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Showing papers on "Asymptotology published in 2008"



Journal ArticleDOI
TL;DR: A survey of results on the asymptotic analysis of autoresonance can be found in this article, where systems of differential equations corresponding to non-linear non-autonomous oscillators with variable excitation frequency are considered.
Abstract: In recent decades new problems have arisen in oscillation theory which are related to the investigation of a?physical phenomenon known as autoresonance. This paper presents a?survey of results on the asymptotic analysis of such problems. Systems of differential equations corresponding to non-linear non-autonomous oscillators with variable excitation frequency are considered. For their solution asymptotic formulae are constructed with respect to a?small parameter or with respect to an independent variable.

56 citations


Journal ArticleDOI
TL;DR: In this article, asymptotic expansions for the exponentially small splitting of separatrices of area preserving maps combining analytical and numerical points of view are studied using high-precision arithmetic, which involves up to several thousands of decimal digits.
Abstract: We study asymptotic expansions for the exponentially small splitting of separatrices of area preserving maps combining analytical and numerical points of view. Using analytic information, we conjecture the basis of functions of an asymptotic expansion and then extract actual values of the coefficients of the asymptotic series numerically. The computations are performed with high-precision arithmetic, which involves up to several thousands of decimal digits. This approach allows us to obtain information which is usually considered to be out of reach of numerical methods. In particular, we use our results to test that the asymptotic series are Gevrey-1 and to study positions and types of singularities of their Borel transform. Our examples are based on generalisations of the standard and Henon maps.

40 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the Euler-Maxwell equations for plasmas with small parameters and showed the existence and uniqueness of an asymptotic expansion up to any order.
Abstract: This work is concerned with the two-fluid Euler-Maxwell equations for plasmas with small parameters. We study, by means of asymptotic expansions, the zero-relaxation limit, the non-relativistic limit and the combined non-relativistic and quasi-neutral limit. For each limit with well-prepared initial data, we show the existence and uniqueness of an asymptotic expansion up to any order. For general data, an asymptotic expansion up to order 1 of the non-relativistic limit is constructed by taking into account the initial layers. Finally, we discuss the justification of the limits.

26 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the Navier-Stokes equations with a time-delayed convective term and a forcing term which contains hereditary features and established the existence and uniqueness of solutions.
Abstract: We consider the two-dimensional Navier-Stokes equations with a time-delayed convective term and a forcing term which contains some hereditary features. Some results on existence and uniqueness of solutions are established. We discuss the asymptotic behaviour of solutions and we also show the exponential stability of stationary solutions.

25 citations


Journal ArticleDOI
TL;DR: In this article, an asymptotic approach to gated ionic models of single-cell cardiac excitability is proposed, which allows a dynamical variable may change its character from fast to slow within a single solution.

19 citations




Journal ArticleDOI
TL;DR: It is proved that the probability that randomly chosen fourth order type (or type of the order not greater than 4), which admits decidable lambda definability problem, is zero.
Abstract: This paper presents a systematic approach for obtaining results from the area of quantitative investigations in logic and type theory. We investigate the proportion between tautologies (inhabited types) of a given length n against the number of all formulas (types) of length n. We investigate an asymptotic behavior of this fraction. Furthermore, we characterize the relation between number of premises of implicational formula (type) and the asymptotic probability of finding such formula among the all ones. We also deal with a distribution of these asymptotic probabilities. Using the same approach we also prove that the probability that randomly chosen fourth order type (or type of the order not greater than 4), which admits decidable lambda definability problem, is zero.

9 citations



01 Jan 2008
TL;DR: In this article, the authors studied the asymptotic behavior of solutions of a dissipative plate equation in R N with periodic coecients and proved that the solutions for the linear model behave as the homogenized heat kernel.
Abstract: In this work we study the asymptotic behavior of solutions of a dissipative plate equation in R N with periodic coecients. We use the Bloch waves decomposition and a convenient Lyapunov function to derive a complete asymptotic expansion of solutions as t ! 1. In a first approximation, we prove that the solutions for the linear model behave as the homogenized heat kernel.

Journal ArticleDOI
TL;DR: In this article, the Brownian motion over the space of fluid velocity configurations driven by the hydrodynamical equations is considered and the Green function is computed in the form of an asymptotic series close to the standard diffusion kernel.

Journal ArticleDOI
TL;DR: In this article, the authors considered the fourth Painleve equation with nonzero values of its two complex parameters a and b and obtained nine families of power asymptotic forms extended by asymythmic expansions.
Abstract: We consider the fourth Painleve equation with nonzero values of its two complex parameters a and b . Asymptotic forms of power, complicated, exotic, and elliptic types for its solutions near the points x = 0 and x = ∞ are sought by methods of two- and three-dimensional power geometry. We obtain nine families of power asymptotic forms extended by asymptotic expansions and one family of elliptic asymptotic forms. Complicated and exotic asymptotic forms are absent.

Journal ArticleDOI
TL;DR: A survey of results on boundary function based models for natural resource modeling using boundary function method is presented in this paper. Detailed reduction procedures as well as conditions under which the reductions are at all possible are discussed.
Abstract: . A survey of results on reduction of models related to problems of natural resource modeling using the boundary function method is presented. Detailed reduction procedures as well as conditions under which the reductions are at all possible are discussed. Particular asymptotic techniques are illustrated by simple examples taken from chemical kinetics, and a realistic example from atmospheric chemistry modeling.

01 Jan 2008
TL;DR: In this article, an analysis of dispersive/dissipative features of the difference schemes used for simulations of the non-linear Burgers' equation is developed based on the travelling wave asymptotic solutions of its differential approximation.
Abstract: An analysis of dispersive/dissipative features of the difference schemes used for simulations of the non-linear Burgers' equation is developed based on the travelling wave asymptotic solutions of its differential approximation. It is shown that these particular solutions describe well deviations in the shock profile even outside the formal applicability of the asymptotic expansions, namely for shocks of moderate amplitudes. Analytical predictions may be used to improve calculations by suitable choice of the parameters of some familiar schemes, i.e., the Lax-Wendroff, Mac-Cormack etc. Moreover, an improvement of the scheme may be developed by adding artificial terms according to the asymptotic solution.

Journal ArticleDOI
TL;DR: A family of asymptotic solutions at infinity for the system of ordinary differential equations is considered in this paper, and the existence of exact solutions which have these exact solutions has been proved.
Abstract: A family of asymptotic solutions at infinity for the system of ordinary differential equations is considered. Existence of exact solutions which have these asymptotics is proved.

Journal ArticleDOI
TL;DR: In this article, the radial distribution of temperature field in a well is used to illustrate the application of a modification of the asymptotic method for solving a number of problems in subterranean thermodynamics.
Abstract: The example of the problem on radial distribution of temperature field in a well is used to illustrate the application of a modification of the asymptotic method for solving a number of problems in subterranean thermodynamics. The problem is represented in the form of a set of equations of mixed types for the respective coefficients of expansion, remainder term, and boundary-layer functions. Analytical expressions are constructed for coefficients of zero-order and first-order expansion and for boundary-layer functions. It is demonstrated that the constructed asymptotic formula provides for vanishing of the solution of the averaged problem for remainder term.

Journal ArticleDOI
TL;DR: In this paper, a multi-scale method was used to construct an asymptotic solution of the auto-resonance arising problem in the domain t ≪ e −1.
Abstract: The problem of auto-resonance arising is investigated. Using the multi-scale method, we construct an asymptotic solution of this problem in the domain t ≪ e −1

Journal ArticleDOI
06 May 2008
TL;DR: In this article, the authors studied the asymptotic behavior of solutions to a semilinear second-order parabolic equation in a cylindrical domain bounded in the spatial variable.
Abstract: We study the asymptotic behavior as t → +∞ of solutions to a semilinear second-order parabolic equation in a cylindrical domain bounded in the spatial variable. We find the leading term of the asymptotic expansion of a solution as t → +∞ and show that each solution of the problem under consideration is asymptotically equivalent to a solution of some nonlinear ordinary differential equation.

Journal ArticleDOI
TL;DR: In this article, the boundary asymptotic behavior of solutions for weighted -Laplacian equations that take infinite value on a bounded domain is proved. But the boundary is not defined.
Abstract: The goal of this paper is to prove the boundary asymptotic behavior of solutions for weighted -Laplacian equations that take infinite value on a bounded domain.

Journal Article
TL;DR: In this article, the existence and nonexistence of asymptotic bifurcation points of nonlinear operators in Banach space was investigated by use of a positively homogeneous operator under the conditions that the asymptonotic Frechet differentiability is not assumed.
Abstract: The existence and nonexistence of asymptotic bifurcation points of nonlinear operators in Banach space was investigated by use of a positively homogeneous operator under the conditions that the asymptotic Frechet differentiability is not assumed.

Journal ArticleDOI
TL;DR: In this paper, an asymptotic expansion of the solution to a system of first order integrodifferential equations taking into account the influence of the roots of the characteristic equation was obtained.
Abstract: We obtain an asymptotic expansion of the solution to a system of first order integrodifferential equations taking into account the influence of the roots of the characteristic equation. We establish exact asymptotics for the remainder in dependence on the asymptotic properties of original functions.

Journal ArticleDOI
TL;DR: In this paper, the exactness of asymptotic expansions of the central limit theorem has been studied, and new explicit estimates of exactness for expansions of this theorem have been given.
Abstract: This paper gives new explicit estimates of exactness for asymptotic expansions in the central limit theorem.

Journal ArticleDOI
TL;DR: In this article, the authors define a notion of system of sets with multiplicative asymptotic density and define a criterion and one necessary condition for a given system i = 1∞ to be a system with multiscale density.
Abstract: The authors define a notion of system of sets with multiplicative asymptotic density in this paper. A criterion and one necessary condition for a given system {Ai}i=1∞ to be a system with multiplicative asymptotic density is given. Properties of certain special types of systems of sets with multiplicative asymptotic density are treated.

Journal ArticleDOI
TL;DR: In this article, the asymptotic behavior of solutions of a second-order semilinear parabolic equation is analyzed in a cylindrical domain that is bounded in the space variables.
Abstract: The asymptotic behaviour of solutions of a second-order semilinear parabolic equation is analyzed in a cylindrical domain that is bounded in the space variables. The dominant term of the asymptotic expansion of the solution as is found. It is shown that the solution of this problem is asymptotically equivalent to the solution of a certain non-linear ordinary differential equation. Bibliography: 8 titles.

Journal ArticleDOI
TL;DR: Asymptotic properties of bias-corrected estimators for small diffusion models from the viewpoint of information geometry were investigated in this article, where the authors obtained results analogous to those for independent and identically distributed (iid) models.
Abstract: Information geometrical quantities such as metric tensors and connection coefficients for small diffusion models are obtained Asymptotic properties of bias-corrected estimators for small diffusion models are investigated from the viewpoint of information geometry Several results analogous to those for independent and identically distributed (iid) models are obtained by using the asymptotic normality of the statistics appearing in asymptotic expansions In contrast to the asymptotic theory for iidmodels, the geometrical quantities depend on the magnitude of noise


Book ChapterDOI
27 May 2008

Journal ArticleDOI
TL;DR: In this paper, lower asymptotic estimates for tachyon fields of open and closed strings as |t| → ∞ were obtained for the first and second terms.
Abstract: We obtain lower asymptotic estimates for tachyon fields of open and closed strings as |t| → ∞. They confirm the expressions in the first asymptotic term that were previously found as solutions of linearized equations; this is not confirmed in the second asymptotic term.

Journal ArticleDOI
TL;DR: In this article, an asymptotic expansion of the solution of an integro-differential equation of order n with the influence of the roots of the characteristic equation taken into account is obtained.
Abstract: We obtain an asymptotic expansion of the solution of an integro-differential equation of order n with the influence of the roots of the characteristic equation taken into account. The exact asymptotics of the remainder is established depending on the asymptotic properties of the kernel and the free term of the equation.