zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
A class of linear differential dynamical systems with fuzzy matrices. (English) Zbl 1193.37025

Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition – an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0,1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1.

This paper investigates the first order linear fuzzy differential dynamical systems with fuzzy matrices. We use a complex number representation of the α-level sets of the fuzzy system, and obtain the solution by employing such representation. It is applicable to practical computations and has also some implications for the theory of fuzzy differential equations. We then present some properties of the two-dimensional dynamical systems and their phase portraits. Some examples are considered to show the richness of the theory and we can clearly see that new behaviors appear. We finally present some conclusions and new directions for further research in the area of fuzzy dynamical systems.

37C10Vector fields, flows, ordinary differential equations
34A30Linear ODE and systems, general
03E72Fuzzy set theory
[1]Bede, B.; Rudas, I. J.; Bencsik, A. L.: First order linear fuzzy differential equations under generalized differentiability, Inform. sci. 177, 1648-1662 (2007) · Zbl 1119.34003 · doi:10.1016/j.ins.2006.08.021
[2]Chen, B.; Liu, X.: Reliable control design of fuzzy dynamical systems with time-varying delay, Fuzzy sets and systems 146, 349-374 (2004) · Zbl 1055.93050 · doi:10.1016/S0165-0114(03)00326-9
[3]Chen, M.; Fu, Y.; Xue, X.; Wu, C.: Two-point boundary value problems of undamped uncertain dynamical systems, Fuzzy sets and systems 159, 2077-2089 (2008) · Zbl 1225.34006 · doi:10.1016/j.fss.2008.03.006
[4]Diamond, P.; Kloeden, P. E.: Metric spaces of fuzzy set: theory and applications, (1994)
[5]Diamond, P.; Watson, P.: Regularity of solution sets for differential inclusions quasiconcave in parameter, Appl. math. Lett. 13, 31-35 (2000) · Zbl 0944.34008 · doi:10.1016/S0893-9659(99)00141-X
[6]Diamond, P.: Brief note on the variation of constants formula for fuzzy differential equations, Fuzzy sets and systems 129, 65-71 (2002) · Zbl 1021.34048 · doi:10.1016/S0165-0114(01)00158-0
[7]Georgiou, D. N.; Nieto, J. J.; Rodriguez-Lopez, R.: Initial value problems for higher-order fuzzy differential equations, Nonlinear anal. 63, 587-600 (2005) · Zbl 1091.34003 · doi:10.1016/j.na.2005.05.020
[8]Bhaskar, T. G.; Lakshmikantham, V.; Devi, V.: Revisiting fuzzy differential equations, Nonlinear anal. 58, 351-358 (2004) · Zbl 1095.34511 · doi:10.1016/j.na.2004.05.007
[9]Goetschel, R.; Voxman, W.: Elementary fuzzy calculus, Fuzzy sets and systems 18, 31-43 (1986) · Zbl 0626.26014 · doi:10.1016/0165-0114(86)90026-6
[10]Hirsch, M. W.; Smale, S.: Differential equation, dynamical systems and linear algebra, (1974)
[11]Hüllermeier, E.: An approach to modeling and simulation of uncertain dynamical systems, Internat. J. Uncertain. fuzziness knowledge-based systems 5, 117-137 (1997) · Zbl 1232.68131 · doi:10.1142/S0218488597000117
[12]Hong, L.; Sun, J.: Bifurcations of fuzzy nonlinear dynamical systems, Commun. nonlinear sci. Numer. simul. 11, 1-12 (2006) · Zbl 1078.37049 · doi:10.1016/j.cnsns.2004.11.001
[13]Kaleva, O.: Fuzzy differential equations, Fuzzy sets and systems 24, 301-317 (1987) · Zbl 0646.34019 · doi:10.1016/0165-0114(87)90029-7
[14]Kaleva, O.: A note on fuzzy differential equations, Nonlinear anal. 64, 895-900 (2006) · Zbl 1100.34500 · doi:10.1016/j.na.2005.01.003
[15]Lakshmikantham, V.; Nieto, J. J.: Differential equations in metric spaces: an introduction and an application to fuzzy differential equations, Dyn. contin. Discrete impuls. Syst. ser. A math. Anal. 10, 991-1000 (2003) · Zbl 1057.34061
[16]Moore, R. E.: Computational functional analysis, (1985)
[17]Nieto, J. J.; Rodriguez-Lopez, R.: Analysis of a logistic differential model with uncertainty, Int. J. Dyn. syst. Differ. equ. 1, 164-176 (2008) · Zbl 1170.34009 · doi:10.1504/IJDSDE.2008.019678
[18]Nieto, J. J.; Rodriguez-Lopez, R.: Bounded solutions for fuzzy differential and integral equations, Chaos solitons fractals 27, 1376-1386 (2006)
[19]Nieto, J. J.; Rodriguez-Lopez, R.; Franco, D.: Linear first-order fuzzy differential equations, Internat. J. Uncertain. fuzziness knowledge-based systems 14, 687-709 (2006) · Zbl 1116.34005 · doi:10.1142/S0218488506004278
[20]Nieto, J. J.; Rodriguez-Lopez, R.; Georgiou, D. N.: Fuzzy differential systems under generalized metric spaces approach, Dynam. systems appl. 17, 1-24 (2008) · Zbl 1168.34005
[21]O’regan, Donal; Lakshmikantham, V.; Nieto, J. J.: Initial and boundary value problems for fuzzy differential equations, Nonlinear anal. 54, 405-415 (2003) · Zbl 1048.34015 · doi:10.1016/S0362-546X(03)00097-X
[22]Pearson, D. W.: A property of linear fuzzy differential equations, Appl. math. Lett. 10, No. 3, 99-103 (1997) · Zbl 0882.34014 · doi:10.1016/S0893-9659(97)00043-8
[23]Perko, L.: Differential equations and dynamical systems, (1991)
[24]Rodriguez-Lopez, R.: Periodic boundary value problems for impulsive fuzzy differential equations, Fuzzy sets and systems 159, 1384-1409 (2008) · Zbl 1225.34008 · doi:10.1016/j.fss.2007.09.005
[25]Roman-Flores, H.; Rojas-Medar, M.: Embedding of level-continuous fuzzy sets on Banach spaces, Inform. sci. 144, 227-247 (2002) · Zbl 1034.46079 · doi:10.1016/S0020-0255(02)00182-2
[26]Seikkala, S.: On the fuzzy initial value problem, Fuzzy sets and systems 24, 319-330 (1987) · Zbl 0643.34005 · doi:10.1016/0165-0114(87)90030-3
[27]Song, S.; Wu, C.: Existence and uniqueness of solutions to Cauchy problem of fuzzy differential equations, Fuzzy sets and systems 110, 55-67 (2000) · Zbl 0946.34054 · doi:10.1016/S0165-0114(97)00399-0
[28]Song, S.; Wu, C.; Xue, X.: Existence and uniqueness of Cauchy problem for fuzzy differential equations under dissipative conditions, Comput. math. Appl. 51, 1483-1492 (2006) · Zbl 1157.34002 · doi:10.1016/j.camwa.2005.12.001
[29]Stefanini, L.; Sorini, L.; Guerra, M. L.: Parametric representation of fuzzy numbers and application to fuzzy calculus, Fuzzy sets and systems 157, 2423-2455 (2006) · Zbl 1109.26024 · doi:10.1016/j.fss.2006.02.002
[30]Xu, J.; Liao, Z.; Hu, Z.: A class of linear differential dynamical systems with fuzzy initial condition, Fuzzy sets and systems 158, 2339-2358 (2007) · Zbl 1128.37015 · doi:10.1016/j.fss.2007.04.016
[31]Zadeh, L. A.: Fuzzy sets, Inform. control 8, 338-353 (1965) · Zbl 0139.24606 · doi:10.1016/S0019-9958(65)90241-X