a class of linear differential dynamical systems with fuzzy initial

Fuzzy Sets and Systems 158 (2007) 2339–2358www.elsevier.com/locate/fss

A class of linear differential dynamical systems withfuzzy initial condition�

Jiuping Xua,∗, Zhigao Liaoa,b, Zhineng Hua

aUncertainty Decision-Making Laboratory, School of Business and Administration, Sichuan University, Chengdu 610065, PR ChinabManagement Department, Guangxi Technology University, Liuzhou 545006, PR China

Received 17 January 2007; received in revised form 13 April 2007; accepted 20 April 2007Available online 22 April 2007

Abstract

This paper investigates linear first-order fuzzy differential dynamical systems where the initial condition is described by a fuzzynumber. We use a complex number representation of the �-level sets of the fuzzy system and prove theorems that provide thesolutions under such representation, which is applicable to practical computations and also has some implications for theory. Thenthe paper shows some properties of the two-dimensional dynamical systems, and their phase portraits are described by means ofexamples. There may be a significant difference between the solutions according to whether the matrix is nonnegative or not; finally,the paper points out future research on the fuzzy dynamical systems.© 2007 Elsevier B.V. All rights reserved.

Keywords: Fuzzy differential equation; Fuzzy dynamical systems; Initial value of fuzzy number

1. Introduction

Since the concept of fuzzy set and the corresponding fuzzy set-theoretic operations was introduced by Zadeh [46],enormous research works have been dedicated on development of various aspects of the theory and applicationsof fuzzy sets. Moreover, in view of the development of the calculus for fuzzy functions, the investigation of fuzzydifferential equations (FDE) has been initiated [23,40] and has naturally formed one of the most important areas infuzzy mathematics. For further investigation, the fuzzy dynamical systems based on FDE are also widely applied tofuzzy control systems [9], bifurcations of fuzzy nonlinear dynamical systems [20], artificial system [37] and manyother fields.

However, linear first-order FDE are one of the simplest FDE with great applicability. Although the form of such anequation is very simple, it raises many problems since under different FDE concepts, the behavior of the solutions isdifferent [7,31]. Among the investigations in this area, the existence and uniqueness of the solution of the problem ofa FDE with fuzzy initial value are mainly studied. In [23,24], the existence and uniqueness of solutions of fuzzy initialvalue problem were considered under a Lipschitz condition or other conditions considered in [42,43,45]. O’Regana

� This work was supported by the National Science Foundation for Distinguished Young Scholars, PR China (Grant no. 70425005) and the Teachingand Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE of PR China (Grant no. 20023834-3).

∗ Corresponding author. Tel.: +86 28 85418522; fax:+86 28 85400222.E-mail address: [email protected] (J. Xu).

0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.fss.2007.04.016

http://www.elsevier.com/locate/fss

mailto:[email protected]

2340 J. Xu et al. / Fuzzy Sets and Systems 158 (2007) 2339–2358

et al. [32] also present a new existence result of Wintner-type for the fuzzy initial value problem and a new existenceresult of superlinear type for the fuzzy boundary value problem, in which the variation t is confined in fixed interval,which is also needed in the research of initial value problems for higher-order FDE discussed by Georgiou et al. [16].But in the traditional theory of fuzzy using the Hukuhara derivative [23,25,29,40], significant problems arise withany attempted development of formulas, mainly because the solutions of FDE have quite different properties fromthose of crisp differential equations, lacking observed properties of physical systems such as stability, periodicity andbifurcation [12]. Hüllermeier [21] largely overcame this undesirable property by interpreting a FDE x′(t) = G(t, x(t))

as a family of differential inclusions

x′�(t) ∈ G�(t, x�(t)), 0��1,

and the approach has been further exploited in [13,14,38,39]. Note that Seikkala and Vorobiev [41] study a similarmethodology. A solution to the �th inclusion is a function x�(·) which is absolutely continuous and satisfies the �thinclusion almost everywhere [11]. Under fairly natural conditions, the solution sets and attainability sets of the family arefuzzy sets and moreover, they have properties similar to solutions of ordinary differential equations. Consequently, thisinterpretation is much better adapted to modelling under uncertainty than the other formalism. The main shortcomingof using differential inclusions is that we do not have a derivative of a fuzzy-number-valued function. Another approachcan be found in [6,7] and it uses the extension principle in order to extend crisp differential equations to the fuzzycase. This approach suffers from the same disadvantage as the approach based on differential inclusions which is theconcept of derivative does not exist. Bede et al. [5], Chalco-Cano and Roman-Flores [8] solved the above mentionedshortcoming under strongly generalized differentiability. But the disadvantage of this approach seems to be that a FDEhas not got a unique solution (two solutions locally).

Many numerical methods for FDE are also considered in [1,4,22], and for fuzzy differential inclusion recently in[2,3]. For example, Abbasbandy et al. [3] have devised a numerical method for computing approximation of the set ofall solutions to a fuzzy differential inclusion, which is named “Tuning of reachable set”.

In this paper we use a complex number representation of the �-level sets of the fuzzy system following Pearson[33], who investigates a property of linear differential equations where the initial state is described by a vector offuzzy numbers. The property is related directly to the matrix defining the original nonfuzzy system by passing to acomplex number representation of the �-level sets of the fuzzy system which may avoid shortcomings mentioned above.Moreover, such approach could be useful in concrete computations and in the future theoretical development of thetopic. Then we present the solutions of linear differential dynamical systems with the initial conditions of fuzzy numberand describe the properties of the eigenvalue–eigenvector problems with nonnegative matrix, which is directly relatedto the matrix defining and the solution of the FDE. Especially, we focus on discussing properties of the two-dimensionaldynamical systems and describe their phase portraits with examples.

This paper is organized as follows. Section 2 provides preliminaries and the formulation of the problem. In Sec-tion 3, the solutions of the linear differential dynamical systems are gained and proofed. The properties of the non-negative matrix are also discussed in this section. The properties of two-dimensional dynamical systems and theirphase portraits are described in Section 4 with some examples. In Section 5, some implications for future research areproposed.

2. Preliminaries

We now recall some definitions needed through the paper.

Definition 2.1. By R we denote the set of all real numbers. A fuzzy number is a mapping � : R → [0, 1] with thefollowing properties:

(a) � is upper semi-continuous.(b) � is fuzzy convex, i.e., �(�x + (1 − �)y)� min{�(x), �(y)} for all x, y ∈ R, � ∈ [0, 1].(c) � is normal, i.e., ∃ x0 ∈ R for which �(x0) = 1.(d) supp � = {x ∈ R | �(x) > 0} is the support of the �, and its closure cl(supp u) is compact.

J. Xu et al. / Fuzzy Sets and Systems 158 (2007) 2339–2358 2341

Let F(R) be the set of all fuzzy numbers on R. The �-level set of a fuzzy number � ∈ F(R), 0��1, denoted by(�)�, is defined as

(�)� ={ {x ∈ R|�(x)��} if 0 < ��1,

cl(supp �) if � = 0.

It is clear that the �-level set of a fuzzy number is a closed and bounded interval [a(�), b(�)], where a(�) denotesthe left-hand endpoint of (�)� and b(�) denotes the right-hand endpoint of (�)�. Since each y ∈ R can be regarded asa fuzzy number y defined by

y(t) ={

1 if t = y,

0 if t �= y.

R can be embedded in F(R).

From this characteristic of fuzzy numbers, we see that a fuzzy number is determined by the endpoints of the intervals(�)�. Thus a fuzzy number � can be identified by a parameterized triple

{(a(�), b(�), �)|0��1}.This leads to the following characteristic representation of a fuzzy number in terms of the two “endpoint” functions

a(�) and b(�).Arithmetic fuzzy addition and scalar multiplication can be defined as

(u + v)(x) = supy∈R

min(u(y), v(x − y)), x ∈ R

and

(ku)(x) ={

u(x/k),

0,

k > 0,

k = 0,

where 0 ∈ F(R). To deal with subtraction, Dubois and Prade [15] define the “opposite” of a fuzzy number � to be thefuzzy number � satisfying �(x) = �(−x).

In other words, if � is represented by the parametric form {(a(�), b(�), �)|0��1}, then � is represented by thecorresponding form {(−b(�), −a(�), �)|0��1}. In this paper we use the vector form to represent the “opposite” of�, given by Goetschel and Voxman [18]

{(−a(�), −b(�), �)|0��1}.That is, −�(x) := (−1)�(x).In this way, from a family of parametric representations of F(R), we can define the corresponding parametric

representations of their “opposites” from subsets of the vector spaceV = {(a(�), b(�), �)|0��1, a : I → R and b : I → R are bounded functions}.We then define a metric D on V by

D(�, �) = sup{max{|a(�) − c(�)| , |b(�) − e(�)|}|0��1},where � = {(a(�), b(�), �)|0��1} and � = {(c(�), e(�), �)|0��1} are members of V.

It is clear that the vector spaceV together with the metric D forms a topological vector space. Let V = {{(a(�), b(�), �)|0��1} ∈ V |a(�) and b(�) are Lebesgue integrable}, then (V , D) is a complete metric space [36].

3. Linear differential dynamical systems

We consider the problem of solving the linear differential dynamical systems{x = Ax,

x(0) = x0,(1)


where x ∈ Rn, A is an n × n matrix,

x = dx

dt=

[dx1

dt,

dx2

dt, . . . ,

dxn

dt

]T

,

and the initial condition x0 is described by a vector made up of n fuzzy numbers.

3.1. The fundamental theorem for linear systems

Lemma 3.1 (Goetschel and Voxman [18]). Assume that I = [0, 1], and a : I → R and b : I → R satisfy theconditions:

(a) a : I → R is a bounded increasing function.(b) b : I → R is a bounded decreasing function.(c) a(1)�b(1).(d) For 0 < k�1, lim�→k− a(�) = a(k) and lim�→k− b(�) = b(k).(e) lim�→0+ a(�) = a(0) and lim�→0+ b(�) = b(0).

Then � : R → I defined by

�(x) = sup{�|a(�)�x�b(�)}is a fuzzy number with parameterization given by {(a(�), b(�), �)|0��1}. Moreover, if � : R → I is a fuzzy numberwith parameterization given by

{(a(�), b(�), �)|0��1},then functions a(�) and b(�) satisfy conditions (a)–(e).

There are several approaches to define a solution for a FDE, such as Hukuhara approach [23,25,29], differentialinclusions [21,39], quasiflows and differential equations in metric spaces [26,27], and other approaches [5,8]. Wechoose to solve (1) levelwise and systems (1) can be written as{

x�(t) = Ax�(t),

x�(0) = x�0, 0��1.(2)

Lemma 3.2 (Seikkala [40]). Assume that each element of the vector x in (1) at the time instant t is a fuzzy numberwhere

xk�(t) = [xk

�(t), xk�(t)], k = 1, . . . , n. (3)

Then, the evolution of the systems (1) can be described by 2n differential equations for the endpoints of the intervals(3), this for each given time t and value of �. The equations for the endpoints of the intervals are⎧⎪⎪⎨

⎪⎪⎩xk

�(t) = min{(Au)k : ui ∈ [xi�(t), x

i�(t)]},

xk�(t) = max{(Au)k : ui ∈ [xi

�(t), xi�(t)]},

x�(0) = x�0,

x�(0) = x�0,

. (4)

where (Au)k := ∑nj=1 akju

j is the kth row of Au. Since the vector field in (1) is linear and so the following ruleapplies in (4):

xk�(t) = akju

j , (5)

where{uj = x

j�(t), akj �0,

uj = xj�(t), akj < 0


and

xk�(t) = akju

j , (6)

where{uj = x

j�(t), akj �0,

uj = xj�(t), akj < 0.

Then systems (2) can be written as{x�(t) = U = AU, U ∈ [x�(t), x�(t)],x�(0) = [x�0, x�0], 0��1.

(7)

As indicated in [33], the same thing can be represented in a more compact way by moving to the field of complexnumbers. Define new complex variables as follows:

xk� := xk

�(t) + ixk�(t) (k = 1, . . . , n), (8)

where i := √−1. Then we have a theorem as follows:

Theorem 1. Let A be an n × n matrix. Then for a given x0 ∈ F(R), the initial value problem{x = Ax,

x(0) = x0(9)

has a unique solution given by{x� + ix� = B(x�(t) + ix�(t)),

x�(0) + ix�(0) = x�0 + ix�0,(10)

where the elements of the matrix B are determined from those of A as follows:

bij ={

eaij , akj �0,

gaij , akj < 0,

in which e is just the identity operation and g corresponds to a flip about the diagonal in the complex plane, i.e.,∀a + bi ∈ C,

e : a + bi → a + bi,g : a + bi → b + ai.

(11)

Proof. It is easily verified that g2 = e and (�g)(a + bi) = (g�)(a + bi) for � ∈ R and we extend g to vectors via

g(x�(t) + ix�(t)) = [g(x1�(t) + ix1

�(t)), . . . , g(xn�(t) + ixn

�(t))]T.

Using (8) and the two operators (11), systems (7) can be written as systems (10). The solution to Eq. (10) is given by

x�(t) + ix�(t) = exp(tB)(x�0 + ix�0). (12)

Now we will validate Eq. (12) is the unique solution of systems (10). Since B commutes with itself, if x�(t)+ ix�(t) =exp(tB)(x�0 + ix�0), then

x�(t) + ix�(t) = d

dtexp(tB)(x�0 + ix�0)

= limh→0

eB(t+h) − eBt

h(x�0 + ix�0)


= limh→0

eBt eBt − I

h(x�0 + ix�0)

= limh→0

limk→∞ eBt

(B + B2h

2! + · · · + Bkhk−1

k!)

(x�0 + ix�0)

= BeBt (x�0 + ix�0) = B(x�(t) + ix�(t)).

Thus exp(tB)(x�0 + ix�0) is a solution. To see that this is the only solution, let x�(t) + ix�(t) be any solution of theinitial value problem (10) and

y�(t) + iy�(t) = exp(−tB)(x�0 + ix�0),

then

y�(t) + iy�(t) = −B exp(−tB)(x� + ix�) + exp(−tB)(x�(t) + ix�(t))

= −B exp(−tB)(x� + ix�) + exp(−tB)B(x� + ix�) = 0,

for all t ∈ R since e−Bt and B commute. Thus, y(t) is a constant. Setting t = 0 shows that y�(t)+ iy�(t) = (x�0 + ix�0)

and therefore any solution of the initial value problem (10) is given by

x�(t) + ix�(t) = exp(tB)(y�

+ iy�) = exp(tB)(x�0 + ix�0).

This completes the proof. �

3.2. The characters of nonnegative matrix A

Let A be a nonnegative matrix, then B = A according to formula (11), the initial value problem (9) can berewritten as{

x�(t) = Ax�(t),

x�(0) = x�0,(13)

and {x�(t) = Ax�(t),

x�(0) = x�0.(14)

Definition 3.1. Let A be a real n × n matrix. Then for t ∈ R,

eAt =∞∑

k=0

Aktk

k! .

Lemma 3.3 (Coddington and Levinson [10], the Jordan Canonical form). Let A be a real matrix with real �j , j =1, . . . , k and complex eigenvalues �j = aj + ibj and �j = aj − ibj , j = k + 1, . . . , n. Then there exists abasis {v1, v2, . . . , vk, vk+1, uk+1, . . . , vn, un} for R2n−k , where vj , j = 1, . . . , k and wj , j = k + 1, . . . , n aregeneralized eigenvectors of A, uj = Re(wj ) and vj = Im(wj ) for j = k + 1, . . . , n, such that the matrix P =[v1, . . . , vk, vk+1, uk+1, . . . , vn, un] is invertible and

P −1AP = diag[B1, . . . , Br ], (15)

where the elementary Jordan blocks B = Bj , j = 1, . . . , r are either of the form

B =

⎡⎢⎢⎢⎢⎣

� 1 0 · · · 00 � 1 · · · 0

· · ·0 · · · � 10 · · · �

⎤⎥⎥⎥⎥⎦ (16)


for � one of the real eigenvalue of A or of the form

B =

⎡⎢⎢⎢⎢⎣

D I2 0 · · · 00 D I2 · · · 0

· · ·0 · · · D I20 · · · D

⎤⎥⎥⎥⎥⎦ (17)

with

D =[

a −b

b a

], I2 =

[1 00 1

], 0 =

[0 00 0

]

for � = a + ib one of the complex eigenvalues of A.

Lemma 3.4 (Hirsch and Smale [19]). Let A be a real n × n matrix with real eigenvalue�j , j = 1, . . . , n repeatedaccording to their multiplicity. Then there exist a basis of generalized eigenvectors for Rn. And if {v1, . . . , vn} is anybasis of generalized eigenvectors for Rn, the matrix P = [v1, . . . , vn] is invertible,

A = S + N,

where

P −1SP = diag[�j ],the matrix N = A − S is nilpotent of order k�n, and s and N commute, i.e., SN = NS.

The Jordan canonical form of A yields some explicit information about the form of the solution of the initial valueproblem (13) or (14) which, according to Theorem 1, is given by

x�(t) = P diag[eBj t ] P −1x�0 (18)

or

x�(t) = P diag[eBj t ] P −1x�0. (19)

3.3. Nonhomogeneous linear dynamical systems

In this section we consider the nonhomogeneous linear dynamical systems{x = Ax + f (t),

x(0) = x0,(20)

where A is an n × n matrix and f (t) is a continuous vector function, the elements of f (t) are fuzzy numbers.Similarly, systems (20) can be rewritten as{

x�(t) + ix�(t) = B(x�(t) + ix�(t)) + (f�(t) + if �(t)),

x�(0) + ix�(0) = x�0 + ix�0,(21)

where the elements of the matrix B are determined from those of A as follows:

bij ={

eaij , akj �0,

gaij , akj < 0,

in which e is just the identity operation and g corresponds to a flip about the diagonal in the complex plane, i.e.,∀a + bi ∈ C,

e : a + bi → a + bi,

g : a + bi → b + ai. (22)


Recall the classical variation of constants formula for a nonhomogeneous n-dimensional system of first-order dif-ferential equations

x = Ax + f (t), x(0) = x0.

The solution may be written as

x(t) = �1(t)x0 +∫ t

0�1(t)�1(s)

−1f (s) ds, (23)

where �1(t, s) = �1(t)�1(s)−1 and �1(t) satisfies the matrix

�1(t) = A�1(t), �1(0) = I. (24)

Combining Eq. (12) with systems (24), from the classical formula (23), we have a theorem as follows:

Theorem 2. If �2(t) is any fundamental matrix solution of (10), then the solution of the nonhomogeneous linearsystems (21) is given by

x�(t) + ix�(t) = �2(t)(x�0 + ix�0) +∫ t

0�2(t − s)(f

�(s) + if �(s)) ds, (25)

where �2(t, s) = �2(t)�2(s)−1 and �2(t) satisfies the matrix

�2(t) = B�2(t), �2(0) = I. (26)

Proof. For the function x�(t) + ix�(t) defined above, and since �2(t) is a fundamental matrix solution of (10), itfollows that

x�(t) + ix�(t) = �2(t)(x�0 + ix�0) + �2(t)

∫ t

0�−1

2 (s)(f�(s) + if �(s)) ds + �2(t)�

−12 (t)(f

�(t) + if �(t))

= B�2(t)(x�0 + ix�0) + B�2(t)

∫ t

0�−1

2 (s)(f�(s) + if �(s)) ds + (f

�(t) + if �(t))

= B(�2(t)(x�0 + ix�0) + B

∫ t

0�2(t − s)(f

�(s) + if �(s)) ds) + (f

�(t) + if �(t))

= B(x�(t) + ix�(t)) + (f�(t) + if �(t))

This completes the proof. �

4. Homogeneous two-dimensional system of first-order differential equations

In this section we discuss various phase portraits that are possible for the linear systems (1) when x ∈ R2and A is a2 × 2 matrix. We begin by describing the phase portraits for the linear system{

x = Cx,

x(0) = x0,(27)

where the matrix C = P −1AP has one of the forms given as follows:

C =[

� 00 �

], C =

[� 10 �

], C =

[a −b

b a

].

Then, the phase portrait for the linear systems{x = Ax,

x(0) = x0(28)

is then obtained from the phase portrait for (27) under the linear transformation of coordinates x = Py, in which P isinvertible.


Similarly, systems (1) can be written as{y = Cy,

y(0) = y0(29)

under the linear transformation of coordinates x0 = Qy0 where the elements of the matrix Q are determined from thoseof A as follows:

qij ={

epij , pkj �0,

gpij , pkj < 0.

The solution of systems (29) is given by

x�(t) + ix�(t) = exp(tC)(x�0 + ix�0) (30)

according to Theorem 1. Especially, if A is nonnegative, then we have{x�(t) = Cx�(t),

x�(0) = x�0(31)

and {x�(t) = Cx�(t),

x�(0) = x�0.(32)

4.1. Nonnegative matrix A

Lemma 4.1 (Perko [34]). Let � = det A and = trace A and consider the linear system

x = Ax. (33)

(a) If � < 0 then (33) has a saddle point at the origin.(b) If � > 0 and 2 − 4��0 then (33) has a node at the origin; it is stable if < 0 and unstable if > 0.(c) If � > 0, 2 − 4� < 0, and �= 0 then (33) has a focus at the origin; it is stable if < 0 and unstable if > 0.(d) If � > 0, = 0 then (33) has a center at the origin.

Lemma 4.2 (Perko [34]). Each coordinate in the solution x(t) of the initial value problem (13) or (14) is a linearcombination of functions of the form

tkeat cos(bt) or tkeat sin(bt)

where � = a + ib is an eigenvalue of the matrix A and 0�k�n − 1.

Let the matrix A be nonnegative, then the linear systems (1) have the following possible properties.

4.1.1. SaddleProperty 4.1. If C =

[�0

0�

]with � < 0 < �, there are two eigenvalues of opposite sign and � = �� < 0. According

to Lemma 4.1, the systems (29) have a saddle at the origin.

The portrait for the linear systems (29) in this case is given in Fig. 1. If � < 0 < �, the arrows in Fig. 1 are reversed.Whenever A has two real eigenvalues of opposite sign, the phase portrait for the linear systems (1) is linearly equivalentto the phase portrait shown in Fig. 1; i.e., it is obtained from Fig. 1 by a linear transformation of coordinates; and thestable and unstable subspaces of (1) are determined by the eigenvectors of A. The shadow part is the possible footprintof the corresponding trajectories.


Example 4.1. Let

C =[

2 00 −2

]

and defining the initial value to be x1(0) = about 1 and x2(0) = about 16 which can be done by setting, for example,

x1(0) =⎧⎨⎩

0, s < 0,

−(x − 1)2 + 1, 0�s�2,

0, s > 2,

and

x2(0) =⎧⎨⎩

0, s < 15,

−(x − 16)2 + 1, 15�s�17,

0, s > 17.

Thus

[x1(0)]� = [1 − √1 − �, 1 + √

1 − �],[x2(0)]� = [16 − √

1 − �, 16 + √1 − �],

x� + ix� =[

e2t 00 e−2t

]x�(0).

Then the phase portrait for the example is given in the first quadrant of Fig. 1 with � = 0.8 and similarly phaseportrait of the other three quadrants can be gained if we define the initial values to be about (1, −16), (−1, 16), or(−1, −16).

4.1.2. NodeProperty 4.2. If C =

[�0

0�

]with �� < 0, then � = �� > 0 and 2 − 4� = (� − �)2 �0; or C =

[�0

1�

]with

� < 0, then � = �2 > 0 and 2 − 4� = 0. According to Lemma 4.1, the systems (29) have a node at the origin.

The phase portrait for the linear system is given in Figs. 2–4, respectively. It is a proper node in the first case with� = � and an improper node in the other two cases.

Example 4.2. Let

C =[ −3 1

0 −3

]

Fig. 1. A saddle at the origin.


Fig. 2. A stable node (� = � < 0).

Fig. 3. A stable node (� < � < 0).

and defining the initial value to be x1(0) =about 1 and x2(0) =about 8 which can be done by setting, for example,

x1(0) =⎧⎨⎩

0, s < 0,

−(x − 1)2 + 1, 0�s�2,

0, s > 2,

and

x2(0) =⎧⎨⎩

0, s < 7,

−(x − 8)2 + 1, 7�s�9,

0, s > 9.


Fig. 4. A stable node (� < 0).

Thus

[x1(0)]� = [1 − √1 − �, 1 + √

1 − �],[x2(0)]� = [8 − √

1 − �, 8 + √1 − �],

(x� + ix�) = e−3t

[1 t

0 1

]x�(0).

Then the phase portrait for the example is given in the first quadrant of Fig. 4 with � = 0.5 and similarly phaseportrait of the 3rd quadrant can be gained if we define the initial values to be about (−1, −8).

4.1.3. FocusProperty 4.3. If C =

[ab

−ba

]with a < 0, then = 2a, � = a2 + b2 > 0, and 2 − 4� = −4b2 < 0. According to

Lemma 4.1, the origin is a stable focus.

The phase portrait for the linear systems (29) in the case is given in Figs. 5 and 6, respectively. If a > 0, the arrowsare reversed in Figs. 5 and 6, respectively; i.e., the trajectories spiral away from the origin with increasing t. The originis called an unstable focus in this case. Whenever A has a pair of complex conjugate eigenvalues with nonzero realpart, the phase portraits for systems (1) are linearly equivalent to the phase portraits shown in Figs. 5 or 6. Note thatthe trajectories in Figs. 5 or 6 do not approach the origin along well-defined tangent lines; i.e., the angle (t) that thevector x(t) makes with the x1-axis does not approach a constant 0 as t → ∞, but rather |x(t)| → 0 as t → ∞ in thiscase.

Example 4.3. Let

C =[ −2 −2

2 −2

]


x1(0) =⎧⎨⎩

0, s < 1,

−(x − 2)2 + 1, 1�s�3,

0, s > 3,


Fig. 5. A stable focus at the origin (b > 0).

Fig. 6. A stable focus at the origin (b < 0).

and

x2(0) =⎧⎨⎩

0, s < 1,

−(x − 2)2 + 1, 1�s�3,

0, s > 3.

Thus

[x1(0)]� = [2 − √1 − �, 2 + √

1 − �],[x2(0)]� = [2 − √

1 − �, 2 + √1 − �],

(x� + ix�) = e−2t

[cos 2t − sin 2t

sin 2t cos 2t

]x�(0).

Then the phase portrait for the example is given in Fig. 5 with � = 0.5 and similarly the phase portrait of caseb = −2 is given in Fig. 6 if we define the initial values to be about (2, −2).


Fig. 7. A center at the origin (b < 0).

Fig. 8. A center at the origin (b > 0).

4.1.4. CenterProperty 4.4. If C =

[0b

−b0

], then = 0, � = b2 > 0. Systems (29) have a center at the origin according to

Lemma 4.1.

The phase portrait for the linear systems (29) in this case is given in Figs. 7 and 8, respectively. Whenever A has apair of pure imaginary complex conjugate eigenvalues, the phase portrait of the linear systems (1) is linearly equivalentto one of the phase portraits shown in Figs. 7 or 8. Note that the trajectories or solution curves lie on circles |x(t)| =constant. The trajectories of the systems (1) will lie on ellipses and the solution x(t) of (1) will satisfy m� |x(t)|�M

for all t ∈ R.

Example 4.4. Let

C =[

0 −22 0

]


Fig. 9. The phase portrait for Case 4.1.


x1(0) =⎧⎨⎩

0, s < 1,

−(x − 2)2 + 1, 1�s�3,

0, s > 3,

and

x2(0) =⎧⎨⎩

0, s < 1,

−(x − 2)2 + 1, 1�s�3,

0, s > 3.

Thus

[x1(0)]� = [2 − √1 − �, 2 + √

1 − �],[x2(0)]� = [2 − √

1 − �, 2 + √1 − �],

(x� + ix�) =[

cos 2t − sin 2t

sin 2t cos 2t

]x�(0).

Then the phase portrait for the example is given in Fig. 7 with � = 0.5 and similarly the phase portrait of caseb = −2 is given in Fig. 8 if we define the initial values to be about (2, −2).

Here, what we should pay attention to is that if the fuzzy number could not keep its sign, positive or negative, therewill have distinct difference between the phase portraits. For example, if the initial value (x10, x20) is about to (0, 0),which means that the left-hand endpoint is negative number, and the right-hand endpoint is positive number, the traceof the phase portrait of the dynamical systems with fuzzy initial number will disobey the rules shown above. If suchconditions cannot be avoided, we could resolve it in two approaches. One is moving the coordinates to right place andthe fuzzy number can keep its sign; the other is controlling the definition of symmetric triangular fuzzy number withthe interval [x�o, x�o] to make the fuzzy number keep its sign.

4.2. General matrix A

Let matrix A be without any restriction, then the linear systems (1) have the following possible properties which isdistinct different from the properties shown in the subsection above.

Case 4.1: If C =[

�0

0�

]with � < 0 < �, the phase portrait is given in the first quadrant of Fig. 9 as t → ∞.

Example 4.5. Let

C =[

2 00 −1

]


Fig. 10. The phase portrait for Case 4.2 (� = � < 0).


x1(0) =⎧⎨⎩

0, s < 0,

−(x − 1)2 + 1, 0�s�2,

0, s > 2.

and

x2(0) =⎧⎨⎩

0, s < 15,

−(x − 16)2 + 1, 15�s�17,

0, s > 17.

Thus

[x1(0)]� = [1 − √1 − �, 1 + √

1 − �],[x2(0)]� = [16 − √

1 − �, 16 + √1 − �].

From Theorem 1, we have the following expression:(x1 + i ˙x1

x2 + i ˙x2

)=

[2 00 −1

] (x1 + ix1x2 + ix2

).

Thus the solution for it is governed by

x1� + ix1� = e2t (1 − √1 − � + i(1 + √

1 − �)),

x2� = et + e−t

2(16 − √

1 − �) + e−t − et

2(16 + √

1 − �),

x2� = e−t − et

2(16 − √

1 − �) + et + e−t

2(16 + √

1 − �).

Then the phase portrait for the example is given in the first quadrant of Fig. 9 with � = 0.8 and similarly phaseportrait of the other three quadrants can be gained if we define the initial values to be about (1, −16), (−1, 16), or(−1, −16).

Case 4.2. If C =[

�0

0�

]with �� < 0, the phase portrait for the linear system is given in Fig. 10 as t → ∞.

Example 4.6. Let

C =[ −2 1

0 −2

]


and defining the initial value to be x1(0) =about 8 and x2(0) = about 8 which can be done by setting, for example,

x1(0) =⎧⎨⎩

0, s < 7,

−(x − 8)2 + 1, 7�s�9,

0, s > 9

and

x2(0) =⎧⎨⎩

0, s < 7,

−(x − 8)2 + 1, 7�s�9,

0, s > 9.

Thus

[x1(0)]� = [8 − √1 − �, 8 + √

1 − �],[x2(0)]� = [8 − √

1 − �, 8 + √1 − �],

x1� = e2t + e−2t

2M + e−2t − e2t

2N + e2t + e−2t

2Mt + e−2t − e2t

2Nt,

x1� = e−2t − e2t

2M + e2t + e−2t

2N + e−2t − e2t

2Mt + e2t + e−2t

2Nt,

x2� = e2t + e−2t

2M + −e2t + e−2t

2N,

x2� = −e2t + e−2t

2M + e2t + e−2t

2N,

where M, N denote 8 − √1 − �, 8 + √

1 − �, respectively.Then the phase portrait for the example is given in the shadow part of Fig. 10 corresponding to t = 1, 2.5, 4 with

� = 0.5 and similarly phase portrait of the initial values to be about (−8, −8), (8, −8), or (−8, 8) can also be shownin Fig. 10.

Case 4.3: If C =[

ab

−ba

]with a < 0, The phase portrait for the linear systems (29) in the case is given in Fig. 11

as t → ∞.

Fig. 11. The phase portrait for Case 4.3 (b > 0).


Example 4.7. Let

C =[ −2 −2

2 −2

]


x1(0) =⎧⎨⎩

0, s < 1,

−(x − 2)2 + 1, 1�s�3,

0, s > 3

and

x2(0) =⎧⎨⎩

0, s < 1,

−(x − 2)2 + 1, 1�s�3,

0, s > 3.

Thus

[x1(0)]� = [2 − √1 − �, 2 + √

1 − �],[x2(0)]� = [2 − √

1 − �, 2 + √1 − �].

⎛⎜⎜⎝

x1�˙x1�x2�˙x2�

⎞⎟⎟⎠ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 + e4t + M

4

−e4t + M − 1

4

e4t − N − 1

4

1 − e4t − N

4−1 − e4t + M

4

e4t + M + 1

4−e4t + N + 1

4

e4t − N − 1

4−1 + e4t + N

4

−e4t + N + 1

4

e4t + M + 1

4−e4t − M + 1

41 − e4t + N

4

e4t + N − 1

4

−e4t + M + 1

4

1 + e4t + M

4

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎝

2 − √1 − �

2 + √1 − �

2 − √1 − �

2 + √1 − �

⎞⎟⎟⎠ ,

where M, N denote 2e−2t cos(2t), 2e−2t sin(2t), respectively.Then the phase portrait for the example is given in Fig. 11 with � = 0.8 which is presented by dash trajectories or

� = 0.2 which is presented by dot trajectories. And the solid line presents the trajectory without fuzziness.

Case 4.4: If C =[

0b

−b0

], the phase portrait for the linear systems (29) in this case is given in Fig. 12 as t → ∞.

Example 4.8. Let

C =[

0 −22 0

]


x1(0) =⎧⎨⎩

0, s < 1,

−(x − 2)2 + 1, 1�s�3,

0, s > 3,

and

x2(0) =⎧⎨⎩

0, s < 1,

−(x − 2)2 + 1, 1�s�3,

0, s > 3.


Fig. 12. The phase portrait for Case 4.4 (b < 0).

Thus

[x1(0)]� = [2 − √1 − �, 2 + √

1 − �],[x2(0)]� = [2 − √

1 − �, 2 + √1 − �].

⎛⎜⎜⎝

x1�˙x1�x2�˙x2�

⎞⎟⎟⎠ =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

cos(2t)

2+ M

cos(2t)

2− M − sin(2t)

2+ N − sin(2t)

2− N

cos(2t)

2− M

cos(2t)

2+ M − sin(2t)

2− N − sin(2t)

2+ N

sin(2t)

2+ N

sin(2t)

2− N

cos(2t)

2+ M

cos(2t)

2− M

sin(2t)

2− N

sin(2t)

2+ N

cos(2t)

2− M

cos(2t)

2+ M

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎛⎜⎜⎝

2 − √1 − �

2 + √1 − �

2 − √1 − �

2 + √1 − �

⎞⎟⎟⎠ ,

where M, N denote (e2t + e−2t )/4, (e2t − e−2t )/4, respectively.Then the phase portrait for the example is given in Fig. 12 responding to t = 1, 5, 10 with � = 0.8 which is presented

by dash trajectories or � = 0.2 which is presented by dot trajectories. And the solid line presents the trajectory withoutfuzziness.

5. Conclusion

Using a new representation of the �-level sets of the fuzzy system, we study the properties of first-order lineardynamical systems. Since this representation is perfectly adapted to the combination of the fuzzy differential equationswith the classical differential equations, the solution of the linear fuzzy equation can be easily inherited from the solutionof the classical differential equations. In this approach, the shortcomings of the previous papers can be avoided. Forfurther research, the homogeneous two-dimensional differential dynamical systems with fuzzy initial value are mainlydiscussed and the phase portraits are also given with the explanation of examples. From the above discussion, we findsignificant difference in the properties of solutions whether the matrix A is nonnegative or not. If the matrix A is notassumed to be nonnegative, the properties of classical differential equations will be lost, such as saddle, node, focusand center. In the future research, every element of matrix A of problem (1) will be supposed to be a fuzzy number. Wewill also investigate the properties of the fuzzy differential dynamical systems under this assumption.

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a class of linear differential dynamical systems with fuzzy initial

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