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1.
Alexander V. Rezounenko 《Comptes Rendus Mathematique》2011,349(3-4):179-183
Parabolic partial differential equations with state-dependent delays (SDDs) are investigated. The delay term presented by Stieltjes integral simultaneously includes discrete and distributed SDDs. The singular Lebesgue–Stieltjes measure is also admissible. The conditions for the corresponding initial value problem to be well-posed are presented. The existence of a compact global attractor is proved. 相似文献
2.
Alexander V. Rezounenko 《Journal of Mathematical Analysis and Applications》2012,385(1):506-516
Parabolic differential equations with discrete state-dependent delay are studied. The approach, based on an additional condition on the delay function introduced in [A.V. Rezounenko, Differential equations with discrete state-dependent delay: uniqueness and well-posedness in the space of continuous functions, Nonlinear Anal. 70 (11) (2009) 3978–3986] is developed. We propose and study an analogue of the condition which is sufficient for the well-posedness of the corresponding initial value problem on the whole space of continuous functions C. The dynamical system is constructed in C and the existence of a compact global attractor is proved. 相似文献
3.
Partial differential equations with discrete (concentrated) state-dependent delays in the space of continuous functions are investigated. In general, the corresponding initial value problem is not well-posed. So we find an additional assumption on the state-dependent delay function to guarantee the well-posedness. For the constructed dynamical system we study the long-time asymptotic behavior and prove the existence of a compact global attractor. 相似文献
4.
Alexander V. Rezounenko Jianhong Wu 《Journal of Computational and Applied Mathematics》2006,190(1-2):99-113
We propose a non-local PDE model for the evolution of a single species population that involves delayed feedback, where the delay such as the maturation time in the delayed birth rate, is selective and the selection depends on the status of the system. This delay selection, in contrast with the usual state-dependent delay widely used in ordinary delay differential equation, ensures the Lipschitz continuity of the nonlinear functional in the classical phase space. We also develop the local theory, and the existence and upper semi-continuity of the global attractor with respect to parameters. 相似文献
5.
Dingshi Li 《Journal of Difference Equations and Applications》2018,24(6):872-897
A system of stochastic discrete complex Ginzburg–Landau equations with time-varying delays is considered. We first prove the existence and uniqueness of random attractor for these equations. Then, we analyze the convergence properties of the solutions as well as the attractors as the length of time delay approaches zero. 相似文献
6.
Attractivity,multistability, and bifurcation in delayed Hopfieldʼs model with non-monotonic feedback
For a system of delayed neural networks of Hopfield type, we deal with the study of global attractivity, multistability, and bifurcations. In general, we do not assume monotonicity conditions in the activation functions. For some architectures of the network and for some families of activation functions, we get optimal results on global attractivity. Our approach relies on a link between a system of functional differential equations and a finite-dimensional discrete dynamical system. For it, we introduce the notion of strong attractor for a discrete dynamical system, which is more restrictive than the usual concept of attractor when the dimension of the system is higher than one. Our principal result shows that a strong attractor of a discrete map gives a globally attractive equilibrium of a corresponding system of delay differential equations. Our abstract setting is not limited to applications in systems of neural networks; we illustrate its use in an equation with distributed delay motivated by biological models. We also obtain some results for neural systems with variable coefficients. 相似文献
7.
Alexander V. Rezounenko 《Mathematical Methods in the Applied Sciences》2008,31(13):1569-1585
A new class of nonlinear partial differential equations with distributed in space and time state‐dependent delay is investigated. We find appropriate assumptions on the kernel function which represents the state‐dependent delay and discuss advantages of this class. Local and long‐time asymptotic properties, including the existence of global attractor and a principle of linearized stability, are studied. Copyright © 2008 John Wiley & Sons, Ltd. 相似文献
8.
LONG-TIME BEHAVIOR OF FINITE DIFFERENCE SOLUTIONS OF THREE-DIMENSIONAL NONLINEAR SCHRODINGER EQUATION WITH WEAKLY DAMPED 总被引:2,自引:0,他引:2
Fa-yongZhang 《计算数学(英文版)》2004,22(4):593-604
The three-dimensional nonlinear SchrSdinger equation with weakly damped that possesses a global attractor are considered. The dynamical properties of the discrete dynamical system which generate by a class of finite difference scheme are analysed. The existence of global attractor is proved for the discrete dynamical system. 相似文献
9.
Alexander V. Rezounenko 《Nonlinear Analysis: Theory, Methods & Applications》2010,73(6):1707-516
We investigate a class of non-linear partial differential equations with discrete state-dependent delays. The existence and uniqueness of strong solutions for initial functions from a Banach space are proved. To get the well-posed initial value problem we restrict our study to a smaller metric space, construct the dynamical system and prove the existence of a compact global attractor. 相似文献
10.
Dalibor Pražák 《Central European Journal of Mathematics》2006,4(4):635-647
We consider a system of ordinary differential equations with infinite delay. We study large time dynamics in the phase space
of functions with an exponentially decaying weight. The existence of an exponential attractor is proved under the abstract
assumption that the right-hand side is Lipschitz continuous. The dimension of the attractor is explicitly estimated.
Research supported by the project LC06052 of the Czech Ministry of Education. 相似文献
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12.
This article studies the probability distributions of solutions in the phase space for the discrete Zakharov equations. The authors first prove that the generated process of the solutions operators possesses a pullback-${\mathcal D}$ attractor, and then they establish that there exists a unique family of invariant Borel probability measures supported by the pullback attractor. 相似文献
13.
John E. Franke Abdul-Aziz Yakubu 《Journal of Difference Equations and Applications》2013,19(3):235-249
A discrete multi-species size-structured competition model is considered. By using decreasing growth functions, we achieve the self-regulation of species. We develop various biologically significant conditions for global convergence to the extinction state of the dominated species in the competitive system. With an example we illustrate coexistence in a chaotic supr transient. The chaotic attractor has an unusual pulsating nature. 相似文献
14.
The article is devoted to the study of the relation between forward and pullback attractors of set-valued nonautonomous dynamical systems (cocycles). Here it is proved that every compact global forward attractor is also a pullback attractor of the set-valued nonautonomous dynamical system. The inverse statement, generally speaking, is not true, but we prove that every global pullback attractor of an α-condensing set-valued cocycle is always a local forward attractor. The obtained general results are applied while studying periodic and homogeneous systems. We give also a new criterion of the absolute asymptotic stability of nonstationary discrete linear inclusions.
Dedicated to our friend Professor Enrico Primo Tomasini on the occasion of his 55th birthdayMathematics Subject Classifications (2000) Primary: 34C35, 34D20, 34D40, 34D45, 58F10,58F12, 58F39; secondary: 35B35, 35B40. 相似文献
15.
Invariant measure and statistical solutions for non-autonomous discrete Klein-Gordon-Schrodinger-type equations
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In this article, we first prove the existence of the pullback attractor for no-autonomous discrete Klein-Gordon-Schrodinger-type equations. Then we construct the invariant measure and statistical solutions for this discrete equations via the generalized Banach limit. 相似文献
16.
Fu-qi Yin Sheng-fan Zhou 《应用数学学报(英文版)》2006,22(3):469-486
In this paper, we establish the existence of a global attractor for a coupled κ-dimensional lattice dynamical system governed by a discrete version of the Klein-Gordon-SchrSdinger Equation. An estimate of the upper bound of the Kohnogorov ε-entropy of the global attractor is made by a method of element decomposition and the covering property of a polyhedron by balls of radii ε in a finite dimensional space. Finally, a scheme to approximate the global attractor by the global attractors of finite-dimensional ordinary differential systems is presented . 相似文献
17.
LONG-TIME BEHAVIOR OF FINITE DIFFERENCE SOLUTIONS OF A NONLINEAR SCHRODINGER EQUATION WITH WEAKLY DAMPED 总被引:2,自引:0,他引:2
Fa-yong Zhang 《计算数学(英文版)》2001,(4)
1. IntroductionThe nonlinear schr~r equation with weakly dampedwhere t = N, o > 0, together with appropriate boUndary and hatal condition, is ared inmany physical fields. The echtence of an attractor is one of the most boortant ~eristiCSfor a dissipative system. The long-tabs dynamics is completely determined by the attractorof the system. J.M. Ghidaglia[1] studied the lOng-the behavior of the nonlineaz Sequation (1.1) and proved the eAstence of a compact global attractor A in H'(n) which… 相似文献
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Alain Miranville 《Applications of Mathematics》2003,48(1):31-47
In this article we introduce the notion of a minimal attractor for families of operators that do not necessarily form semigroups. We then obtain some results on the existence of the minimal attractor. We also consider the nonautonomous case. As an application, we obtain the existence of the minimal attractor for models of Cahn-Hilliard equations in deformable elastic continua. 相似文献
20.
Fa-Yong Zhang & Shu-Juan Lu 《计算数学(英文版)》2001,19(4):393-406
A weakly damped Schrödinger equation possessing a global attractor are considered. The dynamical properties of a class of finite difference scheme are analysed. The existence of global attractor is proved for the discrete system. The stability of the difference scheme and the error estimate of the difference solution are obtained in the autonomous system case. Finally, long-time stability and convergence of the class of finite difference scheme also are analysed in the nonautonomous system case. 相似文献