CC BY
0Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 841987,16 pages doi:10.1155/2012/841987
Research Article
Stability and Hopf Bifurcation in a Computer Virus Model with Multistate Antivirus
Tao Dong,1,2 Xiaofeng Liao,1 and Huaqing Li1
1 State Key Laboratory of Power Transmission Equipment and System Security, College of Computer Science, Chongqing University, Chongqings 400044, China
2 College of Software and Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China
Correspondence should be addressed to Tao Dong, david_312@126.com
Received 9 January 2012; Accepted 6 February 2012
Academic Editor: Muhammad Aslam Noor
Copyright © 2012 Tao Dong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
By considering that people may immunize their computers with countermeasures in susceptible state, exposed state and using anti-virus software may take a period of time, a computer virus model with time delay based on an SEIR model is proposed. We regard time delay as bifurcating parameter to study the dynamical behaviors which include local asymptotical stability and local Hopf bifurcation. By analyzing the associated characteristic equation, Hopf bifurcation occurs when time delay passes through a sequence of critical value. The linerized model and stability of the bifurcating periodic solutions are also derived by applying the normal form theory and the center manifold theorem. Finally, an illustrative example is also given to support the theoretical results.
1. Introduction
As globalization and development of communication networks have made computers more and more present in our daily life, the threat of computer viruses also becomes an increasingly important issue of concern. In 2003, a virus, called worm king, rapidly spread and attacked the global world, which results the network of the internet to be seriously congested and server to be paralyzed [1]. In 2010, the report of pestilence about computer virus in China revealed that more than 90% computers in China are infected computer virus.
Computer viruses are small programs developed to damage the computer systems erasing data, stealing information. Their action throughout a network can be studied by using classical epidemiological models for disease propagation [2-6]. In [7-9], based on SIR classical epidemic model, Mark had proposed the dynamical models for the computer
virus propagation, which provided estimations for temporal evolutions of infected nodes depending on network parameters [10-12]. In [13], Richard and Mark propose a modified propagation model named SEIR (susceptible-exposed-infected-recover) model to simulate virus propagation. In [14], on this basis of the SIR model, Yao et al. proposed a SIDQV model with time delay which add a quarantine state to clean the virus. However, both above models assume the viruses are cleaned in the infective state. In fact, in addition to clean viruses in state I, people may immunize their computers with countermeasures in state S and state E in the real world. Moreover there may be a time lag when the node uses antivirus software to clean the virus.
In this paper, in order to overcome the above-mentioned limitation, we present a new computer virus model with time delay which is depending on the SEIR model [15]; time delay can be considered the period of the node uses antivirus software to clean the virus. This model provides an opportunity for us to study the behaviors of virus propagation in the presence of antivirus countermeasures, which are very important and desirable for understanding of the virus spread patterns, as well as for management and control of the spread. The remainder of this paper is organized as follows. In Section 2, the stability of trivial solutions and the existence of Hopf bifurcation are discussed. In Section 3, a formula for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions will be given by using the normal form and center manifold theorem introduced by Hassard et al. in [16]. In Section 4, numerical simulations aimed at justifying the theoretical analysis will be reported.
2. Mathematical Model Formulation
Our model is based on the traditional SEIR model [7-9, 15, 17]. The SEIR model has four states: susceptible, exposed (infected but not yet infectious), infectious, and recovered. Our assumptions on the dynamical model are as follows.
(1) In the real world, in addition cleaning viruses in state I, people may immunize their computers with countermeasures in state S and state E (after virus being cleaned), which may result in new state transition paths in comparison with SIR model:
S-R: using countermeasure of real-time immunization, E-R: using real-time immunization after virus codes cleaning.
(2) In state S, when people install the antivirus software on their computer, we assume that their computer can be immunized at a unit time.
(3) In state E, since the computer is infected by the virus, the antivirus software may use a period to search the document and clean the viruses.
(4) Denote the period of time of killing viruses when users find that their computers are infected by viruses.
(5) While the computer is installed the antivirus software, it will not be quarantine or replacement. On the basis of the above hypotheses (1)-(5), the dynamical model
can be formulated by the following equations:
^ = uN - pI(t)S(t) - (PSR + n) S{t),
^^ = pI(t)S(t) - (a + ^)E(t) - pEREit - t), dt (2.1)
dIitL = aE(t) - (Y + n)Iit),
= PSRS(t) + pERE(t - t) + jI(t - t) - pR(t),
where pSR describes the impact of implementing real-time immunization, pER describes the impact of cleaning the virus and immunizing the nodes, and ¡1 describes the impact of quarantine or replacement. a is the transition rate from E to I, and j is the recovery rate from I to R. t is the time delay that the node usees antivirus software to clean the virus. ¡5 is the transition rate from S to E.
3. Local Stability of the Equilibrium and Existence of Hopf Bifurcation
We may see that the first three equations in (2.1) are independent of the fourth equation, and therefore, the fourth equation can be omitted without loss of generality. Hence, model (2.1) can be rewritten as
dS(t) = uN - pI(t)S(t) - (psr + ¡i)S(t),
dt dE(t) dt
dI(t) dt
= pI(t)S(t) - (a + ¡i)E(t) - pERE(t - t), (3.1)
= aE(t) - (y + n)I(t).
For the convenience of description, we define the basic reproduction number of the infection
R = fiNpa
0 (Psr + \i)(a + Per + \i)(Y + ' ^2)
Clearly, we have the following results with respect to the stable state of system (3.1). Here, the proof is omitted (see [17] for the details).
Theorem 3.1. If R0 < 1, system (3.1) has only the disease-free equilibrium E0 = (¡N/(pSR+¡¡), 0,0) and is globally asymptotically stable. If R0 > 1, E0 becomes unstable and there exists a unique positive equilibrium Eve, where Eve = (¡N/(pSR + ¡)R0, ¡N(R0 - 1)/R0(a + ¡ + pER), aE*/(j + ¡)). Furthermore, for any t > 0, E0 is asymptotically stable if R0 < 1 and unstable if R0 > 1.
To investigate the qualitative properties of the positive equilibrium E* with t > 0, it is necessary to make the following assumption: (H1) Rq > 1.
Under hypothesis (H1), the Jacobian matrix of the system (3.1) about Eve is given by
J (Eve) =
-a\ 0 -a2
a3 -a4 - a7e~kT a2 a5 -a6_
where ai = ¡51 + pSR + ¡, a2 = ¡5S, a3 = ¡51, a4 = a + ¡, a5 = a, a6 = j + ¡, a7 = pER. We can obtain the following characteristic equation:
I3 + hi2 + fyX + b3 + e-lT(b4X2 + b5X + b^ = 0, (3.4)
b1 = a1 + a4 + a6, b2 = a1a6 + a4(a1 + a6) - a2a5, b3 = a1a4a6 - a1a2a5 + a2a3a5, b4 = a7, b5 = a7(a1 + a6), b6 = a1a6a7.
If iw (w > 0) is a root of (3.4), then
-iw3 - b1w1 + b2iw + b3 + e~lWT(--u)2b4 + b5i^ + b^j = 0. (3.6)
Separating the real and imaginary parts of (3.6), we have
b5w sin wt + (b6 - b4w2^ cos wt = b1w2 - b3, b5w cos wt - (b6 - b4wsin wt = w3 - b2w.
Adding up the squares of (3.7) yields
w6 + (b2 - 2b2 - b2) w4 + (lb2 - 2bb + 2b4b6 - b2) w2 + (b2 - b2) = 0. (3.8)
Letting z = w2, c1 = b2 - 2b2 - b^, c2 = b2 - 2b1b3 + 2b4b6 - b|, c3 = b^ - b^, then (3.8) becomes
z3 + c1z2 + c2 z + c3 = 0. (3.9)
Letting z* = (1/3)(-c1 +yjc2 - 3c2), h(z*) = (z*)3 + c1(z*)2 + c2z* + c3, then we have the following results (see [18-22] for details) about the distributions of the positive roots of (3.9).
Lemma 3.2 (see [18-22]). (i) If c3 < 0, then (3.9) has at least one positive root.
(ii) If c3 > 0 and c2 - 3c2 < 0, then (3.9) has no positive root.
(iii) If c3 > 0 and c2 - 3c2 > 0, then (3.9) has positive roots if and only if z* > 0 and h(z*) < 0.
Suppose (3.9) has positive roots; without loss of generality, we assume that it has three positive roots defined by wk = -Jzk, k = 1,2,3. By (3.7), we have
(b1wk - b3)(b6 - b4wk) + b5Wk (wk - b2) (310) cos(wkT) =-2-. (3.10)
b25w2k + (b6 - b4W2k)
Thus, denoting
Tj = .1arc cos (b1wk - (b6 - b4wk + b5Wk(Wk - + j (3
k W + (b6 - b4W2k ^ , .
Wk b5wk + (b6 - b4wk)z Wk
where k = 1,2,3; j = 0,1,..., then ±iw is a pair of purely imaginary roots of (3.4) with T]k. Define
T0 = Tk0 = Tl], W0 = Wk°. (312)
Note that when t = 0, (3.4) becomes
X3 + (b1 + b4)X2 + (b2 + b5)X + (b3 + b6) = 0. (3.13)
In addition, Routh-Hurwitz criterion [13] implies that, if the following condition holds, then all roots of (3.13) have negative real parts.
(H2) (b1 + b4) > 0, (b1 + b4)(b2 + b5) - (b3 + b6) > 0.
Till now, we can employ a result from Ruan and Wei [23] to analyze (3.4), which is, for the convenience of the reader, stated as follows.
Lemma 3.3 (see [23]). Consider the exponential polynomial
P (l,e-XT.....e~XTm ) = Xn + pf]Xn-1+• • • + p^X + p{? + [p|1)Xn-1 + ••• + p^X + p^] e~XT
+ •••+ [p1m)Xn-1 + •••+ p(m)X + p{m)\e-XTm,
(3.14)
where Ti > 0 (i = 1,2,...,m) and p(1) (j = 1,2,..., m) are constants. As(t1, t2,..., Tm) vary, the sum of the order of the zeros of P (X,e-XT,. ..,e~XTm) on the open right half plane can change only if a zero appears on or crosses the imaginary axis.
Using Lemmas 3.2 and 3.3 we can easily obtain the following results on the distribution of roots of the transcendental (3.4).
Lemma 3.4. (2.1) If c3 > 0 and c2 - 3c2 < 0, then all roots with positive real parts of (3.4) have the same sum as those of the polynomial (3.13) for all t > 0.
(3.1) If either c3 < 0 or c3 > 0 and c2 - 3c2 > 0, z* > 0, h(z*) < 0, then all roots with positive real parts of (3.4) have the same sum as those of the polynomial (3.13) for t e [0,t0).
Lemma 3.5. If 3wk + c\wk + c2 / 0, then the following transversality condition holds:
sgn< Re
= 0 when t = t0.
(3.15)
Proof. Differentiating (3.4) with respect to t yields
[3X2 + 2b\X + b2 + (lb4X + b5 - t(i>4X2 + b5X + h)e~kT)\dX = X^X2 + feX + b^e~XT.
(3.16)
For the sake of simplicity, denoting wQ and tq by w, t respectively, then
3X2 + 2b1X + i
2b4X + b5
X(b4 X2 + b5X + b6)e-XT X(b4X2 + b5X + b6 ) X
2X3 + b1X2 - !
b4 X2 - b6
X2(X3 + b1X2 + b2X + b3) X(b4X2 + b5X + b6) X
-2iw3 - b1^2 - I
b4w2 + b6
w2(b3 - b1w2 - i(w3 - b2w)) w2(b6 - b3w2 + b4iw) iw'
Then we get
Re{( dT
b3 - 2w6 - (b2 - 2b2) w4 (b3 - b1w2)2 + (w3 - b2w)2 (b6 - b4w2)2 + b\w2
b2 - b2w4
2w6 + c1w4 - c3
3w4 + c1w2 + c2
w2((b6 - b4W2)2 + b2w2) ((b6 - b4W2)2 + b2w2)
(3.17)
(3.18)
Then, if 3w4 + c\w2 + c2 = 0, we have sgn{Re{(dX/dT)-1}} = 0, we complete proof. □
Thus from Lemmas 3.2, 3.3, 3.4, and 3.5, and we have the following. Theorem 3.6. Suppose that (H1) and (H2) hold, then the following results hold.
(1) The positive equilibrium of (3.1) is asymptotically stable, if c3 > 0 and c2 - 3c2 < 0;
(2) if either c3 < 0 or c3 > 0 and c2 - 3c2 > 0, z* > 0, h(z*) < 0, system (3.1) is asymptotically stable for t e [0,tq) and system (3.1) undergoes a Hopf bifurcation at the origin when
T = Tq.
4. Direction of the Hopf Bifurcation
In this section, we derive explicit formulae for computing the direction of the Hopf bifurcation and the stability of bifurcation periodic solution at critical values t0 by using the normal form theory and center manifold reduction.
Letting x1 = S - S*, x2 = E - E*, x3 = I - I*, xi(t) = Xi(Tt), t = t0 + f, and dropping the bars for simplification of notation, system (3.1) is transformed into an FDE as
x (t) = L^(xt) + f{p,xt),
L^f = (tq + p) [Biy(0) + #2<p(-1)],
-a\ 0 -a2 "0 0 0"
= CQ a3 -a4 a2 , B2 = 0 -a7ß-lT 0 ,
0 a5 -a6 0 0 0
'-ßfi(0)f2 (Q) f(p,f) = (tq + n) | ßfi(0)f2(0)
Using the Riesz representation theorem, there exists a function n(6,f) of bounded variation for 6 e [-1,0], such that
Lpf = J dty(d,p)f(d) f e C.
In fact, we can choose
n(0,p) = (tq + n) [Bi6(0) + B2Ö(0 + 1)],
where 6(6) is Dirac delta function.
In the next, for y e [-1,0], we define
A(p)f =
' df dд,
0 e [-1,0),
R(p)f =
dn(0,p)f(0), 0 = 0,
0, 0 e [-1,0],
f(p,f), 0 = 0.
Then system (4.2) can be rewritten as
x (t) = A(p)xt + R(n)xt, (4.8)
where xt(9) = x(t + 9).
The adjoint operator A* of A is defined by
r MS) s e (0,1],
A*( ¡)f =
dnT (t, 0)f(-t), s = 0,
where nT is the transpose of the matrix n.
For y e C1 [-1,0] and f e C1 [0,1], we define
f y = f (0) ■ y(0) - - 9)dn(9)y(l)dl (4.10)
J9=-1 Jl=Q
where n(9) = n(9,0). We know that ±ztqwq is an eigenvalue of A(0), so ±iTQwQ is also an eigenvalue of A* (0). We can get
q(9) = ( qx j eiToWo9, -1 <9 < 0. (4.11)
From the above discussion, it is easy to know that
Aq(0) = iTQwQq(0). (4.12)
Hence we obtain
q1 = q2 = -
iwQ + ax
(4.13)
Suppose that the eigenvector q* of A* is
q*(s)= ( q1* jeiT0W0S, (4.14)
Then the following relationship is obtained:
A*q(0) = -iTQwQq*(0).
(4.15)
Hence we obtain
ax - IWq
„ a4 + a7elWQTQ „
q2 = —05— q 1.
(4.16)
(<f,q) = 1. (4.17)
One can obtain
(q*,q) = q(Q) ■ q(Q) - q (S - 0)dn(0)f(i)di
J0=-1 JS=Q
>0=-1J S=Q 1
/si - * - *\
= p (! + W\ + W2 )
f Q f0 1
tq tC1 qr q2*)
J0=-1J S=Q p
-a1 0 -a2 03 -04 02 1 6(0) 0 a5 -a6
/0 0 0\ + ( 0 -07 0 )6(0 + 1) \0 0 0/
(4.18)
q1jeiTQWo0dSd0 W2/
1 _ _ 1 _• _
= p (1 + q1q{ + qiq2*) - pToe lWoToajqq*.
Hence we obtain
p = (1 + q{q{ + q{q2*) - Toe-WoTo a7q{q1*. (4.19)
In the remainder of this section, by using the same notations as in Hassard et al. [16], we first compute the coordinates for describing the center manifold C0 at f = 0. Leting xt be the solution of (4.1) with f = 0, we define
z(t) = (q*,xt), W(t,0) = xt - 2Re{z(t)q(0)}.
(4.20)
On the center manifold C0 we have
W (t,6) = W (z,z,t), (4.21)
z2 zz Z
W (z,z,t)= W20 (6) - + W11 (6) -2- + W02 (6) - + ••• . (4.22)
In fact, z and z are local coordinate for C0 in the direction of q and q*. Note that, if xt is, we will deal with real solutions only. Since n = 0
z (t) = (q*,xt) = (q*,A(p)xt + R(^)Xt) = (q*,Axt) + (q*,Rxt) = iT0W0Z + q*(0) ■ f (0,W(t,0) + 2Re[z(t)q(0)]).
(4.23)
Rewrite (4.23) as
Z (t) = iT0w0z + g (z,z),
(4.24)
, z2 _ z z2z
(z,z)= g20 2 + g11zz + g02y + g2^~2"
(4.25)
From (4.1) and (4.24), we have
W = Xt - z q - z q = <
AW - 2 Re [g* (0) ■ f (z,z)q(d)], 0 e [-2t, 0),
AW - 2Re[q*(0) ■ f (z,z)q(0)] + f0(z,z;)/ 0 = 0.
(4.26)
WV = AW + H (z,z,0),
(4.27)
z2 zz z2
H(z,z,0)= H20 (0) y + Hn(0)zz + H02 (0) zr
(4.28)
Expanding the above series and comparing the corresponding coefficients, we obtain
(A - 2iwo)W2o(e) = -Hw(d), AWn(e) = -Hn(d), (A + 2iwo)Wo2(e) = -H02(6).
(4.29)
Since xt = x(t + 6) = W(z,z, 6)+ zq + z ■ q, we have
'W (X)(z,z,0)s W (2)(z,z/0) W (3)(z/%0),
qx \eiw00 + z\ q1 \e-iw00
(4.30)
Abstract and Applied Analysis Thus, we can obtain
2 —2 <M0) = — + — + (0)—2 + W^ (0)—— + W^ (0)—2,
2 —2 ^2 (0) = —q1 + —qi + W21' (0) y + W^ (0)—— + W01' (0) ^
It follows from (4.24) and (4.25) that
K11 = -ßq1, K12 = -ßq1, K13 = -ßfa + qj, K14 = -ß(w<? + 2 w222)+ W1(1)q1 + 2 W2^q),
K21 = ßqi, K22 = ßq1, K23 = ß(fl\ + q0,
W2(?+ W(1)q1 +1W 20
K24 = ^W12) +1 w$ + w(1)q1 +1 W^q).
Since q*(0) = (1/p)(1,q1,q2)T, we have
K11—2 + K12—— + K13—2 + K14—2—N
g22 = p (K11 + K21q^, g11 = p (K12 + K22qf),
g22 = p (K13 + K23q^, g21 = p (K14 + K24q1) .
(4.31)
^1(0)^2(0) = q1—2 + q1—2 + (q1 + q2)—— + ( W1(2) + 1 W2(2) + W^V + 1 W2(1)^—2—. (4.32)
/K11—2 + K12—— + K13— + K14—2—s = I K21—2 + K22—— + K23—2 + K24—2— \, (4.33)
(4.34)
(—, —) = p (1, q\, q2) I K21—2 + K22—— + K23—2 + K24—2— \. (4.35)
Comparing the coefficients of the above equation with those in (4.27), we have
(4.36)
In what follows, we focus on the computation of W2Q(9) and W11(9). For the expression of g21, we have
H(z,z,9) = -2Re[q*(0) ■ f(z,z)q(9)]
= -(g2Q Z + g11zZ + g02 y + q(9) -(g2Q z + g11zz + g02 \ + q(9).
Comparing the coefficients of the above equation, we can obtain that
W2o(6) = 2iTo WoW2o(6)+ g2oq(6) + g20q(6), Wii(6) =+giiq(6) + g nq(6).
W2o(6) = -ig20- q(0)eiToWo6 q(0)e-iToWo6 + Ei e2iT°w°6,
T0w0 3iT0w0
Wii(6) = — q(0)eiT°w°6 - t^q(0)e-iToW°6 + E2.
iT0W0 iT0W0
(4.37)
H20(6) = -g2oq(6) - go2q(6), 6 e [-1,0), (4.38)
Hii(6) = -giiq(6) - gnq(6), 6 e [-1,0). (4.39)
Substituting (4.39) into (4.27) and (4.38) into (4.27), respectively, we get
(4.40)
(4.41)
In the sequel, we will determine E1 and E2. Form the definition of A in (4.8), we have
J ^ dn(9)Ww(9) = 2zToWqW2q(0) - H2q(0), (4.42) (0
J dn(9)Wn(9) = -Hn(0). (4.43)
From (4.6) and (4.38)-(4.39), we have
H20(6) = -g2oq(6) - go2q(6) + (Kn,K21,0)T, Hn(6) = -gnq(6) - gnq(6) + (K12,K22,0)T.
(4.44)
(4.45)
Substituting (4.41) and (4.44) into (4.42) and noticing that
i^I - | eiw°6dn(6q(0) = 0,
-m>I -j e-ia'06 dn(6)\ q(0) = 0
(4.46)
we can obtain
2iw0I -J e2iT0W06dn(6)^E1 = (Kn K21 0)
(4.47)
which leads to
'2iw0 + a1
-a3 a4 + a7 -a2 0 -as a6
2iw0 + a4 + a7e-2iW0T0 -a2
-as 2i^0 + a6,
\ e23) /
(2) E( 3)
K2s K22
(4.48)
It follows that
(3) = E1=
E(1) = E1=
(2) E1=
K11 - a2E
2iw0 + a1 E(
(2) = 2i^0 + 06r (3)
K22 - Kn/(2iw0 + a1)
(4.49)
a2a3/(2iw0 + a1) + (2i^0 + a4 + a7e-2iW(0T°)((2iw0 + a6)/as) - 2ia>0 - a6'
(1) = K12 - a2E2 E2 = '
e(2) = 06 e(3)
2 as 2 ,
E(3) =_
a2a3as + (a4 + a7)a1a6 - a1 asa6
a1asK22 - asK12
Based on the above analysis, we can see each gij in (4.37) is determined by parameters and delays in (3.1). Thus, we can compute the following quantities:
Re C1(0)
T2 = -
Re A'(T0)' Im C1(0) + f2 Im A'(0)
(4.s0)
fc = -2ReC1(0).
400 350 300 250 200 150 100 50
/\/\/\/Www-
0 200 400 600 800 1000 1200 1400 1600 t
180 160 140 120 100 80 60 40 20
180 160 140 120 100 80 60 40 20
00 (d)
Figure 1: t = 13 < to. The positive equilibrium E* of system (3.1) is asymptotically stable.
Theorem 4.1. In (4.50), the following results hold.
(1) The sign of ¡d2 determines the directions of the Hopf bifurcation: if ¡d2 > 0 (^2 < 0) then the Hopf bifurcation is forward (backward) and the bifurcating periodic solutions exist for T>T0 (t < T0).
(2) The sign of ß2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if ß2 < 0 (ß2 > 0).
(3) The sign of T2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if T2 > 0 (T2 < 0).
5. Numerical Examples
In this section, some numerical results of system (3.1) are presented to justify the Previous theorem above. As an example, considering the following parameters: f = 0.01, N = 10000, y = 0.08, a = 0.1, ¡5 = 0.01, psR = 0.2, per = 0.2, then Rq = 1.706, cs = -3.6284e - 5, and E* = (279,133.6,148.4). According to the Lemma 3.2, (3.9) has one positive real root w = 0.1194. Correspondingly, by (3.13), we obtain tq = 14.05. First, we choose t = 13 < tq, the corresponding wave form and phase plots are shown in Figure 1; it is easy to see from Figure 1 that system (3.1) is asymptotically stable. Finally, we choose t = 14.15 > tq the
corresponding wave form and phase plots are shown in Figure 2; it is easy to see that Figure 2 undergoes a Hopf bifurcation.
6. Conclusions
In this paper, considering that in addition to cleaning viruses in state I, people may immunize their computers with countermeasures in state S and state E, and since using antivirus software will take a period of time, we have constructed a computer virus model with time delay depending on the SEIR model. The theoretical analyses for the computer virus models are given. Furthermore, we have proved that when time cross through the critical value, the system exist a Hopf bifurcation. Finally, simulation clarifies our results.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grant 60973114 and Grant 61170249, in part by the Research Fund of Preferential Development Domain for the Doctoral Program of Ministry of Education of China under Grant 20110191130005, in part by the Natural Science Foundation project of CQCSTC under Grant 2009BA2024, in part by Changjiang Scholars, and in part by the State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, under Grant 2007DA10512711206.
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