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1.
Summary. A system describing an oscillating chemical reaction (known as a Bray—Liebhafsky oscillating reaction) is considered. It
is shown that large amplitude oscillations arise through a homoclinic bifurcation and vanish through a subcritical Hopf bifurcation.
An approximate locus of points corresponding to the homoclinic orbit in a parameter space is calculated using a variation
of the Bogdanov—Takens—Carr method. A special feature of the problem is related to the fact that nonlinear terms in the equations
contain square and cubic roots of expressions depending on the unknowns. For a particular model considered it is possible
to obtain most of the results analytically.
Received September 21, 1998; revised March 29, 1999; accepted June 17, 1999 相似文献
2.
Christiansen 《Constructive Approximation》2008,19(1):1-22
Abstract. We consider the indeterminate Stieltjes moment problem associated with the q -Laguerre polynomials. A transformation of the set of solutions, which has all the classical solutions as fixed points, is
established and we present a method to construct, for instance, continuous singular solutions. The connection with the moment
problem associated with the Stieltjes—Wigert polynomials is studied; we show how to come from q -Laguerre solutions to Stieltjes—Wigert solutions by letting the parameter α —> ∞ , and we explain how to lift a Stieltjes—Wigert solution to a q -Laguerre solution at the level of Pick functions. Based on two generating functions, expressions for the four entire functions
from the Nevanlinna parametrization are obtained. 相似文献
3.
B. Braaksma 《Journal of Nonlinear Science》1998,8(5):457-490
Summary. We study a general nonlinear ODE system with fast and slow variables, i.e., some of the derivatives are multiplied by a small
parameter. The system depends on an additional bifurcation parameter. We derive a normal form for this system, valid close
to equilibria where certain conditions on the derivatives hold. The most important condition concerns the presence of eigenvalues
with singular imaginary parts, by which we mean that their imaginary part grows without bound as the small parameter tends
to zero. We give a simple criterion to test for the possible presence of equilibria satisfying this condition. Using a center
manifold reduction, we show the existence of Hopf bifurcation points, originating from the interaction of fast and slow variables,
and we determine their nature. We apply the theory, developed here, to two examples: an extended Bonhoeffer—van der Pol system
and a predator-prey model. Our theory is in good agreement with the numerical continuation experiments we carried out for
the examples.
Received October 24, 1996; revised October 31, 1997; accepted November 3, 1997 相似文献
4.
Monica E. Bad Dumitrescu 《Annals of the Institute of Statistical Mathematics》1987,39(1):211-218
Summary The Bayesian estimation problem for the parameter θ of an exponential probability distribution is considered, when it is assumed
that θ has a natural conjugate prior density and a loss-function depending on the squared error is used. It is shown that,
with probability one, the posterior density of the Bayesian—centered and scaled parameter converges pointwise to the normal
probability density. The weak convergence of the posterior distributions to the normal distribution follows directly. Both
correct and incorrect models are studied and the asymptotic normality is stated respectively. 相似文献
5.
Vera Mikyoung Hur 《Journal d'Analyse Mathématique》2011,113(1):331-386
The existence of periodic waves propagating downstream on the surface of a two-dimensional infinitely deep body of water under
the force of gravity is established for a general class of vorticities. When reformulated as an elliptic boundary value problem
in a fixed semi-infinite cylinder with a parameter, the operator describing the problem is nonlinear and non-Fredholm. A global
connected set of nontrivial solutions is obtained via singular theory of bifurcation. The proof combines a generalized degree
theory, global bifurcation theory, and Whyburn’s lemma in topology with the Schauder theory for elliptic problems and the
maximum principle. 相似文献
6.
I. L. Pokrovski 《Journal of Mathematical Sciences》2007,146(1):5564-5576
An approach to nonlinear, positively homogeneous eigenvalue problems based on the use of the spectral parameter as a functional
of Euler type is suggested. It allows one to present the spectral-parameter domain as a bifurcation diagram of the problem.
Fučik spectrum problems (classical and for p-Laplacians) and the problem with a nonlinear dependence of the weight function on the spectral parameter are considered.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 169–186, 2005. 相似文献
7.
H. Uecker 《Journal of Nonlinear Science》2001,11(2):89-121
Summary. We show the existence and stability of modulating multipulse solutions for a class of bifurcation problems given by a dispersive
Swift-Hohenberg type of equation with a spatially periodic forcing. Equations of this type arise as model problems for pattern
formation over unbounded weakly oscillating domains and, more specifically, in laser optics. As associated modulation equation,
one obtains a nonsymmetric Ginzburg-Landau equation which possesses exponentially stable stationary n—pulse solutions. The modulating multipulse solutions of the original equation then consist of a traveling pulselike envelope
modulating a spatially oscillating wave train. They are constructed by means of spatial dynamics and center manifold theory.
In order to show their stability, we use Floquet theory and combine the validity of the modulation equation with the exponential
stability of the n—pulses in the modulation equation. The analysis is supplemented by a few numerical computations.
In addition, we also show, in a different parameter regime, the existence of exponentially stable stationary periodic solutions
for the class of systems under consideration.
Received November 30, 1999; accepted December 4, 2000 Online publication March 23, 2001 相似文献
8.
Gauss—Seidel type relaxation techniques are applied in the context of strictly convex pure networks with separable cost functions.
The algorithm is an extension of the Bertsekas—Tseng approach for solving the linear network problem and its dual as a pair
of monotropic programming problems. The method is extended to cover the class of generalized network problems. Alternative
internal tactics for the dual problem are examined. Computational experiments —aimed at the improved efficiency of the algorithm
— are presented.
This research was supported in part by National Science Foundation Grant No. DCR-8401098-A0l. 相似文献
9.
Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving
a semilinear elliptic equation depending on a parameter. The domain is star-shaped and such that a Poincaré inequality holds
but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach
can be seen, in some cases, as an extension of non-existence results for non-trivial solutions. It is based on Rellich–Pohožaev
type estimates. Semilinear elliptic equations naturally arise in many applications, for instance in astrophysics, hydrodynamics
or thermodynamics. We simplify the proof of earlier results by K. Schmitt and R. Schaaf in the so-called local multiplicative
case, extend them to the case of a non-local dependence on the bifurcation parameter and to the additive case, both in local
and non-local settings. 相似文献
10.
I. Dobson 《Journal of Nonlinear Science》1993,3(1):307-327
Summary Engineering and physical systems are often modeled as nonlinear differential equations with a vector λ of parameters and operated
at a stable equilibrium. However, as the parameters λ vary from some nominal value λ0, the stability of the equilibrium can be lost in a saddle-node or Hopf bifurcation. The spatial relation in parameter space
of λ0 to the critical set of parameters at which the stable equilibrium bifurcates determines the robustness of the system stability
to parameter variations and is important in applications. We propose computing a parameter vector λ* at which the stable equilibrium bifurcates which is locally closest in parameter space to the nominal parameters λ0. Iterative and direct methods for computing these locally closest bifurcations are described. The methods are extensions
of standard, one-parameter methods of computing bifurcations and are based on formulas for the normal vector to hypersurfaces
of the bifurcation set. Conditions on the hypersurface curvature are given to ensure the local convergence of the iterative
method and the regularity of solutions of the direct method. Formulas are derived for the curvature of the saddle node bifurcation
set. The methods are extended to transcritical and pitchfork bifurcations and parametrized maps, and the sensitivity to λ0 of the distance to a closest bifurcation is derived. The application of the methods is illustrated by computing the proximity
to the closest voltage collapse instability of a simple electric power system. 相似文献
11.
Differential Equations - For nonlinear evolution equations in a Banach space that depend in two ways—regularly and singularly—on a small parameter, we construct $$\varepsilon $$... 相似文献
12.
13.
T. J. Bridges 《Journal of Nonlinear Science》1994,4(1):221-251
Summary The governing equations for three-dimensional time-dependent water waves in a moving frame of reference are reformulated in
terms of the energy and momentum flux. The novelty of this approach is that time-independent motions of the system—that is,
motions that are steady in a moving frame of reference—satisfy a partial differential equation, which is shown to be Hamiltonian.
The theory of Hamiltonian evolution equations (canonical variables, Poisson brackets, symplectic form, conservation laws)
is applied to the spatial Hamiltonian system derived for pure gravity waves. The addition of surface tension changes the spatial
Hamiltonian structure in such a way that the symplectic operator becomes degenerate, and the properties of this generalized
Hamiltonian system are also studied. Hamiltonian bifurcation theory is applied to the linear spatial Hamiltonian system for
capillary-gravity waves, showing how new waves can be found in this framework. 相似文献
14.
Generalized Chaplygin’s transformation and explicit integration of a system with a spherical support
Alexey V. Borisov Alexander A. Kilin Ivan S. Mamaev 《Regular and Chaotic Dynamics》2012,17(2):170-190
We discuss explicit integration and bifurcation analysis of two non-holonomic problems. One of them is the Chaplygin’s problem
on no-slip rolling of a balanced dynamically non-symmetric ball on a horizontal plane. The other, first posed by Yu. N. Fedorov,
deals with the motion of a rigid body in a spherical support. For Chaplygin’s problem we consider in detail the transformation
that Chaplygin used to integrate the equations when the constant of areas is zero. We revisit Chaplygin’s approach to clarify
the geometry of this very important transformation, because in the original paper the transformation looks a cumbersome collection
of highly non-transparent analytic manipulations. Understanding its geometry seriously facilitate the extension of the transformation
to the case of a rigid body in a spherical support — the problem where almost no progress has been made since Yu.N. Fedorov
posed it in 1988. In this paper we show that extending the transformation to the case of a spherical support allows us to
integrate the equations of motion explicitly in terms of quadratures, detect mostly remarkable critical trajectories and study
their stability, and perform an exhaustive qualitative analysis of motion. Some of the results may find their application
in various technical devices and robot design. We also show that adding a gyrostat with constant angular momentum to the spherical-support
system does not affect its integrability. 相似文献
15.
Douglas José AlemJr. Pedro Augusto MunariJr. Marcos Nereu Arenales Paulo Augusto Valente Ferreira 《Annals of Operations Research》2010,179(1):169-186
This paper addresses the one-dimensional cutting stock problem when demand is a random variable. The problem is formulated
as a two-stage stochastic nonlinear program with recourse. The first stage decision variables are the number of objects to
be cut according to a cutting pattern. The second stage decision variables are the number of holding or backordering items
due to the decisions made in the first stage. The problem’s objective is to minimize the total expected cost incurred in both
stages, due to waste and holding or backordering penalties. A Simplex-based method with column generation is proposed for
solving a linear relaxation of the resulting optimization problem. The proposed method is evaluated by using two well-known
measures of uncertainty effects in stochastic programming: the value of stochastic solution—VSS—and the expected value of perfect information—EVPI. The optimal two-stage solution is shown to be more effective than the alternative wait-and-see and expected value approaches,
even under small variations in the parameters of the problem. 相似文献
16.
T. B. Boffey 《TOP》1998,6(2):205-221
In recent years, interest has been shown in the optimal location of ‘extensive’ facilities in a network. Two such problems—the
Maximal Direct and Indirect Covering Tree problems—were introduced by Hutson and ReVelle. Previous solution techniques are
extended to provide an efficient algorithm for the Indirect Covering Tree problem and the generalization in which demand covered
is attenuated by distance. It is also shown that the corresponding problem is NP-hard when the underlying network is not a
tree. 相似文献
17.
Shiwei He Sohail S. Chaudhry Zhonglin Lei Wang Baohua 《Annals of Operations Research》2009,168(1):169-179
We study a vendor selection problem in which the buyer allocates an order quantity for an item among a set of suppliers such
that the required aggregate quality, service, and lead time requirements are achieved at minimum cost. Some or all of these
characteristics can be stochastic and hence, we treat the aggregate quality and service as uncertain. We develop a class of
special chance-constrained programming models and a genetic algorithm is designed for the vendor selection problem. The solution
procedure is tested on randomly generated problems and our computational experience is reported. The results demonstrate that
the suggested approach could provide managers a promising way for studying the stochastic vendor selection problem.
The authors would like to thank the referees for providing constructive comments that led to an improved version of the paper.
Also, this research was partially supported by grants from National Natural Science Foundation (60776825)—China, 863 Programs
(2007AA11Z208)—China, Doctorate Foundation (20040004012)—China, Villanova University Research Sabbatical Fall 2006, and the
National Science Foundation (0332490)—USA. 相似文献
18.
V. N. Paimushin T. V. Polyakova N. V. Polyakova R. K. Gazizullin 《Russian Mathematics (Iz VUZ)》2020,64(5):56-65
The formulation of the acoustoelasticity problem is given on the basis of refined motion equations of orthotropic plates. These equations are constructed in the first approximation by reducing the three-dimensional equations of the theory of elasticity to the two-dimensional equations of the theory of plates, where the approximation of the transverse tangential stresses and the transverse reduction stress is made with the help of trigonometric basis functions in the thickness direction. Wherein at the points of the boundary (front) surfaces, the static boundary conditions of the problem for tangential stresses are satisfied exactly and for transverse normal stress — approximately. Accounting for internal energy dissipation in the plate material is based on the Thompson—Kelvin—Voigt hysteresis model. In case of formulating problems on dynamic processes of plate deformation in vacuum, the equations are divided into two separate systems of equations. The first of these systems describes non-classical shear-free, longitudinal-transverse forms of movement, accompanied by a distortion of the flat form of cross sections, and the second system describes transverse bending-shear forms of movement. The latter are practically equivalent in quality and content to the analogous equations of the well-known variants of refined theories, but, unlike them, with a decrease in the relative thickness parameter, they lead to solutions according to the classical theory of plates. The motion of the surrounding the plate acoustic media is described by the generalized Helmholtz wave equations, constructed with account of energy dissipation by introducing into consideration the complex sound velocity according to Skudrzyk. 相似文献
19.
L. J. Alvarez-Vázquez N. García-Chan A. Martínez M. E. Vázquez-Méndez 《Computational Optimization and Applications》2010,46(1):135-157
This work treats, within a multi-objective framework, of an economical-ecological problem related to the optimal management
of a wastewater treatment system consisting of several purifying plants. The problem is formulated as a multi-objective parabolic
optimal control problem and it is studied from a cooperative point of view, looking for Pareto-optimal solutions. The weighting
method is used here to characterize the Pareto solutions of our problem. To obtain them, a numerical algorithm—based in a
characteristics-Galerkin discretization—is proposed and numerical results for a real world situation in the estuary of Vigo
(NW Spain) are also presented. 相似文献
20.
A.Yu. Morozov 《Theoretical and Mathematical Physics》2010,162(1):1-33
We briefly review the basic properties of unitary matrix integrals, using three matrix models to analyze their properties:
the ordinary unitary, the Brezin—Gross—Witten, and the Harish—Chandra—Itzykson—Zuber models. We especially emphasize the nontrivial
aspects of the theory, from the De Witt’Hooft anomaly in unitary integrals to the problem of calculating correlators with
the Itzykson-Zuber measure. We emphasize the method of character expansions as a technical tool. Unitary integrals are still
insufficiently investigated, and many new results should be expected as this field attracts increased attention. 相似文献