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 \(\alpha \)-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.


37C10 Dynamics induced by flows and semiflows
34A30 Linear ordinary differential equations and systems
03E72 Theory of fuzzy sets, etc.
Full Text: DOI


[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
[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
[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
[4] Diamond, P.; Kloeden, P.E., Metric spaces of fuzzy set: theory and applications, (1994), World Scientific Singapore
[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
[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
[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
[8] Bhaskar, T.G.; Lakshmikantham, V.; Devi, V., Revisiting fuzzy differential equations, Nonlinear anal., 58, 351-358, (2004) · Zbl 1095.34511
[9] Goetschel, R.; Voxman, W., Elementary fuzzy calculus, Fuzzy sets and systems, 18, 31-43, (1986) · Zbl 0626.26014
[10] Hirsch, M.W.; Smale, S., Differential equation, dynamical systems and linear algebra, (1974), Academic Press New York · Zbl 0309.34001
[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
[12] Hong, L.; Sun, J., Bifurcations of fuzzy nonlinear dynamical systems, Commun. nonlinear sci. numer. simul., 11, 1-12, (2006) · Zbl 1078.37049
[13] Kaleva, O., Fuzzy differential equations, Fuzzy sets and systems, 24, 301-317, (1987) · Zbl 0646.34019
[14] Kaleva, O., A note on fuzzy differential equations, Nonlinear anal., 64, 895-900, (2006) · Zbl 1100.34500
[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), Ellis Horwood Limited New York · Zbl 0574.46001
[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
[18] Nieto, J.J.; Rodriguez-Lopez, R., Bounded solutions for fuzzy differential and integral equations, Chaos solitons fractals, 27, 1376-1386, (2006) · Zbl 1330.34039
[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
[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
[22] Pearson, D.W., A property of linear fuzzy differential equations, Appl. math. lett., 10, 3, 99-103, (1997) · Zbl 0882.34014
[23] Perko, L., Differential equations and dynamical systems, (1991), Springer-Verlag New York · Zbl 0717.34001
[24] Rodriguez-Lopez, R., Periodic boundary value problems for impulsive fuzzy differential equations, Fuzzy sets and systems, 159, 1384-1409, (2008) · Zbl 1225.34008
[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
[26] Seikkala, S., On the fuzzy initial value problem, Fuzzy sets and systems, 24, 319-330, (1987) · Zbl 0643.34005
[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
[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
[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
[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
[31] Zadeh, L.A., Fuzzy sets, Inform. control, 8, 338-353, (1965) · Zbl 0139.24606
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.