Fuzzy differential invariant (FDI) M. Nadjafikhah * , R. Bakhshandeh-Chamazkoti School of Mathematics, Iran University of Science and Technology, Narmak, 16 Tehran, Iran article info Article history: Accepted 13 March 2009 abstract In this paper, we have tried to apply the concepts of fuzzy set to Lie groups and fuzzy dif- ferential invariant (FDI) in order to provide suitable conditions for applying Lie symmetry method in solving fuzzy differential equations (FDEs). For this, we define a C 1 -fuzzy sub- manifold and fuzzy immersion with some examples. In main section, we defined the fuzzy Lie group and some its relative concepts such as fuzzy transformation group and fuzzy G- invariant. The goal of this paper is to introduce and study new defining for fuzzy Lie group and fuzzy differential invariant (FDI). Also, some illustrative examples are given. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction The fuzzy differential equation (FDE) was first introduced by Kandel and Byatt [13] and then was formulated by Kaleva [12]. Its topics have been rapidly growing in recent years [4,14,16]. The theory of fuzzy differential equations has attracted much attention in recent times because this theory represents a natural way to model dynamical systems under uncertainty [21]. The fuzzy differential equation is a very important topic from the theoretical point of view [8,12,17] as well as the ap- plied point of view [1,4,14,16]; for example, in population models [8], civil engineering [17], and in hydraulics modeling [2]. Fuzzy differential equations were considered by many papers. Numerical techniques were developed in [11,13] and others. Pederson and Sambandham [19,20] study the Euler and Runge–Kutta numerical methods, respectively for hybrid fuzzy dif- ferential equations. In some sense, Pederson and Sambandham [12,13] ‘‘rewrite the whole literature on numerical solutions of ODEs” in the hybrid fuzzy setting, focusing on the Euler and Runge–Kutta methods, respectively. Lie symmetry method has long been used to study differential equations. It has been developed into a powerful tool to solve differential equations, to classify them and to establish properties of their solution space. These aspects of Lie group theory have been described in many books and papers [10,18]. Now we want to applicable Lie symmetry method as a ana- lytical method in solving fuzzy differential equations (FDEs). For this, we require to define new fuzzy concepts such as fuzzy Lie group, fuzzy transformation group, fuzzy differential invariant (FDI) and other their relative concepts. The notion of a fuzzy Lie group is depend on the basic concepts in fuzzy topology [3,5,7], C 1 -fuzzy manifold and fuzzy differentiable function between two C 1 -fuzzy manifolds [9]. 2. Preliminaries A fuzzy topology s on a group G is said to be compatible if the mappings m : ðG  G; s  sÞ!ðG; sÞ ðx; yÞ#xy ð1Þ i : ðG; sÞ!ðG; sÞ x#x 1 ð2Þ are fuzzy continuous [3,15]. A group G equipped with a compatible fuzzy topology s on G is called a fuzzy topological group [6]. 0960-0779/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2009.03.070 * Corresponding author. Tel.: +9821 73913426; fax: +9821 77240472. E-mail addresses: m_nadjafikhah@iust.ac.ir (M. Nadjafikhah), r_bakhshandeh@iust.ac.ir (R. Bakhshandeh-Chamazkoti). Chaos, Solitons and Fractals 42 (2009) 1677–1683 Contents lists available at ScienceDirect Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos