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1.
The article presents a new universal theory of dynamical chaos in nonlinear dissipative systems of differential equations,
including autonomous and nonautonomous ordinary differential equations (ODE), partial differential equations, and delay differential
equations. The theory relies on four remarkable results: Feigenbaum’s period doubling theory for cycles of one-dimensional
unimodal maps, Sharkovskii’s theory of birth of cycles of arbitrary period up to cycle of period three in one-dimensional
unimodal maps, Magnitskii’s theory of rotor singular point in two-dimensional nonautonomous ODE systems, acting as a bridge
between one-dimensional maps and differential equations, and Magnitskii’s theory of homoclinic bifurcation cascade that follows
the Sharkovskii cascade. All the theoretical propositions are rigorously proved and illustrated with numerous analytical examples
and numerical computations, which are presented for all classical chaotic nonlinear dissipative systems of differential equations. 相似文献
2.
On the Growth of Components of Meromorphic Solutions of Systems of Complex Differential Equations 总被引:3,自引:0,他引:3
Ling-yun Gao 《应用数学学报(英文版)》2005,21(3):499-504
This paper investigates the problem of the growth of the components of meromorphic solutions of a class of a system of complex algebraic differential equations, and generalized some of N. Toda's results concerning the growth of differential equations to the case of systems of differential equations. The paper considers the existence of admissible solutions of the system of differential equations. 相似文献
3.
In this paper we use measure theory to solve a wide range of second-order boundary value ordinary differential equations. First, we transform the problem to a first order system of ordinary differential equations (ODE’s) and then define an optimization problem related to it. The new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; the optimal measure is then approximated by a finite combination of atomic measures and the problem converted approximatly to a finite-dimensional linear programming problem. The solution to this problem is used to construct the approximate solution of the original problem. Finally we get the error functionalE (we define in this paper) for the approximate solution of the ODE’s problems. 相似文献
4.
Aleksandr Ivanovich Dvirny Vitalii Ivanovich Slyn’ko 《Journal of Mathematical Sciences》2011,179(2):245-260
We obtained the sufficient conditions for the stability of solutions of a class of nonlinear differential equations with fixed
instant impulsive effects in the Banach space. With the use of the Slyusarchuk’s condition and methods of the theory of operators
in a partially ordered Banach space, the problem is reduced to the study of the stability of a linear system of second-order
impulsive differential equations. 相似文献
5.
A. M. Meirmanov 《Journal of Mathematical Sciences》2009,163(2):111-150
A linear system of differential equations describing a joint motion of elastic porous body and fluid occupying porous space
is considered. Although the problem is linear, it is very hard to tackle due to the fact that its main differential equations
involve nonsmooth oscillatory coefficients, both big and small, under the differentiation operators. The rigorous justification,
under various conditions imposed on physical parameters, is fulfilled for homogenization procedures as the dimensionless size
of the pores tends to zero, while the porous body is geometrically periodic. As the results for different ratios between physical
parameters, we derive Biot’s equations of poroelasticity, a system consisting of nonisotropic Lamé’s equations for the solid
component and acoustic equations for the liquid component, nonisotropic Lamé’s equations or equations of viscoelasticity for
one-velocity continuum, decoupled system consisting of Darcy’s system of filtration or acoustic equations for the liquid component
(first approximation) and nonisotropic Lamé’s equations for the solid component (second approximation), a system consisting
of nonisotropic Stokes equations for the liquid component and acoustic equations for the solid component, nonisotropic Stokes
equations for one-velocity continuum, or, finally a different type of acoustic equations for one- or two-velocity continuum.
The proofs are based on Nguetseng’s two-scale convergence method of homogenization in periodic structures. 相似文献
6.
A. V. Shanin 《Journal of Mathematical Sciences》2008,148(5):769-783
The problems of diffraction by a slit or a strip having ideal boundary conditions, and some other problems, can be reduced
to the problem of wave propagation on a multisheet surface by applying the method of reflections. Further simplifications
of the problem can be achieved by applying an embedding formula. As a result, the solution of the problem with a plane wave
incidence becomes expressed in terms of the edge Green’s functions, i.e., in terms of the fields generated by dipole sources
localized at branchpoints of the surface.
The present paper is devoted to finding the edge Green’s functions. For this problem, two sets of differential equations,
namely, the coordinate and spectral equations, are used. The properties of solutions of these equations are studied. Bibliography:
9 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 342, 2007, pp. 233–256. 相似文献
8.
Levon Andreevich Beklaryan 《Journal of Mathematical Sciences》2006,135(2):2813-2954
In this work, we give an introduction to the theory of nonlinear functional differential equations of pointwise type on a
finite interval, semi-axis, or axis. This approach is based on the formalism using group peculiarities of such differential
equations. For the main boundary-value problem and the Euler-Lagrange boundary-value problem, we consider the existence and
uniqueness of the solution, the continuous dependence of the solution on boundary-value and initial-value conditions, and
the “roughness” of functional differential equations in the considered boundary-value problems. For functional differential
equations of pointwise type we also investigate the pointwise completeness of the space of solutions for given boundary-value
conditions, give an estimate of the rank for the space of solutions, describe types of degeneration for the space of solutions,
and establish conditions for the “smoothness” of the solution. We propose the method of regular extension of the class of
ordinary differential equations in the class of functional differential equations of pointwise type.
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions),
Vol. 8, Functional Differential Equations, 2004. 相似文献
9.
M. C. Nucci 《Theoretical and Mathematical Physics》2007,151(3):851-862
In addition to the reduction method, we present a novel application of Jacobi’s last multiplier for finding Lie symmetries
of ordinary differential equations algorithmically. These methods and Lie symmetries allow unveiling the hidden linearity
of certain nonlinear equations that are relevant in physics. We consider the Einstein-Yang-Mills equations and Calogero’s
many-body problem in the plane as examples.
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 495–509, June, 2007. 相似文献
10.
S. M. Chuiko 《Russian Mathematics (Iz VUZ)》2019,63(12):80-95
We study the Cauchy problem for a linear differential-algebraic system of equations with concentrated delay. Our research continues investigation of solvability of linear Noether boundary value problems for systems of functional-differential equations given in the monographs by A.D. Myshkis, N.V. Azbelev, V.P. Maksimov, L.F. Rakhmatullina, A.M. Samoilenko, and A.A. Boichuk; meanwhile, we use essentially the tool of Moore-Penrose inverse matrices. For a linear differential-algebraic system with concentrated delay, we find sufficient conditions for its solvability and give a construction of generalized Green’s operator for Cauchy’s problem. We also give some examples which illustrate in detail the solvability conditions and the suggested construction. 相似文献
11.
In recent times, optimal control theory for distributed parameter systems has been actively studied; among them, an important
place is occupied by the class of systems describing oscillation processes. This work studies linear control distributed parameter
systems of hyperbolic type. The minimization problem of a quadratic functional on the trajectories of the system is considered.
By using the Fourier method, the problem is reduced to studying optimal solutions for a countable control system of ordinary
differential equations. For Galerkin’s approximations of this system, it is proved that the optimal control is a chattering
control, i.e., it has infinitely many switchings on a finite interval of time. The construction of the optimal synthesis uses
the results of the theory of singular regimes and regimes with with more and more frequent switchings.
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions),
Vol. 19, Optimal Control, 2006. 相似文献
12.
V. A. Gorelov 《Mathematical Notes》2000,67(2):138-151
We prove a generalization of Shidlovskii’s theorem on the algebraic independence of the values ofE-functions satisfying a system of linear differential equations that is well known in the theory of transcendental numbers.
We consider the case in which the values ofE-functions are taken at singular points of these systems. Using the obtained results, we prove Siegel’s conjecture that, for
the case of first-order differential equations, anyE-function satisfying a linear differential equation is representable as a polynomial in hypergeometricE-functions.
Translated fromMatematicheskie Zametki, Vol. 67, No. 2, pp. 174–190, February, 2000. 相似文献
13.
We present a numerical-analytic method for solution of the problem of nonlinear periodic oscillation of a string pendulum
(unstretched string suspended from a point in a gravitational field). The method is based on successive linearization of the
problem with the Lindstadt-Poincare method, separation of time dependencies with Fourier’s method, reduction of differential
equations to recurrence relations for the coefficients of power series in the spatial coordinate, and computer realization
of the convergence of the power series.
Translated fromDinamicheskie Sistemy, Vol. 12. pp. 44–51, 1993. 相似文献
14.
D. V. Valovik 《Computational Mathematics and Mathematical Physics》2011,51(9):1622-1632
A boundary value problem for Maxwell’s equations describing propagation of TM waves in a nonlinear dielectric layer with arbitrary
nonlinearity is considered. The layer is located between two linear semi-infinite media. The problem is reduced to a nonlinear
boundary eigenvalue problem for a system of second-order nonlinear ordinary differential equations. A dispersion equation
for the eigenvalues of the problem (propagation constants) is derived. For a given nonlinearity function, the dispersion equation
can be studied both analytically and numerically. A sufficient condition for the existence of at least one eigenvalue is formulated. 相似文献
15.
V. I. Maksimov 《Proceedings of the Steklov Institute of Mathematics》2010,268(1):188-203
The problem of a stable dynamical reconstruction of unknown characteristics in a distributed system described by a pair of differential equations is considered. The aim of the paper is to construct a solution algorithm for this problem. The proposed algorithm is based on Yu.S. Osipov’s theory of dynamic inversion and N.N. Krasovskii’s extremal shift method. 相似文献
16.
Pee Choon Toh 《The Ramanujan Journal》2011,25(2):179-194
Ramanujan’s differential equations for the classical Eisenstein series are of great importance to many areas in number theory
and special functions. H.H. Chan recently demonstrated that these differential equations can be derived from the triple product
identity and the quintuple product identity in an elementary manner. In this article, we extend this method in a uniform manner
to derive corresponding differential equations for the Eisenstein series of level 2. Several applications of these differential
equations are also given. 相似文献
17.
We study the problem of finding conditions for the existence of solutions of weakly nonlinear Noetherian boundary-value problems
for systems of ordinary differential equations and the construction of these solutions. A new iterative procedure with accelerated
convergence is proposed for the construction of solutions of a weakly nonlinear Noetherian boundary-value problem for a system
of ordinary differential equations in the critical case.
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1587–1601, December, 2008. 相似文献
18.
A. I. Perov 《Differential Equations》2012,48(5):670-680
We obtain new conditions for the existence of bounded solutions of higher-order nonlinear differential equations. In addition to the classical contraction mapping principle, A.N. Tikhonov’s fixed-point principle is used in the proof of existence theorems. Assertions dealing with the stability of a bounded solution are derived directly from the corresponding results obtained by M.A. Krasnosel’skii and A.V. Pokrovskii. 相似文献
19.
Motivated by boundary problems for linear differential equations, we define an abstract boundary problem as a pair consisting
of a surjective linear map (“differential operator”) and an orthogonally closed subspace of the dual space (“boundary conditions”).
Defining the composition of boundary problems corresponding to their Green’s operators in reverse order, we characterize and
construct all factorizations of a boundary problem from a given factorization of the defining operator. For the case of ordinary
differential equations, the main results can be made algorithmic. We conclude with a factorization of a boundary problem for
the wave equation.
This work was supported by the Austrian Science Fund (FWF) under the SFB grant F1322. 相似文献
20.
Lenore Blum Felipe Cucker Tomaso Poggio James Renegar Michael Shub 《Foundations of Computational Mathematics》2005,5(4):349-349
Steve Smale set the agenda for FoCM in his call for the 1995 conference in Park City, Utah. No stranger he to ambitious agendas
and extraordinary accomplishments. He is one of the dominant figures in the mathematics of the second half of the twentieth
century. Smale’s theory of immersions, the generalized Poincare conjecture, and H-cobordism theorems with their far-reaching
consequences are the bedrock of differential topology. His horseshoe is the hallmark of chaos, and his hyperbolic dynamics
the rejuvenation of the geometric theory of dynamical systems. He is one of the pioneers in the introduction of infinite-dimensional
manifolds for the study of nonlinear problems in the calculus of variations and partial differential equations. The list goes
on: the systematic use of differential techniques in microeconomics, electrical circuit theory, chaos in predator–prey equations
and, finally, for the twentieth century, the foundations of computational mathematics and complexity theory, and now, in the
twenty-first century, the theory of learning. It has been our privilege to be among his collaborators and students in the
broadest sense of the word. With these issues (Volume 5 Number 4 and Volume 6 Number 1, as well as an earlier article appearing
in Volume 5 Number 2, are dedicated to Steve Smale’s 75th Birthday) we celebrate Steve’s 75th birthday and continuing vitality.
He sets the bar high. We do our best. 相似文献