Time-dependent differential inclusions, cocycle attractors and fuzzy differential equations

Diamond, Phil

doi:10.1109/91.811243

Cited by 131 publications

(98 citation statements)

References 18 publications

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“…where [33] proved that the attainable solution sets A α (X α , t), α ∈ [0, 1] of the family of inclusions (4) on [0, t] are the level sets of fuzzy set A(X 0 , t) ∈ D n and then extended existing results of stability and periodicity to time-dependent differential inclusions [34]. The idea was to solve these differential inclusions and using the stacking theorem of Negoita and Ralescu [53] to bunch these solutions into a fuzzy solution.…”

Section: Basic Conceptsmentioning

confidence: 99%

Fuzzy Complex Dynamical Networks and Its Synchronization

Mahdavi

Menhaj

Kurths

et al. 2013

IEEE Trans. Cybern.

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Abstract-In this paper, the robust synchronization problem of fuzzy complex dynamical networks is investigated. A fuzzy complex dynamical network is an extension to an uncertain complex dynamical network in which all sources of parametric uncertainties are modeled with fuzzy numbers, i.e., all nodes' dynamics are described by fuzzy differential equations (FDEs) that permit a better description of a real process occurring in the presence of inaccuracy. To resolve the synchronization problem, this paper introduces new adaptive and impulsive controllers in which globally exponential synchronization of fuzzy dynamical networks under easily verified conditions is guaranteed. Moreover, we propose an efficient method that helps to find certain suitable nodes to be impulsively controlled via pinning, noting that these nodes, in general, vary at distinct impulsive time instants. Therefore, by using adaptive controllers and applying impulsive controllers to only a small portion of nodes, the whole network will completely be synchronized to a certain objective state. Finally, two numerical examples are given to illustrate the effectiveness of the proposed controllers.

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Section: Basic Conceptsmentioning

confidence: 99%

Fuzzy Complex Dynamical Networks and Its Synchronization

Mahdavi

Menhaj

Kurths

et al. 2013

IEEE Trans. Cybern.

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“…Subsequently, using the H-derivative, Kaleva [19] started to develop a theory for FDE. In the last few years, many works have been done by several authors in theoretical and applied fields (see [3,1,8,10,4,13,19,23]. A variety of exact, approximate, and purely numerical methods are available to find the solution of a fuzzy initial value problem (FIVP).…”

Section: Introductionmentioning

confidence: 99%

Solution of the Fuzzy Boundary Value Differential Equations Under Generalized Differentiability By Shooting Method

Jamshidi¹,

Avazpour²

2012

Journal of Fuzzy Set Valued Analysis

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In this paper, we apply the shooting method for solving the Fuzzy Boundary Value Differential Equations (FBVDEs) of the second order under generalized differentiability. By this method an FBVDE of the second order will be replaced with two fuzzy initial value differential equations and the answers of each of them are obtained by the Adomian method. Finally via linear combination of their solutions, the fuzzy solution will be obtained.

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“…, where U = U 1 × U 2 , the attainable sets associated with problem (10) and it is defined, for each α ∈ [0, 1], by 1] satisfies the conditions of the Theorem 1 (see [1,7,12]) and so there exists an interval fuzzy A(t, X 0 , U ) which become a fuzzy solution X(t) = A(t, X 0 , U ) of (9) via differential inclusion and…”

Section: Fuzzy Differential Equations Via Differential Inclusionsmentioning

confidence: 99%

“…If we have a function whose parameters are fuzzy intervals in the right-hand side of (1), what is the appropriate extension within fuzzy theory?. To obtain a fuzzy differential equation from such a problem (1), we consider For problem (2) there are at least three possibilities for representing a fuzzy solution: the first involves the derivative of fuzzy functions [3,4,8,9,10,24]; the second is obtained by applying Zadeh's extension principle to the deterministic solution [9,20] and the last one is based on a family of differential inclusions [2,7,12,13,16,17,18]. In this article we devote to the last approach, obtaining a fuzzy solution of (2) via a family of differential inclusions.…”

Section: Introductionmentioning

confidence: 99%