobi equations

Unlike the previous investigation of the sufficient conditions for the convergence of minimax solutions of singularly perturbed Hamilton-Jacobi (H-J) equations, a typical example of which would be the Bellman-Isaacs (B-I) equations, convergence conditions are formulated not in terms of auxiliary constructs [1], but in terms of the Hamiltonian, the boundary function, assumptions regarding their continuity, Lipschitz continuity, etc. In addition, an asymptotic equation is derived, that is, a H-J equation whose minimax solution is the limit of solutions of H-J equations in which some of the momentum variables have coefficients whose denominators contain a small parameter which is made to approach zero.     

Asymptotic analysis of a singularly perturbed linear system with a singular limit pencil of matrices

With the help of perturbation methods and Newton diagrams, an asymptotic analysis is conducted of the general solution of a linear singularly perturbed system of ordinary differential equations in the case of degeneracy of a matrix multiplying the derivative in the approach of a small parameter to zero. It is assumed that the pencil of limit matrices of the system is singular and possesses a minimal index for rows and columns.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 1, pp. 106–122, January, 1992.