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Bifurcation analysis of a class of neural networks with delays
Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252 www.elsevier.com/locate/na
Bifurcation analysis of a class of neural networks with delays夡 Junjie Weia,∗ , Chunrui Zhangb a Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China b Department of Mathematics, Northeast Forestry University, Harbin 150040, PR China
Received 21 February 2007; accepted 3 August 2007
Abstract A class of n-dimensional neural network model with multi-delay is considered. Firstly, a group of sufficient conditions for the existence of Hopf bifurcation are obtained via employing the polynomial theorem to analyze the distribution of the roots of the associated characteristic equation. Secondly, the stability and direction of the Hopf bifurcation are determined by applying the normal form method and center manifold theorem. Finally, the multiple stability is investigated and pitchfork bifurcation is also found. The results are illustrated by some numerical simulations. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Neural network; Delay; Hopf bifurcation; Pitchfork bifurcation
1. Introduction Hopfield [17] proposed a simplified neural network model in which each neuron is represented by a linear circuit consisting of a resistor and capacitor, which is connected to the other neurons via nonlinear sigmoidal activation functions instantaneously. In reality, neural networks often have time delays. For example, it is due to the finite switching speed of amplifiers in electronic neural networks, or due to finite signal propagation time in biological networks. After Hopfield’s work, many neural network models with delays have been proposed and studied, we refer to Wu [37]. Baldi and Atiya [3] particularly constructed a network model that consists of a ring of neurons connected cyclically with delayed interactions. Different delays are introduced for the communication between the adjacent neurons. This leads to the system dui ui = − + Tii−1 fi−1 (ui−1 (t − ii−1 )), dt Ti
i (mod n).
(1.1)
Campbell [5] generalized Baldi and Atiya’s model (1.1) to a network that consists of a ring of neurons where the j th element receives two time delayed inputs: One from the current element itself and the other from the previous element. In the last few years, in order to obtain a deep and clear understanding of the dynamics of the model, 夡 This research was supported by the National Natural Science Foundations of China and Young Innovative Programme of Northeast Forestry University. ∗ Corresponding author. Tel.: +86 451 86417440; fax: +86 451 86412549. E-mail address: [email protected] (J. Wei).
1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.08.008
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several researchers have focused on the bifurcation analysis in the model (1.1) or Campbell’s model with n = 2 and 3. For n = 2, we refer to [2,4,9–12,14,19,21,26,27,32,35], for n = 3, see [6,31,33,38,42], and for n = 4, see [20]. Another important models such as Cohen–Grossberg and BAM neural networks with delays have been studied extensively, we refer to [1,7,8,13,18,24,25,29,30] and references therein for stability analysis and existence of periodic solutions by constructing Liapunov functions. For Campbell’s models with delays, there are several articles on synchronized Hopf bifurcation analysis, we refer to [34,41]. The purpose of the present paper is to study the model (1.1). For convenience, we rewrite it in the following form: ⎧ u˙ 1 (t) = −a1 u1 (t) + F1 (un (t − 1 )), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u˙ 2 (t) = −a2 u2 (t) + F2 (u1 (t − 2 )), .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎩ u˙ n (t) = −an un (t) + Fn (un−1 (t − n )).
(1.2)
Employing the Routh–Hurwitz criterion and the result on distribution of the zeros of transcendental functions due to Ruan and Wei [28], we get a set of conditions to describe the stability of the fixed point of model (1.2) and the existence of Hopf bifurcations. Furthermore, we derive an algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcating periodic solutions by using the normal form method and center manifold theorem introduced by Hassard et al. [16]. We particularly obtain the explicit conclusions on the Hopf bifurcation properties when the transfer functions Fj (u) is similar to tanh(u). Meanwhile, we investigate the multiple stability and the pitchfork bifurcation to the model (1.2) is also found. We would like to mention that there are several papers related to multi-stability of neural networks with delay, see [6,35,39,40]. Our paper is organized as follows: In Section 2, we present some preliminary results which will be used later to analyze the distribution of the roots of the characteristic equation associated with system (1.2). In Section 3, we analyze the distribution of the eigenvalues and obtain the stability of the fixed point and existence of Hopf bifurcations. In Section 4, an algorithm for determining the properties of Hopf bifurcations is derived by using the normal form method and center manifold theorem. Section 5 deals with the multiple stability. An example is considered and some numerical simulations are carried out in Section 6 to support the analysis results. 2. Preliminary results In this section, we provide some preliminary results to be used in the subsequent section. Let x1 (t) = u1 (t − (2 + 3 + · · · + n )),
x2 (t) = u2 (t − (3 + 4 + · · · + n )),
.. . xn−1 (t) = un−1 (t − n ) and
xn (t) = un (t).
Then Eq. (1.2) becomes the following equivalent system: ⎧ x˙1 (t) = −a1 x1 (t) + F1 (xn (t − )), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x˙2 (t) = −a2 x2 (t) + F2 (x1 (t)), .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎩ x˙n (t) = −an xn (t) + Fn (xn−1 (t)), where = 1 + 2 + · · · + n . We make the following assumptions. (H1 ) For i = 1, 2, . . . , n, constants ai > 0, Fi ∈ C 2 and Fi (0) = 0.
(2.1)
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From the assumption it follows that the origin (0, 0, . . . , 0) is an equilibrium of Eq. (2.1). The linearization of Eq. (2.1) around the origin is given by ⎧ x˙ (t) = −a x (t) + F (0)x (t − ), 1 1 1 n 1 ⎪ ⎪ ⎪ ⎪ (t) + F (0)x x ˙ (t) = −a x ⎨ 2 1 (t), 2 2 2 (2.2) .. ⎪ ⎪ . ⎪ ⎪ ⎩ x˙n (t) = −an xn (t) + Fn (0)xn−1 (t), whose characteristic equation is n +
n
ai n−1 +
1 i
i=1
+
n
ai −
n
ai aj n−2 +
ai aj ak n−3 + · · · +
ai aj ak · · · al
1 i
1 i
Fi (0)e− = 0.
i=1
i=1
Denote A1 =
n
n
i=1 ai .
ai aj ak , . . . , An−1 =
1 i
ai aj ak · · · al ,
1 i
Then the characteristic equation becomes
+ A1 n
ai aj , A3 =
1 i
i=1
and An =
ai , A2 =
n−1
+ A2
n−2
+ A3
n−2
+ · · · + An −
n
Fi (0)e− = 0.
(2.3)
i=1
Lemma 2.1. For a1 , a2 , . . . , an ∈ R, let Aj (j = 1, . . . , n) be defined as above, and k ∈ N, 2k n. Then A2k − 2Ak−1 Ak+1 + 2Ak−2 Ak+2 − · · · + (−1)k 2A2k = ai21 ai22 · · · ai2k .
(2.4)
1 i1
· · · ai2k is a symmetric homogeneous polynomial of a1 , a2 , . . . , an , and its
principle term (according to dictionary order) is (22 · · · 2 00000). Let ai21 ai22 · · · ai2k = bk A2k + bk−1 Ak−1 Ak+1 + bk−2 Ak−2 Ak+2 + · · · + b0 A2k . (2.5) Proof. Clearly,
2 2 1 i1
1 i1
By the polynomial theorem, comparing the coefficients of a12 a22 · · · ak2 , we have that bk = 1. Similarly, by using 2 a a 2 2 2 coefficients comparison of a12 a22 · · · ak−1 k k+1 , and of a1 a2 · · · ak−2 ak−1 ak ak+1 ak+2 , we obtain bk−1 = −2 and bk−2 = 2, respectively. By the induction, we assume that bk−l = (−1)l 2 for l 1. 2 Using coefficient comparison of a12 a22 · · · ak−(l+1) ak−l ak−l+1 · · · ak+l+1 , it follows that l+1 l−1 l l C2(l+1) − 2C2(l+1) + 2C2(l+1) − · · · + (−1)l 2C2(l+1) + bk−(l+1) = 0.
Thus,
2(l+1) 2(l+1) 0 1 2 0 bk−(l+1) = (−1)l C2(l+1) − C2(l+1) + C2(l+1) − · · · + C2(l+1) + (−1)l+1 C2(l+1) + C2(l+1) = (−1)l (1 − 1)2(l+1) + (−1)l+1 2 = (−1)l+1 2.
This completes the proof.
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3. Stability analysis In this section we shall analyze the distribution of the roots of Eq. (2.3) to discuss the stability and existence of Hopf bifurcation of system (2.1). Lemma 3.1. (1) If A2n > ( ni=1 Fi (0))2 , then Eq. (2.3) has no roots with zero real parts for all 0. (2) If A2n < ( ni=0 Fi (0))2 , then Eq. (2.3) has a pair of purely imaginary roots ±i0 when = j , j = 0, 1, 2, . . . , where ⎧ 1 n−2 A +···−2 A (−1)m n +(−1)m−1 2 n−2 +An + 2j , n = 2m, ⎨ 0 arccos n i=1 Fi (0) j = j = 0, 1, . . . , (3.1) n−2 A +···−3 A ⎩ 1 (−1)m n +(−1)m−1 2 n−3 +An−1 + 2j , n = 2m + 1, n 0 arcsin − i=1 Fi (0) √ and 0 = z0 , z0 is the unique positive zero of the function n 2 n n−1 n−2 + B4 z + · · · + B2n−2 z + B2n − Fi (0) , g(z) = z + B2 z i=1
with
B2 = A21 +
(−1)i+1 2Ai Aj ,
i+j =2,i=j
B6 = A23 +
(−1)i+2 2Ai Aj ,
i+j =4,i=j
(−1)i+3 2Ai Aj , . . . ,
i+j =6,i=j
B2n−2 = A2n−1 +
B4 = A22 +
(−1)i+n 2Ai Aj
and
B2n = A2n , i, j = 0, 1, . . . , n, A0 = 1.
i+j =2n−2,i=j
Proof. Let i( > 0) be a root of Eq. (2.3), then n
in n + in−1 n−1 A1 + in−2 n−2 A2 + · · · + iAn−1 + An =
Fi (0)e−i .
i=1
Separating the real and imaginary parts gives, when n = 2m, ⎧ n m n m−1 n−2 ⎪ A2 + · · · − 2 An−2 + An = Fi (0) cos , ⎨ (−1) + (−1) i=1
n ⎪ ⎩ (−1)m+1 n−1 A1 + (−1)m n−3 A3 + · · · − 3 An−3 + An−1 = − F (0) sin , i
(3.2)
i=1
and when n = 2m + 1, ⎧ n m n m−1 n−2 ⎪ A2 + · · · − 3 An−3 + An−1 = − Fi (0) sin , ⎨ (−1) + (−1) i=1
n ⎪ ⎩ (−1)m n−1 A1 + (−1)m−1 n−3 A3 + · · · − 2 An−2 + An = F (0) cos . i
(3.3)
i=1
Hence,
2n
+ B2
2n−2
+ B4
2n−4
+ · · · + B2n−2 + B2n = 2
n
2 Fi (0)
.
(3.4)
i=1
By Lemma 2.1, B2k > 0, k = 0, 1, . . . , n. Denote g(z) = z + B2 z n
n−1
+ B4 z
n−2
+ · · · + B2n−2 z + B2n −
n i=1
2 Fi (0)
.
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n
(1) From A2n > (
2 i=1 Fi (0)) ,
g(0) = B2n −
n
it follows that
2 Fi (0)
> 0.
i=1
Hence, by g (z) = nzn−1 + B2 (n − 1)zn−2 + · · · + B2n−2 > 0
for z > 0,
the function g(z) has no positive zero. This implies that Eq. (3.4) has no real root, and hence Eq. (2.3) has no purely imaginary root. Clearly of (1) is complete. = 0 is not a root of Eq. (2.3). The proof (2) From A2n < ( ni=1 Fi (0))2 , it follows that g(0) = B2n − ( ni=1 Fi (0))2 < 0. Then, by limz→+∞ g(z) = +∞, and g (z) > 0 for z > 0, there exists a unique positive number z0 such that g(z0 ) = 0. √ Let 0 = z0 , where z0 is the positive root of the equation g(z) = z + B2 z n
n−1
+ B4 z
n−2
+ · · · + B2n −
n
2 Fi (0)
= 0.
(3.5)
i=1
Then 0 is a root of Eq. (3.4). Define j as (3.1). Hence, (0 , j ) is a root of (3.2) or (3.3). This implies that ±i0 is a pair of purely imaginary roots of Eq. (2.3) when = j (j = 0, 1, . . .). The proof of (2) is complete. Now let us consider the behavior of the roots of Eq. (2.3) near the values = j (j = 0, 1, . . .). To do this we assume that () = () + i() is a solution of Eq. (2.3) satisfying (j ) = 0 and (j ) = 0 . Lemma 3.2. d/d|=j > 0. Proof. For convenience, set P () = n + A1 n−1 + A2 n−2 + A3 n−2 + · · · + An . Then Eq. (2.3) is rewritten in the following form: P () =
n
Fi (0)e− .
i=1
Differentiating both sides of Eq. (2.3) with respect to gives ni=1 Fi (0)e− d() =− . d P () + ni=1 Fi (0)e− Replacing by j , and noting P () = Re
n
− , i=1 Fi (0)e
it follows that
20 g (z0 ) d = > 0, d =j =0
where =
nn−1 0
− (n − 2)n−3 0 A2
+ · · · − 20 An−2 ∓ 0
2 Fi (0) sin 0
i=1
+
n
(n − 1)n−2 0 A1
− (n − 3)n−4 0 A3
+ · · · − An−1 ∓ 0
n i=1
2 Fi (0) cos 0
J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
or
=
nn−1 0
− (n − 2)n−3 0 A2
+ · · · − 20 An−2 ± 0
2 Fi (0) cos 0
i=1
+
n
(n − 1)n−2 0 A1
− (n − 3)n−4 0 A3
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+ · · · − An−1 ± 0
n
2 Fi (0) sin 0
.
i=1
For A1 , A2 , . . . , An , we define A1 1 = A1 , 2 = A3 A1 A 3 . n = . . A 2n−1
A1 1 , 3 = A3 A2 A 5
1 A2 A4
0 A1 , . . . , A 3
1
0
0
···
0
A2 .. .
A1 .. .
1 .. .
···
A2n−2
A2n−3
A2n−4
0 .. . n
··· · · · An −
i=1
n = An − Fi (0) n−1 , i=1 Fi (0)
and make the following assumptions on 1 , 2 , . . . , n . (H2 ) 1 > 0, 2 > 0, . . . , n−1 > 0. (H3 ) An > ni=1 Fi (0). Theorem 3.3. Suppose that (H1 ) is satisfied. (1) If A2n > ( ni=1 Fi (0))2 , (H2 ) and (H3 ) hold, then the zero solution of system (2.1) is asymptotically stable for all 0. (2) If A2n < ( ni=1 Fi (0))2 , (H2 ) and (H3 ) hold, then the zero solution of system (2.1) is asymptotically stable for ∈ [0, 0 ), and unstable when > 0 . Furthermore, system (2.1) undergoes a Hopf bifurcation at the origin when = j (j = 0, 1, . . .). Here = j is defined as (3.1). (3) If A2n > ( ni=1 Fi (0))2 , and either (H2 ) or n > 0 is not satisfied, then the zero solution of system (2.1) is unstable for all 0. Proof. When = 0, Eq. (2.3) becomes n + A1 n−1 + A2 n−2 + A3 n−2 + · · · + An −
n
Fi (0) = 0.
(3.6)
i=1
(1) By the Routh–Hurwitz criterion, all the roots of Eq. (3.6) have negative real parts if and only if (H2 ) and (H3 ) are satisfied. Hence, all roots of Eq. (2.3) with = 0 have negative real parts. From the conclusion (1) in Lemma 3.1, we know that Eq. (2.3) has no root with zero real part for all > 0. Applying the Corollary 2.4 in Ruan and Wei [29], all roots of Eq. (2.3) have negative real parts for all 0. Hence, the conclusion of (1) follows. (2) Similarly, all the roots of Eq. (2.3) have negative real parts when ∈ [0, 0 ). Applying Lemma 3.2 we know that Eq. (2.3) has at least a couple of roots with positive real parts when > 0 . Hence the first conclusion of (2) follows. The second conclusion of (2) follows from Lemmas 3.1 and 3.2, and the Hopf bifurcation theorem for functional differential equations [15, Theorem 1.1, pp. 246–247]. (3) From the (1) in Lemma 3.1 we know that, under the inequality A2n > ( ni=1 Fi (0))2 , Eq. (2.3) has no roots with zero real part for all 0. Hence Eq. (3.6) has at least one root with positive real part by the Routh–Hurwitz criterion when either (H2 ) or n > 0 does not hold. Applying Corollary 2.4 in Ruan and Wei [29], Eq. (2.3) has at least one root with positive real part. Hence the conclusion of (3) follows. The proof is complete.
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4. Direction and stability of the Hopf bifurcation In Section 3, we have obtained some conditions which guarantee that system (2.1) undergoes the Hopf bifurcation at a sequence values of . In this section, we shall study the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. The method we used is based on the normal form theory and the center manifold theorem introduced by Hassard et al. [16]. For convenience, let us take xi (t) = xi (t) and = ∗ + , where ∗ = j for fixed j, and ∈ R. Then system (2.1) becomes ⎧ x˙1 (t) = (∗ + )[−a1 x1 (t) + F1 (xn (t − 1))], ⎪ ⎪ ⎪ ⎪ ⎨ x˙2 (t) = (∗ + )[−a2 x2 (t) + F2 (x1 (t))], (4.1) .. ⎪ ⎪ . ⎪ ⎪ ⎩ x˙n (t) = (∗ + )[−an xn (t) + Fn (xn−1 (t))]. The characteristic equation associated with the linearization of Eq. (4.1) around the origin is n n n−1 2 n−2 3 n−2 n − + A1 (∗ + ) = 0, + A2 (∗ + ) + A3 (∗ + ) + · · · + (∗ + ) An − Fi (0)e i=1
(4.2) where the definitions of Aj (j = 1, 2, . . .) are same as in Eq. (2.3). Comparing Eq. (2.3) with Eq. (4.2), it follows that = (∗ + ). Thus, from the discussion above, we know that Eq. (4.1) undergoes a Hopf bifurcation at the origin when = 0, and ±i∗ 0 are simple roots of Eq. (4.2). Let () be the root of Eq. (4.2) satisfying Re (0) = 0 and Im (0) = ∗ 0 . Then Re (0) = ∗ (∗ ) > 0.
In order to study the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions, we need to assume that Fi ∈ C 3 (i = 1, 2, . . . , n), further. Then Eq. (4.1) can be rewritten as ⎧ x˙1 (t) = (∗ + )[−a1 x1 (t) + F1 (0)xn (t − 1) + 21 F1 (0)xn2 (t − 1) + 16 F1 (0)xn3 (t − 1) + O(xn4 )], ⎪ ⎪ ⎪ ⎪ ⎨ x˙2 (t) = (∗ + )[−a2 x2 (t) + F2 (0)x1 (t) + 21 F2 (0)x12 (t) + 16 F2 (0)x13 (t) + O(x14 )], (4.3) ⎪ .. ⎪ . ⎪ ⎪ ⎩ 2 (t) + 1 F (0)x 3 (t) + O(x 4 )]. x˙n (t) = (∗ + )[−an xn (t) + Fn (0)xn−1 (t) + 21 Fn (0)xn−1 n−1 n−1 6 n Choosing the phase space as C = C([−1, 0], R n ). For ∈ C, let Lv = B1 (0) + B2 (−1), where
⎛ −a 1 ⎜ F2 (0) ⎜ 0 B1 = (∗ + ) ⎜ ⎜ . ⎝ . . 0 ⎛ 0 0 ⎜0 0 ⎜ ⎜0 0 B2 = (∗ + ) ⎜ ⎜0 0 ⎜. . ⎝ .. .. 0
0
0 0 −a2 0 F3 (0) −a3 .. .. . . 0 0 0 0 0 0 .. . 0
··· ··· ··· ···
0 0 0 0 . · · · .. ··· 0
··· ··· ···
0 0 0 .. ··· . · · · Fn (0) ⎞ F1 (0) 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ .. ⎟ . ⎠ 0
0 ⎞ 0 ⎟ 0 ⎟ ⎟, .. ⎟ ⎠ . −an
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and ⎛ ( + v)[ 1 F (0) 2 (−1) + 1 F (0) 3 (−1)) + O(| |4 )] ⎞ ∗ n n 2 1 6 1 ⎜ ( + v)[ 1 F (0)) 2 (0) + 1 F (0) 3 (0) + O(| |4 )] ⎟ ∗ ⎜ ⎟ 1 1 6 2 2 2 ⎟. F˜ (v, ) = ⎜ ⎜ ⎟ .. ⎝ ⎠ . 2 3 1 1 4 (∗ + v)[ 2 Fn (0) n−1 (0) + 6 Fn (0) n−1 (0) + O(| | )] By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions ( , ): [−1, 0] → R n , such that L =
0 −1
d ( , ) ( ).
In fact, we choose ⎧ ⎨ −B2 , = −1, ( , ) = 0, ∈ (−1, 0), ⎩ B1 , = 0, the last equation is satisfied. For ∈ C 1 ([−1, 0], R n ), define ⎧ ⎨ d ( ) , ∈ [−1, 0), d A() = ⎩ 0 −1 d (t, ) (t), = 0, and
R() =
0, ∈ [−1, 0), F˜ (, ), = 0.
Hence (4.3) can be rewritten in the following form: u˙ t = A()ut + R()ut ,
(4.4)
where u = (u1 , u2 , . . . , un )T ,
ut = u(t + ), ∈ [−1, 0].
For ∈ C 1 ([0, 1], R n ) define ⎧ ⎨ − d (s) , s ∈ (0, 1], ∗ ds A (s) = ⎩ 0 −1 d (t, 0) (−t), s = 0. For ∈ C([−, 0], C n ) and ∈ C([0, 1], (C n )∗ ), define the bilinear form ¯
, = (0) (0) −
0
=−1 =0
¯ − ) d ( ) () d,
(
where ( ) = ( , 0), then A = A(0) and A∗ are adjoint operators. By the results of above, ± i0 ∗ are eigenvalues of A(0), thus they are also eigenvalues of A∗ .
(4.5)
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J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
By direct computation, we obtain that q( ) = q0 ei0 ∗ , with ⎛ ⎞ 1 ⎜ ⎟ F2 (0) ⎜ ⎟ ⎜ ⎟ a2 + i0 ⎜ ⎟ ⎜ ⎟ F3 (0)F2 (0) ⎜ ⎟ ⎜ ⎟ q0 = ⎜ ⎟ (a + i )(a + i ) 3 0 2 0 ⎜ ⎟ ⎜ ⎟ . ⎜ ⎟ .. ⎜ ⎟ ⎝ ⎠ Fn (0) · · · F3 (0)F2 (0) (an + i0 ) · · · (a3 + i0 )(a2 + i0 ) is the eigenvector of A corresponding to i0 ∗ , and q ∗ (s) = q0∗ ei0 ∗ s with ⎛
⎞T 1 a1 − i0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ F2 (0) ⎜ ⎟ ⎜ ⎟ (a2 − i0 )(a1 − i0 ) ⎜ ⎟ ∗ ⎜ ⎟ ¯ q0 = K ⎜ F3 (0)F2 (0) ⎟ ⎜ ⎟ ⎜ ⎟ . ⎜ ⎟ .. ⎜ ⎟ ⎝ (an−1 − i0 ) · · · (a2 − i0 )(a1 − i0 ) ⎠ Fn (0) · · · F3 (0)F2 (0) is the eigenvector of A∗ corresponding to −i0 ∗ , where −1 n 1 ∗ K = (a1 + i0 ) + . ai + i0 i=1
q ∗ , q = 1
and q ∗ , q ¯ = 0. Moreover, Using the same notation as in [16], we compute the coordinates to describe the center manifold C0 at v = 0. Let ut be the solution of Eq. (4.4) when v = 0. Define z(t) = q ∗ , ut , W (t, ) = ut ( ) − 2 Re{z(t)q( )}. On the center manifold C0 we have W (t, ) = W (z(t), z¯ (t), ), where W (z, z¯ , ) = W20 ( )
z3 z2 z¯ 2 + W11 ( )z¯z + W02 ( ) + W30 + · · · , 2 2 6
z and z¯ are local coordinates for center manifold C0 in the direction of q ∗ and q¯ ∗ . Note that W is real if ut is also real. We only consider real solutions. For solution ut ∈ C0 of (4.4), since v = 0, z˙ = i∗ 0 z + q ∗ ( ), F˜ (W + 2 Re{z(t)q( )}) = i∗ 0 z + q¯ ∗ (0)F˜ (W (z, z¯ , 0) + 2 Re{z(t)q( )}) = i∗ 0 z + q¯ ∗ (0)F˜0 (z, z¯ ).
def
We rewrite this as z˙ (t) = i∗ 0 z(t) + g(z, z¯ ),
(4.6)
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where g(z, z¯ ) = q¯ ∗ (0)F˜ (W (z, z¯ , 0) + 2 Re{z(t)q( )}) z¯ 2 z2 z¯ z2 + ···. = g20 + g11 z¯z + g02 + g21 2 2 2 By (4.4) and (4.6), we have
(4.7)
W˙ = u˙ t − z˙ q − z˙¯ q¯ AW − 2 Re{q¯ ∗ (0)F˜0 q( )}, ∈ [−1, 0), = AW − 2 Re{q¯ ∗ (0)F˜0 q( )} + F˜0 , = 0 def
= AW + H (z, z¯ , ),
where z2 z¯ 2 + H11 ( )z¯z + H02 ( ) + · · · . 2 2 Expanding the above series and comparing the coefficients, we obtain H (z, z¯ , ) = H20 ( )
(4.8)
(A − 2i∗ 0 )W20 ( ) = −H20 ( ), AW 11 ( ) = −H11 ( ), .. .
(4.9)
Denote the jth element of q(0) by qj , and the jth element of W (z, z¯ , ) by j
W j (z, z¯ , ) = W20 ( )
z2 j + W11 ( )z¯z + · · · . 2
Then it follows that xi (t) = W i (z, z¯ , 0) + zq i + z¯ q¯i ,
i = 1, 2, . . . , n − 1
and xn (t − 1) = W n (z, z¯ , −1) + zq n e−i∗ 0 + z¯ q¯n ei∗ 0 . Hence F˜ (0, W + zq + z¯ q) ¯ ⎛ 1 F (0)[q 2 e−2i0 ∗ z2 + 2q q¯ z¯z + q¯ 2 e2i0 ∗ z¯ 2 ] ⎞ n n n n 2 1 ⎜ +[ 1 F (0)(W n (−1)q¯ ei0 ∗ + 2W n (−1)q e−i0 ∗ ) + 1 F (0)q 2 q¯ e−i0 ∗ ]z2 z¯ ⎟ n n ⎜ ⎟ n n 20 11 2 1 2 1 ⎜ ⎟ ⎜ + · · · + 1 F (0)[q 2 z2 + 2q1 q¯1 z¯z + q¯ 2 z¯ 2 ] ⎟ ⎜ ⎟ 1 1 2 2 = ∗ ⎜ ⎟. 1 (0)q¯ + 2W 1 (0)q ) + 1 F (0)q 2 q¯ ]z2 z¯ ⎜ +[ 1 F2 (0)(W20 ⎟ 1 1 11 1 1 2 2 2 ⎜ ⎟ ⎜ ⎟ 1 2 2 2 2 ⎝ + · · · + 2 Fn (0)[qn−1 z + 2qn−1 q¯n−1 z¯z + q¯n−1 z¯ ] ⎠ n−1 n−1 2 q¯ 2 +[ 21 Fn (0)(W20 (0)q¯n−1 + 2W11 (0)qn−1 ) + 21 Fn (0)qn−1 n−1 ]z z¯ + · · ·
Notice that
⎛
⎞T 1 a1 − i0 ⎜ ⎟ ⎜ ⎟ F2 (0) ⎜ ⎟ ⎜ ⎟ (a2 − i0 )(a1 − i0 ) ⎜ ⎟ i s ∗ ⎜ ⎟ e 0 ∗ ¯ q (s) = K ⎜ F3 (0)F2 (0) ⎟ ⎜ ⎟ .. ⎜ ⎟ ⎜ ⎟ . ⎝ (an−1 − i0 ) · · · (a2 − i0 )(a1 − i0 ) ⎠ Fn (0) · · · F3 (0)F2 (0)
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J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
and g(z, z¯ ) = q¯ ∗ (0)F˜0 , we have
! F3 (0) F1 (0) F2 (0) Fn (0) −2i0 ∗ qn−1 , Me q1 + q2 + · · · + qn + g20 = ∗ (a1 + i0 )K Fn (0) F1 (0) F2 (0) F3 (0) ! F3 (0) F1 (0) F2 (0) Fn (0) q ¯ g11 = ∗ (a1 + i0 )K + + · · · + M q ¯ q ¯ q ¯ + 1 2 n−1 , n Fn (0) F2 (0) F3 (0) F1 (0) F3 (0) (a1 + i0 )(a2 + i0 ) F (0) a1 + i0 F1 (0) ¯ q¯n + 2 q¯2 q ¯ + M 1 F1 (0) F2 (0) a1 − i0 F3 (0) (a1 − i0 )(a2 − i0 ) ⎤ n−1 Fn (0) aj + i0 +··· + q¯n−1 ⎦ , aj − i0 Fn (0)
g02 = ∗ (a1 − i0 )K
j =1
n g21 = ∗ (a1 − i0 )K W20 (−1)
F1 (0) F (0) a1 + i0 1 ¯ i0 ∗ + W20 Me (0) 2 F1 (0) F2 (0) a1 − i0
⎤ (0) (0) n−1 F a + i F + i )(a + i ) (a j 0⎦ 1 0 2 0 n−1 2 (0) n (0) 3 + · · · + W20 +W20 aj − i0 Fn (0) F3 (0) (a1 − i0 )(a2 − i0 )
j =1 F (0) F (0) F (0) n 1 2 Me−i0 ∗ + W11 + W11 (−1) 1 (0) 2 (0) 3 + 2∗ K(a1 + i0 ) W11 F1 (0) F2 (0) F3 (0) ! F (0) F (0) F (0) F (0) n−1 (0) n + ∗ K(a1 + i0 ) 1 |qn |2 + 2 |q1 |2 + 3 |q2 |2 + · · · + W11 Fn (0) F1 (0) F2 (0) F3 (0) ! Fn (0) 2 |qn−1 | , +··· + Fn (0)
where M=
n Fj (0) j =1
aj + i0
.
We still need to compute W20 ( ) and W11 ( ). For ∈ [−1, 0), we have H (z, z¯ , ) = 2 Re{q¯ ∗ (0)F˜0 q( )} = −gq( ) − g¯ q( ) ¯ % $ % $ 2 2 z z¯ z¯ 2 z2 ¯ = − g20 + g11 z¯z + g02 + · · · q( ) − g¯ 20 + g¯ 11 z¯z + g¯ 02 + · · · q( ). 2 2 2 2 Comparing the coefficients with (4.6) gives that H20 ( ) = −g20 q( ) − g¯ 02 q( ) ¯
and
H11 ( ) = −g11 q( ) − g¯ 11 q( ). ¯
It follows from the definition of W that −i0 W˙ 20 ( ) = 2i∗ 0 W20 ( ) − g20 q(0)ei0 − g¯ 02 q(0)e ¯ .
Solving for this, we obtain W20 ( ) =
g20 g¯ 20 −i∗ 0 q(0)ei∗ 0 − q(0)e ¯ + E1 e2i∗ 0 , i∗ 0 3i∗ 0
(4.10)
J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
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and similarly W11 ( ) =
g¯ 11 g11 −i∗ 0 q(0)e ¯ + E2 , q(0)ei∗ 0 − i∗ 0 i∗ 0
where E1 and E2 are both n-dimensional vectors, and can be determined by setting = 0 in H. In fact, H (z, z¯ , 0) = −2 Re{q¯ ∗ (0)F˜0 q(0)} + F˜0 , we have
⎛
⎜ ⎜ ⎜ H20 = −g20 q(0) − g¯ 20 q(0) ¯ + ∗ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ H11 = −g11 q(0) − g¯ 11 q(0) ¯ ⎜ ⎜ ⎝
F1 (0)qn2 e−2i0 ∗ F2 (0)q12
F3 (0)q22 .. . 2 Fn (0)qn−1 ⎞ F (0)q q¯
⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠
n n
1
F2 (0)q1 q¯1
F3 (0)q2 q¯2 .. .
⎟ ⎟ ⎟ ⎟, ⎟ ⎠
Fn (0)qn−1 q¯n−1 and
⎛
2i0 + a1
⎜ −F (0) 2 ⎜ ⎜ 0 ⎜ ⎜ ⎜ 0 ⎜ ⎝ ··· 0
0
0
2i0 + a2 −F3 (0) 0
0 2i0 + a3 −F4 (0)
··· 0
··· 0 (1)
· · · −F1 (0)e−2i0 ∗ ··· ···
0 0
··· ··· ···
0 ··· 2i0 + an
(2)
(n)
⎞
⎛
⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ E1 = ⎜ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠
F1 (0)qn2 e−2i0 ∗ F2 (0)q12 F3 (0)q22 .. . 2 Fn (0)qn−1
⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠
(1)
(2)
(n)
Solving this we can obtain E1 = (E1 , E1 , . . . , E1 )T . Similarly, we can obtain E2 = (E2 , E2 , . . . , E2 )T from ⎛ ⎞ 0 0 · · · −F1 (0) a1 ⎞ ⎛ F (0)q q¯ n n 1 ⎜ −F (0) ⎟ a 0 · · · 0 2 ⎜ ⎟ 2 ⎜ F (0)q1 q¯1 ⎟ ⎜ ⎟ 2 ⎟ ⎜ ⎜ 0 −F3 (0) a3 ··· 0 ⎟ ⎟ (0)q q¯ ⎜ ⎟ E2 = ⎜ F ⎟. ⎜ 2 2 3 (0) · · · ⎜ 0 ⎟ ⎟ ⎜ 0 −F 0 ⎜ ⎟ 4 . ⎠ ⎝ .. ⎜ ⎟ . . . . ⎝ ⎠ .. .. .. . ··· . Fn (0)qn−1 q¯n−1 0 0 0 ··· an Based on this analysis, we can observe that each gij is determined by the parameters and delay in (2.1). Thus, we can compute the following quantities: $ % g21 i 1 g20 g11 − 2|g11 |2 − |g02 |2 + c1 (0) = , 20 3 2 2 = −
Re c1 (0) , Re (0 )
2 = 2 Re c1 (0).
(4.11)
We know that (see [16]) 2 determines the directions of the Hopf bifurcation: if 2 > 0 (< 0), then the direction of the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for > 0 (< 0 ); 2
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determines the stability of the bifurcating periodic solutions on the center manifold: the bifurcating periodic solutions are orbitally asymptotically stable (unstable) if 2 < 0 (> 0). Remark. If Fj (0) = 0, then g11 = g20 = g02 = 0, and
! F3 (0) F2 (0) F1 (0) Fn (0) 2 2 2 2 |qn−1 | . |qn | + |q1 | + |q2 | + · · · + g21 = ∗ K(a1 + i0 ) Fn (0) F2 (0) F3 (0) F1 (0)
Hence 2 and 2 can be computed directly. Consequently, the direction of the Hopf bifurcation and stability of he bifurcating periodic solutions can be determined. 5. Pitchfork bifurcation In this section, we shall study the multiple stabilities of system (2.1). Initially, we make the following assumptions on Fi (x). (H1 ) Fi (x) ∈ C 2 (i = 1, 2, . . . , n) are all monotonically increasing functions, xF i (x) > 0 (i = 1, 2, . . . , n) for x = 0, and there exists an L > 0 such that |Fi (x)| L for x ∈ R. (H4 ) xF i (x) < 0 (i = 1, 2, . . . , n) for x = 0. Clearly, under the assumption (H1 ), the origin is an equilibrium of system (2.1). Theorem 5.1. Suppose (H1 ) and (H4 ) are satisfied. (i) If An > ni=1 Fi (0), then system (2.1) has only one equilibrium at the origin. n (ii) If An < i=1 Fi (0), then system (2.1) has just three equilibria (x1+ , x2+ , . . . , xn+ ), (x1− , x2− , . . . , xn− ) and the origin, where xj+ > 0 and xj− < 0 for j = 1, 2, . . . , n. Remark. The conclusions tell us that the line An =
n
i=1 Fi (0)
is a pitchfork bifurcation curve of system (2.1).
Proof. The assumption (H1 ) implies that the origin (0, 0, . . . , 0) is an equilibrium of system (2.1). Since Fi (x) ∈ C 2 (i = 1, 2, . . . , n) are all monotonically increasing functions, then Fi (x) 0. Furthermore, by the assumption xF i (x) < 0 (i = 1, 2, . . . , n) for x = 0 we have Fi (0) = maxx∈R Fi (x) (i = 1, 2, . . . , n). (x1∗ , x2∗ , . . . , xn∗ ) is an equilibrium of system (2.1) if and only if (x1∗ , x2∗ , . . . , xn∗ ) satisfies the a1 x1 = F1 xn , a2 x2 = F2 (x1 ), . . . , an xn = Fn (xn−1 ).
(5.1)
x1∗
This leads to the condition that satisfies $ $ $ $ % %%% 1 1 1 1 1 x − F1 Fn Fn−1 · · · F3 F2 (x) · · · = 0. a1 an an−1 a3 a2 Let 1 h(x) = x − F1 a1
$
1 Fn an
$
1 an−1
$ Fn−1
1 · · · F3 a3
$
(5.2)
% %%% 1 F2 (x) · · · . a2
Under the inequality An −
n
Fi (0) > 0,
i=1
by the definition of h(x) we have $ $ $ %%% 1 1 1 1 h (x) = 1 − F Fn Fn−1 · · · F2 (x) · · · A n 1 an a a2 $ $ n−1 $ $ %%% % 1 1 1 1 · Fn Fn−1 Fn−2 · · · F2 ((x)) · · · · · · · · F3 F2 (x) · F2 (x). an−1 an−2 a2 a2
(5.3)
J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
From (H1 ) we know that Fi (0) = 0 for i = 1, 2, . . . , n. Hence n j =1 Fj (0) h (0) = 1 − . An
2247
(5.4)
Observing Fi (0) = maxx∈R Fi (x) (i = 1, 2, . . . , n), we have h (x) > 0, for x ∈ R, and hence x = 0 is the unique zero of h(x). Thus the origin is the unique equilibrium of (2.1). Under the inequality An −
n
Fi (0) < 0,
(5.5)
i=1
from (5.4) it follows that h (0) < 0. Hence, there exists a neighborhood, N (0), of x =0 such that h (x) < 0, for x ∈ N (0). From h(0) = 0 we have h(x) < 0 when x ∈ N (0) and x > 0. On the other hand, from |Fi (x)| L (i = 1, 2, . . . , n), we have h(x) → +∞ when x → +∞. There exists x1+ > 0 such that h(x1+ ) = 0, and h(x) < 0 when x ∈ (0, x1+ ). So h (x1+ )0. By (H4 ), we know that Fi (x) (i = 1, 2, . . . , n) are monotonically decreasing for x > 0. Thus, from (H1 ), $ $ $ %%% 1 1 1 1 h (x) = 1 − F1 Fn Fn−1 · · · F2 (x) · · · An an a a2 $ $ n−1 $ $ %%% % 1 1 1 1 · Fn Fn−1 Fn−2 · · · F2 ((x)) · · · · · · · · F3 F2 (x) · F2 (x) > 0, an−1 an−2 a2 a2 when x > x1+ . This implies that x1+ is a unique positive zero of the function h(x). Let xi+ =
1 + Fi (xi−1 ) ai
(i = 2, . . . , n),
then (x1+ , x2+ , . . . , xn+ ) is a positive equilibrium to system (2.1), and (2.1) has no other positive equilibrium. Similarly, there exists x1− < 0 such that x1− is the unique negative zero of the function h(x). Hence (x1− , x2− , . . . , xn− ) − ) < 0 (i = 2, . . . , n), and (2.1) has no other negative is an equilibrium of system (2.1), where xi− = (1/ai )Fi (xi−1 equilibrium. The proof is complete. Corollary 5.2. Suppose that (H1 ), (H4 ) and (5.5) are satisfied. Then 1 + F (x + )F (x + ) · · · Fn (xn−1 ) < 1, a1 a2 · · · an 1 n 2 1
(5.6)
1 − F (x − )F (x − ) · · · Fn (xn−1 ) < 1, a1 a2 · · · an 1 n 2 1
(5.7)
where (x1+ , x2+ , . . . , xn+ ) and (x1− , x2− , . . . , xn− ) are defined as in Theorem 5.1. Proof. By the proof of Theorem 5.1, we know that h (x1+ ) 0 and h(x) < 0 when x ∈ (0, x1+ ). Hence, from h(x1+ ) = 0, there exists an x0 ∈ (0, x1+ ) such that h (x0 ) > 0. By the monotonically increasing property of Fi (x) and decreasing property of Fi (x) for i = 1, 2, . . . , n, we have h (x) > h (x0 ) > 0 when x > x0 . Then h (x11 ) > 0, that + ) < 1. The proof of (5.7) is similar, and we shall omit it. is (1/a1 a2 · · · an )F1 (xn+ )F2 (x1+ ) · · · Fn (xn−1 On the stability of the equilibria, we have the following results. Theorem 5.3. Suppose that (H1 ), (H2 ), (H4 ) are satisfied. (i) If (H3 ) holds, then the unique equilibrium, the origin, of system (2.1) is asymptotically stable for all 0. (ii) If (5.5) holds, then the zero solution of system (2.1) is unstable for all 0; and the equilibria (x1+ , x2+ , . . . , xn+ ) and (x1− , x2− , . . . , xn− ) are both asymptotically stable for all 0.
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Proof. (i) From Theorem 5.1 we know that the origin is the unique equilibrium of system (2.1). Then the conclusion follows from (1) of Theorem 3.3. (ii) Clearly, (5.5) implies that Eq. (2.3) with = 0 has at least one positive real root. By the increasing property of Fi (x) and (5.5), we have that Fi (0) > 0. Hence, A2n < ( ni=1 Fi (0))2 is satisfied. Applying Lemma 3.1, Eq. (2.3) with > 0 has purely imaginary roots if and only if = j . Applying Lemma 3.2, Eq. (2.3) has at least one positive real root when ∈ [0, 0 ], and at least three roots with positive real parts when > 0 . Therefore, the zero solution of system (2.1) is unstable for all 0. From Theorem 5.1 we know that (x1+ , x2+ , . . . , xn+ ) and (x1− , x2− , . . . , xn− ) are equilibria of system (2.1). Noticing that the linearization of system (2.1) around the equilibrium (x1+ , x2+ , . . . , xn+ ) is given by ⎧ y˙ (t) = −a y (t) + F (x + )y (t − ), 1 1 1 n 1 n ⎪ ⎪ ⎪ ⎨ y˙2 (t) = −a2 y2 (t) + F (x + )y1 (t), 2 1 .. ⎪ ⎪ . ⎪ ⎩ + )yn−1 (t), y˙n (t) = −an yn (t) + Fn (xn−1
(5.8)
and its characteristic equation is n + A1 n−1 + A2 n−2 + A3 n−2 + · · · + An −
n
+ Fi (xi−1 )e− = 0,
(5.9)
i=1
with x0+ = xn+ . Applying (5.6), similar to the proof of (1) in Theorem 3.3, we can obtain the second conclusion. We shall omit the detail. 6. Examples In Section 4, we have derived an algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcating periodic solutions. Even though the formula is quite long, one can compute the important quality such as Re c1 (0) for some special transfer functions. For example, consider the following network model: u˙ i (t) = −aui (t) + bi F (ui−1 (t − i )),
(6.1)
i (mod n),
where a > 0 and bi (i = 1, 2, . . . , n) are constants. In view of the properties of the function tanh(x), we make the following assumptions: (P1 ) F ∈ C 3 , xF (x) > 0 for x = 0, F (0) = 1, F (0) = 0, and F (0) = 0. Denote 1 C a 1 = na, 2 = 3n 3 C a n
1 Cn a 3 3 1 C a , = 3 n 2 2 Cn a 5 5 Cn a
1 Cn1 a 3 3 2 Cn a Cn a 2 . .. n = .. . 2n−1 2n−1 2n−2 Cn a 2n−2 Cn a n n = a − bi n−1 . i=1
1 Cn2 a 2 Cn4 a 4
0 1 , . . . , Cn3 a 3
0
0
1 .. .
0 .. .
Cn2n−3 a 2n−3
Cn2n−4 a 2n−4
··· 0 .. ··· . n n n · · · Cn a − bi
···
0
i=1
J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
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We make the following assumptions on 1 , 2 , . . . , n : 0, 2 > 0, . . . , n−1 > 0. (P2 ) 1 > (P3 ) a n > ni=1 bi . Applying Theorem 3.3 we have the following. Theorem 6.1. Suppose that (P1 ) is satisfied. (1) If a 2n > ( ni=1 bi )2 , (P2 ) and (P3 ) hold, then the zero solution of system (6.1) is asymptotically stable for all 0. (2) If a 2n < ( ni=1 bi )2 , (P2 ) and (P3 ) hold, then the zero solution of system (6.1) is asymptotically stable for ∈ [0, 0 ), and unstable when > 0 . Furthermore, system (6.1) undergoes a Hopf bifurcation at the origin when = j (j = 0, 1, . . .). (3) If a 2n > ( ni=1 bi )2 , and either (P2 ) or n > 0 is not satisfied, then the zero solution of system (6.1) is unstable for all 0. Here ⎧ 2 n−2 n−2 +a n 2 2 (−1)m n0 +(−1)m−1 n−2 0 Cn a +···−0 Cn a ⎨ 1 (arccos + 2j ), n = 2m, n 0 i=1 bi j = n−2 ⎩ 1 (−1)m n0 +(−1)m−1 0 Cn2 a 2 +···−30 Cnn−3 a n−3 +0 Cnn−1 a n−1 + 2j ), n = 2m + 1, 0 (arcsin − ni=1 bi √ j = 0, 1, . . . , and 0 = z0 , z0 is the unique positive zero of the function n 2 n n−1 n−2 h(z) = z + B2 z + B4 z + · · · + B2n−2 z + B2n − bi , ⎡
B2 = a 2 ⎣n2 + ⎡
⎤
⎡
j (−1)i+1 2Cni Cn ⎦ , B4
i+j =2,i=j
B6 = a 6 ⎣(Cn3 )2 + ⎡
= a 4 ⎣(Cn2 )2 +
⎤ j (−1)i+2 2Cni Cn ⎦ ,
i+j =4,i=j
⎤ (−1)i+3 2Cni Cn ⎦ , . . . , j
i+j =6,i=j
B2n−2 = a 2n−2 ⎣(Cnn−1 )2 +
i=1
⎤ j (−1)i+n 2Cni Cn ⎦ ,
B2n = a 2n , i, j = 0, 1, . . . , n.
i+j =2n−2,i=j
Applying Theorems 5.1 and 5.3, we have the following. Theorem 6.2. Suppose that xF (x) > 0 and xF (x) < 0 for x = 0, F (x) > 0 and F (0) = 1, bi > 0 (i = 1, 2, . . . , n) are satisfied. (i) If a n > nj=1 bj , then the model (6.1) has only one equilibrium at the origin, and the origin is asymptotically stable for all 0. (ii) If a n < nj=1 bj , then the model (6.1) has just three equilibria (x1+ , x2+ , . . . , xn+ ), (x1− , x2− , . . . , xn− ) and the origin, and the origin is unstable, the other two equilibria are both asymptotically stable for all 0, where xj+ > 0 and xj− < 0 solve % % $ $ b1 bn−1 bn bn b2 ± ± ± ± ± x1 = F = ) and xn± = F (xn−1 ), ···F F (x1 ) · · · , . . . , xn−1 F (xn−2 a a a a a respectively. The conclusions of Theorems 6.1 and 6.2 are illustrated in Fig. 1.
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J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
b b = an G3
G2
a
G1 b = −an
Fig. 1. The bifurcation set. Here we set b = nj=1 bj . The curves b = ±a n divide the right half plane into three regions: G1 is a conditional stable region; G2 is an absolutely stable region; and G3 is an unstable region. b = a n is a pitchfork bifurcation curve.
Now, let the conditions of (2) in Theorem 6.1 be satisfied. From F (0) = 0 and the formulae on gij derived in Section 4, we have g11 = g20 = g02 = 0, and
g21 = ∗ F (0)
(a + i0 )(n + ∗ a − i∗ 0 )
1+
(n + ∗ a)2 + 2∗ 20
b22 a 2 + 20
+
b22 b32 (a 2 + 20 )2
+ ··· +
b22 b32 · · · bn2
.
(a 2 + 20 )n−1
It follows that Re c1 (0) = Re
g21 2
= ∗ F (0)
[a(n + ∗ a) + ∗ 20 ] (n + ∗ a)2 + 2∗ 20
1+
b22 a 2 + 20
+
b22 b32 (a 2 + 20 )2
+ ··· +
b22 b32 · · · bn2 (a 2 + 20 )n−1
.
Hence, from 2 =2 Re c1 (0) and 2 =−Rec1 (0)/Re (∗ ), and applying Lemma 3.2, we have that sign 2 =sign F (0) and sign 2 = −sign F (0). Consequently, we have the following. Theorem 6.3. Suppose that the conditions of (2) in Theorem 6.1 are satisfied. If F (0) > 0 (< 0), then, at each bifurcation value j , the direction of the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions on the center manifold are orbitally asymptotically stable (unstable). Particularly, the stability of the bifurcating periodic solutions in the phase space and on the center manifold are coincidence at the first bifurcation value 0 . To illustrate the analytical results found, let us consider the following special case of Eq. (6.1): ⎧ ⎨ u˙1 (t) = −au1 (t) + b1 tanh(u3 (t − 1 )), (6.2) u˙ (t) = −au2 (t) + b2 tanh(u1 (t − 2 )), ⎩ 2 u˙3 (t) = −au3 (t) + b3 tanh(u1 (t − 3 )). √ . Let a = 1, b1 = −2 2, b2 = b3 = 1, then 0 = 0.782, and 0 = 1. Under this group of data, from Theorem 6.3, . we have that the origin is asymptotically stable when ∈ [0, 0.782), and unstable when > 0 = 0.782, the Hopf bifurcation is supercritical at 0 , as well as the bifurcating periodic solutions are asymptotically stable. . . In Fig. 2, = 0.6 < 0 = 0.782, we show that the origin is asymptotically stable. In Fig. 3, = 1 > 0 = 0.782, we . show that the origin is unstable, and a periodic solution bifurcates from the origin for > 0 = 0.782, as well as the bifurcating periodic solution is asymptotically stable.
J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
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0.1 0.05 0 −0.05 −0.1 0.2
0.1
0 −0.1 −0.2
−0.2
−0.1
0.1
0
0.2
√ Fig. 2. A Matlab simulation of the asymptotically stable origin to system (6.2) with a = 1, b1 = −2 2, b2 = b3 = 1, 1 = 2 = 3 = 0.2, and . = 3i=1 i = 0.6 < 0 = 0.782.
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 1 0.5 0
−0.5 −1
−1
−0.5
0
0.5
1
√ Fig. 3 3. A Matlab .simulation of a periodic solution to system (6.2) with a = 1, b1 = −2 2, b2 = b3 = 1. 1 = 2 = 0.3, 3 = 0.4, and i=1 i = 1 > 0 = 0.782.
Acknowledgment We wish to thank the reviewer for his or her valuable comments and suggestions that led to truly improvement of the manuscript. References [1] S. Arik, Global asymptotic stability analysis of bidirectional associative memory neural networks with time delays, IEEE Trans. Neural Networks 16 (2005) 580–586. [2] K.L. Babcock, R.M. Westervelt, Dynamics of simple electronic neural networks, Physica D: Nonlinear Phenom. 28 (1987) 305–359. [3] P. Baldi, A.F. Atiya, How delays affect neural dynamics and learning, IEEE Trans. Neural Networks 5 (1994) 612–621. [4] D.W. Berns, J.L. Moiola, G. Chen, Predicting period-doubling bifurcations and multiple oscillations in nonlinear time-delayed feedback systems, IEEE Trans. Circuits Syst. I—Fund. Theory Appl. 45 (1998) 759–763. [5] S.A. Campbell, Stability and bifurcation of a simple neural network with multiple time delays, Fields Inst. Commun. 21 (1999) 65–79. [6] S.A. Campbell, I. Ncube, J. Wu, Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system, Physica D: Nonlinear Phenom. 214 (2006) 101–119. [7] J. Cao, L. Wang, Exponential stability and periodic oscillatory solution in BAM networks with delays, IEEE Trans. Neural Networks 13 (2002) 457–463.
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[8] J. Cao, J. Wang, Global asymptotic stability of a general class of recurrent neural networks with time-varying delays, IEEE Trans. Circuits Syst. I 50 (2003) 34–44. [9] Y. Chen, J. Wu, Slowly oscillating periodic solutions for a delayed frustrated network of two neurons, J. Math. Anal. Appl. 259 (2001) 188–208. [10] T. Faria, On a planar system modelling a neuron network with memory, J. Differential Equations 168 (2001) 129–149. [11] F. Giannakopoulos, A. Zapp, Bifurcations in a planar system of differential delay equations modeling neural activity, Physica D: Nonlinear Phenom. 159 (2001) 215–232. [12] S. Guo, L. Huang, Periodic solutions in an inhibitory two-neuron network, J. Comput. Appl. Math. 161 (2003) 217–229. [13] S. Guo, L. Huang, Stability analysis of Cohen–Grossberg neural networks, IEEE Trans. Neural Networks 17 (2006) 106–117. [14] S. Guo, L. Huang, J. Wu, Regular dynamics in a delayed network of two neurons with all-or-none activation functions, Physica D: Nonlinear Phenom. 206 (2005) 32–48. [15] J. Hale, Theory of Functional Differential Equations, Springer, Berlin, 1977. [16] B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. [17] J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proc. Natl. Acad. Sci. USA 81 (1984) 3088–3092. [18] H. Jiang, L. Zhang, Z. Teng, Existence and global exponential stability of almost periodic solution for cellular neural networks with variable coefficients and time-varying delays, IEEE Trans. Neural Networks 16 (2005) 1340–1351. [19] C. Li, G. Chen, Local stability and bifurcation in small-world delayed networks, Chaos Solitons Fractals 20 (2004) 353–361. [20] X. Li, J. Wei, Stability and bifurcation analysis on a delayed neural network model, Int. J. Bifur. Chaos 15 (2005) 2883–2893. [21] X. Liao, S. Li, G. Chen, Bifurcation analysis on a two-neuron system with distributed delays in the frequency domain, Neural Networks 17 (2004) 545–561. [24] X. Liao, J. Yu, G. Chen, Novel stability criteria for bidirectional associative memory neural networks with time delays, Int. J. Circuit Theory Appl. 30 (2002) 519–546. [25] H. Lu, R. Shen, C. Fu, Global exponential convergence of Cohen–Grossberg neural networks with time delays, IEEE Trans. Neural Networks 16 (2005) 1694–1696. [26] S. Ruan, R.S. Filfil, Dynamics of a two-neuron system with discrete and distributed delays, Physica D: Nonlinear Phenom. 191 (2004) 323–342. [27] S. Ruan, J. Wei, Periodic solutions of planar systems with two delays, Proc. R. Soc. Edinburgh 129A (1999) 1017–1032. [28] S. Ruan, J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Continuous Discrete Impulsive Syst. Ser. A: Math. Anal. 10 (2003) 863–874. [29] L. Wang, X. Zou, Harmless delays in Cohen–Grossberg neural networks, Physica D: Nonlinear Phenom. 170 (2002) 162–173. [30] L. Wang, X. Zou, Hopf bifurcation in bidirectional associative memory neural networks with delays: analysis and computation, J. Comput. Appl. Math. 167 (2004) 73–90. [31] J. Wei, M.Y. Li, Global existence of periodic solutions in a tri-neuron network model with delays, Physica D: Nonlinear Phenom. 198 (2004) 106–119. [32] J. Wei, S. Ruan, Stability and bifurcation in a neural network model with two delays, Physica D: Nonlinear Phenom. 130 (1999) 255–272. [33] J. Wei, M. Velarde, Bifurcation analysis and existence of periodic solutions in a simple neural network with delays, Chaos 14 (2004) 940–953. [34] J. Wei, Y. Yuan, Synchronized Hopf bifurcation analysis in a neural network model with delays, J. Math. Anal. Appl. 312 (2005) 205–229. [35] J. Wei, C. Zhang, X. Li, Bifurcation in a two-dimensional neural network model with delay, Appl. Math. Mech. 26 (2005) 210–217. [37] J. Wu, Introduction to Neural Dynamics and Signal Transmission Delay, Walter de Gruyter, Berlin, New York, 2001. [38] X.P. Yan, Hopf bifurcation and stability for a delayed tri-neuron network model, J. Comput. Appl. Math. 196 (2006) 579–595. [39] Z. Yi, K.K. Tan, Multistability analysis of discrete-time recurrent neural networks with unsaturating piecewise linear activation functions, IEEE Trans. Neural Networks 15 (2004) 329–336. [40] Z. Yi, K.K. Tan, T.H. Lee, Multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions, Neural Comput. 15 (2003) 639–662. [41] Y. Yuan, Dynamics in a delayed-neural network, Chaos Solitons Fractals 33 (2) (2007) 443–454. [42] Z. Zhang, S. Guo, Periodic oscillation for a three-neuron network with delays, Appl. Math. Lett. 16 (2003) 1251–1255.
Bifurcation analysis of a class of neural networks with delays夡 Junjie Weia,∗ , Chunrui Zhangb a Department of Mathematics, Harbin Institute of Technology, Harbin 150001, PR China b Department of Mathematics, Northeast Forestry University, Harbin 150040, PR China
Received 21 February 2007; accepted 3 August 2007
Abstract A class of n-dimensional neural network model with multi-delay is considered. Firstly, a group of sufficient conditions for the existence of Hopf bifurcation are obtained via employing the polynomial theorem to analyze the distribution of the roots of the associated characteristic equation. Secondly, the stability and direction of the Hopf bifurcation are determined by applying the normal form method and center manifold theorem. Finally, the multiple stability is investigated and pitchfork bifurcation is also found. The results are illustrated by some numerical simulations. 䉷 2007 Elsevier Ltd. All rights reserved. Keywords: Neural network; Delay; Hopf bifurcation; Pitchfork bifurcation
1. Introduction Hopfield [17] proposed a simplified neural network model in which each neuron is represented by a linear circuit consisting of a resistor and capacitor, which is connected to the other neurons via nonlinear sigmoidal activation functions instantaneously. In reality, neural networks often have time delays. For example, it is due to the finite switching speed of amplifiers in electronic neural networks, or due to finite signal propagation time in biological networks. After Hopfield’s work, many neural network models with delays have been proposed and studied, we refer to Wu [37]. Baldi and Atiya [3] particularly constructed a network model that consists of a ring of neurons connected cyclically with delayed interactions. Different delays are introduced for the communication between the adjacent neurons. This leads to the system dui ui = − + Tii−1 fi−1 (ui−1 (t − ii−1 )), dt Ti
i (mod n).
(1.1)
Campbell [5] generalized Baldi and Atiya’s model (1.1) to a network that consists of a ring of neurons where the j th element receives two time delayed inputs: One from the current element itself and the other from the previous element. In the last few years, in order to obtain a deep and clear understanding of the dynamics of the model, 夡 This research was supported by the National Natural Science Foundations of China and Young Innovative Programme of Northeast Forestry University. ∗ Corresponding author. Tel.: +86 451 86417440; fax: +86 451 86412549. E-mail address: [email protected] (J. Wei).
1468-1218/$ - see front matter 䉷 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.nonrwa.2007.08.008
J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
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several researchers have focused on the bifurcation analysis in the model (1.1) or Campbell’s model with n = 2 and 3. For n = 2, we refer to [2,4,9–12,14,19,21,26,27,32,35], for n = 3, see [6,31,33,38,42], and for n = 4, see [20]. Another important models such as Cohen–Grossberg and BAM neural networks with delays have been studied extensively, we refer to [1,7,8,13,18,24,25,29,30] and references therein for stability analysis and existence of periodic solutions by constructing Liapunov functions. For Campbell’s models with delays, there are several articles on synchronized Hopf bifurcation analysis, we refer to [34,41]. The purpose of the present paper is to study the model (1.1). For convenience, we rewrite it in the following form: ⎧ u˙ 1 (t) = −a1 u1 (t) + F1 (un (t − 1 )), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ u˙ 2 (t) = −a2 u2 (t) + F2 (u1 (t − 2 )), .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎩ u˙ n (t) = −an un (t) + Fn (un−1 (t − n )).
(1.2)
Employing the Routh–Hurwitz criterion and the result on distribution of the zeros of transcendental functions due to Ruan and Wei [28], we get a set of conditions to describe the stability of the fixed point of model (1.2) and the existence of Hopf bifurcations. Furthermore, we derive an algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcating periodic solutions by using the normal form method and center manifold theorem introduced by Hassard et al. [16]. We particularly obtain the explicit conclusions on the Hopf bifurcation properties when the transfer functions Fj (u) is similar to tanh(u). Meanwhile, we investigate the multiple stability and the pitchfork bifurcation to the model (1.2) is also found. We would like to mention that there are several papers related to multi-stability of neural networks with delay, see [6,35,39,40]. Our paper is organized as follows: In Section 2, we present some preliminary results which will be used later to analyze the distribution of the roots of the characteristic equation associated with system (1.2). In Section 3, we analyze the distribution of the eigenvalues and obtain the stability of the fixed point and existence of Hopf bifurcations. In Section 4, an algorithm for determining the properties of Hopf bifurcations is derived by using the normal form method and center manifold theorem. Section 5 deals with the multiple stability. An example is considered and some numerical simulations are carried out in Section 6 to support the analysis results. 2. Preliminary results In this section, we provide some preliminary results to be used in the subsequent section. Let x1 (t) = u1 (t − (2 + 3 + · · · + n )),
x2 (t) = u2 (t − (3 + 4 + · · · + n )),
.. . xn−1 (t) = un−1 (t − n ) and
xn (t) = un (t).
Then Eq. (1.2) becomes the following equivalent system: ⎧ x˙1 (t) = −a1 x1 (t) + F1 (xn (t − )), ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x˙2 (t) = −a2 x2 (t) + F2 (x1 (t)), .. ⎪ ⎪ . ⎪ ⎪ ⎪ ⎩ x˙n (t) = −an xn (t) + Fn (xn−1 (t)), where = 1 + 2 + · · · + n . We make the following assumptions. (H1 ) For i = 1, 2, . . . , n, constants ai > 0, Fi ∈ C 2 and Fi (0) = 0.
(2.1)
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J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
From the assumption it follows that the origin (0, 0, . . . , 0) is an equilibrium of Eq. (2.1). The linearization of Eq. (2.1) around the origin is given by ⎧ x˙ (t) = −a x (t) + F (0)x (t − ), 1 1 1 n 1 ⎪ ⎪ ⎪ ⎪ (t) + F (0)x x ˙ (t) = −a x ⎨ 2 1 (t), 2 2 2 (2.2) .. ⎪ ⎪ . ⎪ ⎪ ⎩ x˙n (t) = −an xn (t) + Fn (0)xn−1 (t), whose characteristic equation is n +
n
ai n−1 +
1 i
i=1
+
n
ai −
n
ai aj n−2 +
ai aj ak n−3 + · · · +
ai aj ak · · · al
1 i
1 i
Fi (0)e− = 0.
i=1
i=1
Denote A1 =
n
n
i=1 ai .
ai aj ak , . . . , An−1 =
1 i
ai aj ak · · · al ,
1 i
Then the characteristic equation becomes
+ A1 n
ai aj , A3 =
1 i
i=1
and An =
ai , A2 =
n−1
+ A2
n−2
+ A3
n−2
+ · · · + An −
n
Fi (0)e− = 0.
(2.3)
i=1
Lemma 2.1. For a1 , a2 , . . . , an ∈ R, let Aj (j = 1, . . . , n) be defined as above, and k ∈ N, 2k n. Then A2k − 2Ak−1 Ak+1 + 2Ak−2 Ak+2 − · · · + (−1)k 2A2k = ai21 ai22 · · · ai2k .
(2.4)
1 i1
· · · ai2k is a symmetric homogeneous polynomial of a1 , a2 , . . . , an , and its
principle term (according to dictionary order) is (22 · · · 2 00000). Let ai21 ai22 · · · ai2k = bk A2k + bk−1 Ak−1 Ak+1 + bk−2 Ak−2 Ak+2 + · · · + b0 A2k . (2.5) Proof. Clearly,
2 2 1 i1
1 i1
By the polynomial theorem, comparing the coefficients of a12 a22 · · · ak2 , we have that bk = 1. Similarly, by using 2 a a 2 2 2 coefficients comparison of a12 a22 · · · ak−1 k k+1 , and of a1 a2 · · · ak−2 ak−1 ak ak+1 ak+2 , we obtain bk−1 = −2 and bk−2 = 2, respectively. By the induction, we assume that bk−l = (−1)l 2 for l 1. 2 Using coefficient comparison of a12 a22 · · · ak−(l+1) ak−l ak−l+1 · · · ak+l+1 , it follows that l+1 l−1 l l C2(l+1) − 2C2(l+1) + 2C2(l+1) − · · · + (−1)l 2C2(l+1) + bk−(l+1) = 0.
Thus,
2(l+1) 2(l+1) 0 1 2 0 bk−(l+1) = (−1)l C2(l+1) − C2(l+1) + C2(l+1) − · · · + C2(l+1) + (−1)l+1 C2(l+1) + C2(l+1) = (−1)l (1 − 1)2(l+1) + (−1)l+1 2 = (−1)l+1 2.
This completes the proof.
J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
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3. Stability analysis In this section we shall analyze the distribution of the roots of Eq. (2.3) to discuss the stability and existence of Hopf bifurcation of system (2.1). Lemma 3.1. (1) If A2n > ( ni=1 Fi (0))2 , then Eq. (2.3) has no roots with zero real parts for all 0. (2) If A2n < ( ni=0 Fi (0))2 , then Eq. (2.3) has a pair of purely imaginary roots ±i0 when = j , j = 0, 1, 2, . . . , where ⎧ 1 n−2 A +···−2 A (−1)m n +(−1)m−1 2 n−2 +An + 2j , n = 2m, ⎨ 0 arccos n i=1 Fi (0) j = j = 0, 1, . . . , (3.1) n−2 A +···−3 A ⎩ 1 (−1)m n +(−1)m−1 2 n−3 +An−1 + 2j , n = 2m + 1, n 0 arcsin − i=1 Fi (0) √ and 0 = z0 , z0 is the unique positive zero of the function n 2 n n−1 n−2 + B4 z + · · · + B2n−2 z + B2n − Fi (0) , g(z) = z + B2 z i=1
with
B2 = A21 +
(−1)i+1 2Ai Aj ,
i+j =2,i=j
B6 = A23 +
(−1)i+2 2Ai Aj ,
i+j =4,i=j
(−1)i+3 2Ai Aj , . . . ,
i+j =6,i=j
B2n−2 = A2n−1 +
B4 = A22 +
(−1)i+n 2Ai Aj
and
B2n = A2n , i, j = 0, 1, . . . , n, A0 = 1.
i+j =2n−2,i=j
Proof. Let i( > 0) be a root of Eq. (2.3), then n
in n + in−1 n−1 A1 + in−2 n−2 A2 + · · · + iAn−1 + An =
Fi (0)e−i .
i=1
Separating the real and imaginary parts gives, when n = 2m, ⎧ n m n m−1 n−2 ⎪ A2 + · · · − 2 An−2 + An = Fi (0) cos , ⎨ (−1) + (−1) i=1
n ⎪ ⎩ (−1)m+1 n−1 A1 + (−1)m n−3 A3 + · · · − 3 An−3 + An−1 = − F (0) sin , i
(3.2)
i=1
and when n = 2m + 1, ⎧ n m n m−1 n−2 ⎪ A2 + · · · − 3 An−3 + An−1 = − Fi (0) sin , ⎨ (−1) + (−1) i=1
n ⎪ ⎩ (−1)m n−1 A1 + (−1)m−1 n−3 A3 + · · · − 2 An−2 + An = F (0) cos . i
(3.3)
i=1
Hence,
2n
+ B2
2n−2
+ B4
2n−4
+ · · · + B2n−2 + B2n = 2
n
2 Fi (0)
.
(3.4)
i=1
By Lemma 2.1, B2k > 0, k = 0, 1, . . . , n. Denote g(z) = z + B2 z n
n−1
+ B4 z
n−2
+ · · · + B2n−2 z + B2n −
n i=1
2 Fi (0)
.
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J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
n
(1) From A2n > (
2 i=1 Fi (0)) ,
g(0) = B2n −
n
it follows that
2 Fi (0)
> 0.
i=1
Hence, by g (z) = nzn−1 + B2 (n − 1)zn−2 + · · · + B2n−2 > 0
for z > 0,
the function g(z) has no positive zero. This implies that Eq. (3.4) has no real root, and hence Eq. (2.3) has no purely imaginary root. Clearly of (1) is complete. = 0 is not a root of Eq. (2.3). The proof (2) From A2n < ( ni=1 Fi (0))2 , it follows that g(0) = B2n − ( ni=1 Fi (0))2 < 0. Then, by limz→+∞ g(z) = +∞, and g (z) > 0 for z > 0, there exists a unique positive number z0 such that g(z0 ) = 0. √ Let 0 = z0 , where z0 is the positive root of the equation g(z) = z + B2 z n
n−1
+ B4 z
n−2
+ · · · + B2n −
n
2 Fi (0)
= 0.
(3.5)
i=1
Then 0 is a root of Eq. (3.4). Define j as (3.1). Hence, (0 , j ) is a root of (3.2) or (3.3). This implies that ±i0 is a pair of purely imaginary roots of Eq. (2.3) when = j (j = 0, 1, . . .). The proof of (2) is complete. Now let us consider the behavior of the roots of Eq. (2.3) near the values = j (j = 0, 1, . . .). To do this we assume that () = () + i() is a solution of Eq. (2.3) satisfying (j ) = 0 and (j ) = 0 . Lemma 3.2. d/d|=j > 0. Proof. For convenience, set P () = n + A1 n−1 + A2 n−2 + A3 n−2 + · · · + An . Then Eq. (2.3) is rewritten in the following form: P () =
n
Fi (0)e− .
i=1
Differentiating both sides of Eq. (2.3) with respect to gives ni=1 Fi (0)e− d() =− . d P () + ni=1 Fi (0)e− Replacing by j , and noting P () = Re
n
− , i=1 Fi (0)e
it follows that
20 g (z0 ) d = > 0, d =j =0
where =
nn−1 0
− (n − 2)n−3 0 A2
+ · · · − 20 An−2 ∓ 0
2 Fi (0) sin 0
i=1
+
n
(n − 1)n−2 0 A1
− (n − 3)n−4 0 A3
+ · · · − An−1 ∓ 0
n i=1
2 Fi (0) cos 0
J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
or
=
nn−1 0
− (n − 2)n−3 0 A2
+ · · · − 20 An−2 ± 0
2 Fi (0) cos 0
i=1
+
n
(n − 1)n−2 0 A1
− (n − 3)n−4 0 A3
2239
+ · · · − An−1 ± 0
n
2 Fi (0) sin 0
.
i=1
For A1 , A2 , . . . , An , we define A1 1 = A1 , 2 = A3 A1 A 3 . n = . . A 2n−1
A1 1 , 3 = A3 A2 A 5
1 A2 A4
0 A1 , . . . , A 3
1
0
0
···
0
A2 .. .
A1 .. .
1 .. .
···
A2n−2
A2n−3
A2n−4
0 .. . n
··· · · · An −
i=1
n = An − Fi (0) n−1 , i=1 Fi (0)
and make the following assumptions on 1 , 2 , . . . , n . (H2 ) 1 > 0, 2 > 0, . . . , n−1 > 0. (H3 ) An > ni=1 Fi (0). Theorem 3.3. Suppose that (H1 ) is satisfied. (1) If A2n > ( ni=1 Fi (0))2 , (H2 ) and (H3 ) hold, then the zero solution of system (2.1) is asymptotically stable for all 0. (2) If A2n < ( ni=1 Fi (0))2 , (H2 ) and (H3 ) hold, then the zero solution of system (2.1) is asymptotically stable for ∈ [0, 0 ), and unstable when > 0 . Furthermore, system (2.1) undergoes a Hopf bifurcation at the origin when = j (j = 0, 1, . . .). Here = j is defined as (3.1). (3) If A2n > ( ni=1 Fi (0))2 , and either (H2 ) or n > 0 is not satisfied, then the zero solution of system (2.1) is unstable for all 0. Proof. When = 0, Eq. (2.3) becomes n + A1 n−1 + A2 n−2 + A3 n−2 + · · · + An −
n
Fi (0) = 0.
(3.6)
i=1
(1) By the Routh–Hurwitz criterion, all the roots of Eq. (3.6) have negative real parts if and only if (H2 ) and (H3 ) are satisfied. Hence, all roots of Eq. (2.3) with = 0 have negative real parts. From the conclusion (1) in Lemma 3.1, we know that Eq. (2.3) has no root with zero real part for all > 0. Applying the Corollary 2.4 in Ruan and Wei [29], all roots of Eq. (2.3) have negative real parts for all 0. Hence, the conclusion of (1) follows. (2) Similarly, all the roots of Eq. (2.3) have negative real parts when ∈ [0, 0 ). Applying Lemma 3.2 we know that Eq. (2.3) has at least a couple of roots with positive real parts when > 0 . Hence the first conclusion of (2) follows. The second conclusion of (2) follows from Lemmas 3.1 and 3.2, and the Hopf bifurcation theorem for functional differential equations [15, Theorem 1.1, pp. 246–247]. (3) From the (1) in Lemma 3.1 we know that, under the inequality A2n > ( ni=1 Fi (0))2 , Eq. (2.3) has no roots with zero real part for all 0. Hence Eq. (3.6) has at least one root with positive real part by the Routh–Hurwitz criterion when either (H2 ) or n > 0 does not hold. Applying Corollary 2.4 in Ruan and Wei [29], Eq. (2.3) has at least one root with positive real part. Hence the conclusion of (3) follows. The proof is complete.
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4. Direction and stability of the Hopf bifurcation In Section 3, we have obtained some conditions which guarantee that system (2.1) undergoes the Hopf bifurcation at a sequence values of . In this section, we shall study the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions. The method we used is based on the normal form theory and the center manifold theorem introduced by Hassard et al. [16]. For convenience, let us take xi (t) = xi (t) and = ∗ + , where ∗ = j for fixed j, and ∈ R. Then system (2.1) becomes ⎧ x˙1 (t) = (∗ + )[−a1 x1 (t) + F1 (xn (t − 1))], ⎪ ⎪ ⎪ ⎪ ⎨ x˙2 (t) = (∗ + )[−a2 x2 (t) + F2 (x1 (t))], (4.1) .. ⎪ ⎪ . ⎪ ⎪ ⎩ x˙n (t) = (∗ + )[−an xn (t) + Fn (xn−1 (t))]. The characteristic equation associated with the linearization of Eq. (4.1) around the origin is n n n−1 2 n−2 3 n−2 n − + A1 (∗ + ) = 0, + A2 (∗ + ) + A3 (∗ + ) + · · · + (∗ + ) An − Fi (0)e i=1
(4.2) where the definitions of Aj (j = 1, 2, . . .) are same as in Eq. (2.3). Comparing Eq. (2.3) with Eq. (4.2), it follows that = (∗ + ). Thus, from the discussion above, we know that Eq. (4.1) undergoes a Hopf bifurcation at the origin when = 0, and ±i∗ 0 are simple roots of Eq. (4.2). Let () be the root of Eq. (4.2) satisfying Re (0) = 0 and Im (0) = ∗ 0 . Then Re (0) = ∗ (∗ ) > 0.
In order to study the direction of the Hopf bifurcation and stability of the bifurcating periodic solutions, we need to assume that Fi ∈ C 3 (i = 1, 2, . . . , n), further. Then Eq. (4.1) can be rewritten as ⎧ x˙1 (t) = (∗ + )[−a1 x1 (t) + F1 (0)xn (t − 1) + 21 F1 (0)xn2 (t − 1) + 16 F1 (0)xn3 (t − 1) + O(xn4 )], ⎪ ⎪ ⎪ ⎪ ⎨ x˙2 (t) = (∗ + )[−a2 x2 (t) + F2 (0)x1 (t) + 21 F2 (0)x12 (t) + 16 F2 (0)x13 (t) + O(x14 )], (4.3) ⎪ .. ⎪ . ⎪ ⎪ ⎩ 2 (t) + 1 F (0)x 3 (t) + O(x 4 )]. x˙n (t) = (∗ + )[−an xn (t) + Fn (0)xn−1 (t) + 21 Fn (0)xn−1 n−1 n−1 6 n Choosing the phase space as C = C([−1, 0], R n ). For ∈ C, let Lv = B1 (0) + B2 (−1), where
⎛ −a 1 ⎜ F2 (0) ⎜ 0 B1 = (∗ + ) ⎜ ⎜ . ⎝ . . 0 ⎛ 0 0 ⎜0 0 ⎜ ⎜0 0 B2 = (∗ + ) ⎜ ⎜0 0 ⎜. . ⎝ .. .. 0
0
0 0 −a2 0 F3 (0) −a3 .. .. . . 0 0 0 0 0 0 .. . 0
··· ··· ··· ···
0 0 0 0 . · · · .. ··· 0
··· ··· ···
0 0 0 .. ··· . · · · Fn (0) ⎞ F1 (0) 0 ⎟ ⎟ 0 ⎟ ⎟ 0 ⎟ .. ⎟ . ⎠ 0
0 ⎞ 0 ⎟ 0 ⎟ ⎟, .. ⎟ ⎠ . −an
J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
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and ⎛ ( + v)[ 1 F (0) 2 (−1) + 1 F (0) 3 (−1)) + O(| |4 )] ⎞ ∗ n n 2 1 6 1 ⎜ ( + v)[ 1 F (0)) 2 (0) + 1 F (0) 3 (0) + O(| |4 )] ⎟ ∗ ⎜ ⎟ 1 1 6 2 2 2 ⎟. F˜ (v, ) = ⎜ ⎜ ⎟ .. ⎝ ⎠ . 2 3 1 1 4 (∗ + v)[ 2 Fn (0) n−1 (0) + 6 Fn (0) n−1 (0) + O(| | )] By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions ( , ): [−1, 0] → R n , such that L =
0 −1
d ( , ) ( ).
In fact, we choose ⎧ ⎨ −B2 , = −1, ( , ) = 0, ∈ (−1, 0), ⎩ B1 , = 0, the last equation is satisfied. For ∈ C 1 ([−1, 0], R n ), define ⎧ ⎨ d ( ) , ∈ [−1, 0), d A() = ⎩ 0 −1 d (t, ) (t), = 0, and
R() =
0, ∈ [−1, 0), F˜ (, ), = 0.
Hence (4.3) can be rewritten in the following form: u˙ t = A()ut + R()ut ,
(4.4)
where u = (u1 , u2 , . . . , un )T ,
ut = u(t + ), ∈ [−1, 0].
For ∈ C 1 ([0, 1], R n ) define ⎧ ⎨ − d (s) , s ∈ (0, 1], ∗ ds A (s) = ⎩ 0 −1 d (t, 0) (−t), s = 0. For ∈ C([−, 0], C n ) and ∈ C([0, 1], (C n )∗ ), define the bilinear form ¯
, = (0) (0) −
0
=−1 =0
¯ − ) d ( ) () d,
(
where ( ) = ( , 0), then A = A(0) and A∗ are adjoint operators. By the results of above, ± i0 ∗ are eigenvalues of A(0), thus they are also eigenvalues of A∗ .
(4.5)
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By direct computation, we obtain that q( ) = q0 ei0 ∗ , with ⎛ ⎞ 1 ⎜ ⎟ F2 (0) ⎜ ⎟ ⎜ ⎟ a2 + i0 ⎜ ⎟ ⎜ ⎟ F3 (0)F2 (0) ⎜ ⎟ ⎜ ⎟ q0 = ⎜ ⎟ (a + i )(a + i ) 3 0 2 0 ⎜ ⎟ ⎜ ⎟ . ⎜ ⎟ .. ⎜ ⎟ ⎝ ⎠ Fn (0) · · · F3 (0)F2 (0) (an + i0 ) · · · (a3 + i0 )(a2 + i0 ) is the eigenvector of A corresponding to i0 ∗ , and q ∗ (s) = q0∗ ei0 ∗ s with ⎛
⎞T 1 a1 − i0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ F2 (0) ⎜ ⎟ ⎜ ⎟ (a2 − i0 )(a1 − i0 ) ⎜ ⎟ ∗ ⎜ ⎟ ¯ q0 = K ⎜ F3 (0)F2 (0) ⎟ ⎜ ⎟ ⎜ ⎟ . ⎜ ⎟ .. ⎜ ⎟ ⎝ (an−1 − i0 ) · · · (a2 − i0 )(a1 − i0 ) ⎠ Fn (0) · · · F3 (0)F2 (0) is the eigenvector of A∗ corresponding to −i0 ∗ , where −1 n 1 ∗ K = (a1 + i0 ) + . ai + i0 i=1
q ∗ , q = 1
and q ∗ , q ¯ = 0. Moreover, Using the same notation as in [16], we compute the coordinates to describe the center manifold C0 at v = 0. Let ut be the solution of Eq. (4.4) when v = 0. Define z(t) = q ∗ , ut , W (t, ) = ut ( ) − 2 Re{z(t)q( )}. On the center manifold C0 we have W (t, ) = W (z(t), z¯ (t), ), where W (z, z¯ , ) = W20 ( )
z3 z2 z¯ 2 + W11 ( )z¯z + W02 ( ) + W30 + · · · , 2 2 6
z and z¯ are local coordinates for center manifold C0 in the direction of q ∗ and q¯ ∗ . Note that W is real if ut is also real. We only consider real solutions. For solution ut ∈ C0 of (4.4), since v = 0, z˙ = i∗ 0 z + q ∗ ( ), F˜ (W + 2 Re{z(t)q( )}) = i∗ 0 z + q¯ ∗ (0)F˜ (W (z, z¯ , 0) + 2 Re{z(t)q( )}) = i∗ 0 z + q¯ ∗ (0)F˜0 (z, z¯ ).
def
We rewrite this as z˙ (t) = i∗ 0 z(t) + g(z, z¯ ),
(4.6)
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where g(z, z¯ ) = q¯ ∗ (0)F˜ (W (z, z¯ , 0) + 2 Re{z(t)q( )}) z¯ 2 z2 z¯ z2 + ···. = g20 + g11 z¯z + g02 + g21 2 2 2 By (4.4) and (4.6), we have
(4.7)
W˙ = u˙ t − z˙ q − z˙¯ q¯ AW − 2 Re{q¯ ∗ (0)F˜0 q( )}, ∈ [−1, 0), = AW − 2 Re{q¯ ∗ (0)F˜0 q( )} + F˜0 , = 0 def
= AW + H (z, z¯ , ),
where z2 z¯ 2 + H11 ( )z¯z + H02 ( ) + · · · . 2 2 Expanding the above series and comparing the coefficients, we obtain H (z, z¯ , ) = H20 ( )
(4.8)
(A − 2i∗ 0 )W20 ( ) = −H20 ( ), AW 11 ( ) = −H11 ( ), .. .
(4.9)
Denote the jth element of q(0) by qj , and the jth element of W (z, z¯ , ) by j
W j (z, z¯ , ) = W20 ( )
z2 j + W11 ( )z¯z + · · · . 2
Then it follows that xi (t) = W i (z, z¯ , 0) + zq i + z¯ q¯i ,
i = 1, 2, . . . , n − 1
and xn (t − 1) = W n (z, z¯ , −1) + zq n e−i∗ 0 + z¯ q¯n ei∗ 0 . Hence F˜ (0, W + zq + z¯ q) ¯ ⎛ 1 F (0)[q 2 e−2i0 ∗ z2 + 2q q¯ z¯z + q¯ 2 e2i0 ∗ z¯ 2 ] ⎞ n n n n 2 1 ⎜ +[ 1 F (0)(W n (−1)q¯ ei0 ∗ + 2W n (−1)q e−i0 ∗ ) + 1 F (0)q 2 q¯ e−i0 ∗ ]z2 z¯ ⎟ n n ⎜ ⎟ n n 20 11 2 1 2 1 ⎜ ⎟ ⎜ + · · · + 1 F (0)[q 2 z2 + 2q1 q¯1 z¯z + q¯ 2 z¯ 2 ] ⎟ ⎜ ⎟ 1 1 2 2 = ∗ ⎜ ⎟. 1 (0)q¯ + 2W 1 (0)q ) + 1 F (0)q 2 q¯ ]z2 z¯ ⎜ +[ 1 F2 (0)(W20 ⎟ 1 1 11 1 1 2 2 2 ⎜ ⎟ ⎜ ⎟ 1 2 2 2 2 ⎝ + · · · + 2 Fn (0)[qn−1 z + 2qn−1 q¯n−1 z¯z + q¯n−1 z¯ ] ⎠ n−1 n−1 2 q¯ 2 +[ 21 Fn (0)(W20 (0)q¯n−1 + 2W11 (0)qn−1 ) + 21 Fn (0)qn−1 n−1 ]z z¯ + · · ·
Notice that
⎛
⎞T 1 a1 − i0 ⎜ ⎟ ⎜ ⎟ F2 (0) ⎜ ⎟ ⎜ ⎟ (a2 − i0 )(a1 − i0 ) ⎜ ⎟ i s ∗ ⎜ ⎟ e 0 ∗ ¯ q (s) = K ⎜ F3 (0)F2 (0) ⎟ ⎜ ⎟ .. ⎜ ⎟ ⎜ ⎟ . ⎝ (an−1 − i0 ) · · · (a2 − i0 )(a1 − i0 ) ⎠ Fn (0) · · · F3 (0)F2 (0)
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and g(z, z¯ ) = q¯ ∗ (0)F˜0 , we have
! F3 (0) F1 (0) F2 (0) Fn (0) −2i0 ∗ qn−1 , Me q1 + q2 + · · · + qn + g20 = ∗ (a1 + i0 )K Fn (0) F1 (0) F2 (0) F3 (0) ! F3 (0) F1 (0) F2 (0) Fn (0) q ¯ g11 = ∗ (a1 + i0 )K + + · · · + M q ¯ q ¯ q ¯ + 1 2 n−1 , n Fn (0) F2 (0) F3 (0) F1 (0) F3 (0) (a1 + i0 )(a2 + i0 ) F (0) a1 + i0 F1 (0) ¯ q¯n + 2 q¯2 q ¯ + M 1 F1 (0) F2 (0) a1 − i0 F3 (0) (a1 − i0 )(a2 − i0 ) ⎤ n−1 Fn (0) aj + i0 +··· + q¯n−1 ⎦ , aj − i0 Fn (0)
g02 = ∗ (a1 − i0 )K
j =1
n g21 = ∗ (a1 − i0 )K W20 (−1)
F1 (0) F (0) a1 + i0 1 ¯ i0 ∗ + W20 Me (0) 2 F1 (0) F2 (0) a1 − i0
⎤ (0) (0) n−1 F a + i F + i )(a + i ) (a j 0⎦ 1 0 2 0 n−1 2 (0) n (0) 3 + · · · + W20 +W20 aj − i0 Fn (0) F3 (0) (a1 − i0 )(a2 − i0 )
j =1 F (0) F (0) F (0) n 1 2 Me−i0 ∗ + W11 + W11 (−1) 1 (0) 2 (0) 3 + 2∗ K(a1 + i0 ) W11 F1 (0) F2 (0) F3 (0) ! F (0) F (0) F (0) F (0) n−1 (0) n + ∗ K(a1 + i0 ) 1 |qn |2 + 2 |q1 |2 + 3 |q2 |2 + · · · + W11 Fn (0) F1 (0) F2 (0) F3 (0) ! Fn (0) 2 |qn−1 | , +··· + Fn (0)
where M=
n Fj (0) j =1
aj + i0
.
We still need to compute W20 ( ) and W11 ( ). For ∈ [−1, 0), we have H (z, z¯ , ) = 2 Re{q¯ ∗ (0)F˜0 q( )} = −gq( ) − g¯ q( ) ¯ % $ % $ 2 2 z z¯ z¯ 2 z2 ¯ = − g20 + g11 z¯z + g02 + · · · q( ) − g¯ 20 + g¯ 11 z¯z + g¯ 02 + · · · q( ). 2 2 2 2 Comparing the coefficients with (4.6) gives that H20 ( ) = −g20 q( ) − g¯ 02 q( ) ¯
and
H11 ( ) = −g11 q( ) − g¯ 11 q( ). ¯
It follows from the definition of W that −i0 W˙ 20 ( ) = 2i∗ 0 W20 ( ) − g20 q(0)ei0 − g¯ 02 q(0)e ¯ .
Solving for this, we obtain W20 ( ) =
g20 g¯ 20 −i∗ 0 q(0)ei∗ 0 − q(0)e ¯ + E1 e2i∗ 0 , i∗ 0 3i∗ 0
(4.10)
J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
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and similarly W11 ( ) =
g¯ 11 g11 −i∗ 0 q(0)e ¯ + E2 , q(0)ei∗ 0 − i∗ 0 i∗ 0
where E1 and E2 are both n-dimensional vectors, and can be determined by setting = 0 in H. In fact, H (z, z¯ , 0) = −2 Re{q¯ ∗ (0)F˜0 q(0)} + F˜0 , we have
⎛
⎜ ⎜ ⎜ H20 = −g20 q(0) − g¯ 20 q(0) ¯ + ∗ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ H11 = −g11 q(0) − g¯ 11 q(0) ¯ ⎜ ⎜ ⎝
F1 (0)qn2 e−2i0 ∗ F2 (0)q12
F3 (0)q22 .. . 2 Fn (0)qn−1 ⎞ F (0)q q¯
⎞ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎠
n n
1
F2 (0)q1 q¯1
F3 (0)q2 q¯2 .. .
⎟ ⎟ ⎟ ⎟, ⎟ ⎠
Fn (0)qn−1 q¯n−1 and
⎛
2i0 + a1
⎜ −F (0) 2 ⎜ ⎜ 0 ⎜ ⎜ ⎜ 0 ⎜ ⎝ ··· 0
0
0
2i0 + a2 −F3 (0) 0
0 2i0 + a3 −F4 (0)
··· 0
··· 0 (1)
· · · −F1 (0)e−2i0 ∗ ··· ···
0 0
··· ··· ···
0 ··· 2i0 + an
(2)
(n)
⎞
⎛
⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎟ E1 = ⎜ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠
F1 (0)qn2 e−2i0 ∗ F2 (0)q12 F3 (0)q22 .. . 2 Fn (0)qn−1
⎞ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠
(1)
(2)
(n)
Solving this we can obtain E1 = (E1 , E1 , . . . , E1 )T . Similarly, we can obtain E2 = (E2 , E2 , . . . , E2 )T from ⎛ ⎞ 0 0 · · · −F1 (0) a1 ⎞ ⎛ F (0)q q¯ n n 1 ⎜ −F (0) ⎟ a 0 · · · 0 2 ⎜ ⎟ 2 ⎜ F (0)q1 q¯1 ⎟ ⎜ ⎟ 2 ⎟ ⎜ ⎜ 0 −F3 (0) a3 ··· 0 ⎟ ⎟ (0)q q¯ ⎜ ⎟ E2 = ⎜ F ⎟. ⎜ 2 2 3 (0) · · · ⎜ 0 ⎟ ⎟ ⎜ 0 −F 0 ⎜ ⎟ 4 . ⎠ ⎝ .. ⎜ ⎟ . . . . ⎝ ⎠ .. .. .. . ··· . Fn (0)qn−1 q¯n−1 0 0 0 ··· an Based on this analysis, we can observe that each gij is determined by the parameters and delay in (2.1). Thus, we can compute the following quantities: $ % g21 i 1 g20 g11 − 2|g11 |2 − |g02 |2 + c1 (0) = , 20 3 2 2 = −
Re c1 (0) , Re (0 )
2 = 2 Re c1 (0).
(4.11)
We know that (see [16]) 2 determines the directions of the Hopf bifurcation: if 2 > 0 (< 0), then the direction of the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for > 0 (< 0 ); 2
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determines the stability of the bifurcating periodic solutions on the center manifold: the bifurcating periodic solutions are orbitally asymptotically stable (unstable) if 2 < 0 (> 0). Remark. If Fj (0) = 0, then g11 = g20 = g02 = 0, and
! F3 (0) F2 (0) F1 (0) Fn (0) 2 2 2 2 |qn−1 | . |qn | + |q1 | + |q2 | + · · · + g21 = ∗ K(a1 + i0 ) Fn (0) F2 (0) F3 (0) F1 (0)
Hence 2 and 2 can be computed directly. Consequently, the direction of the Hopf bifurcation and stability of he bifurcating periodic solutions can be determined. 5. Pitchfork bifurcation In this section, we shall study the multiple stabilities of system (2.1). Initially, we make the following assumptions on Fi (x). (H1 ) Fi (x) ∈ C 2 (i = 1, 2, . . . , n) are all monotonically increasing functions, xF i (x) > 0 (i = 1, 2, . . . , n) for x = 0, and there exists an L > 0 such that |Fi (x)| L for x ∈ R. (H4 ) xF i (x) < 0 (i = 1, 2, . . . , n) for x = 0. Clearly, under the assumption (H1 ), the origin is an equilibrium of system (2.1). Theorem 5.1. Suppose (H1 ) and (H4 ) are satisfied. (i) If An > ni=1 Fi (0), then system (2.1) has only one equilibrium at the origin. n (ii) If An < i=1 Fi (0), then system (2.1) has just three equilibria (x1+ , x2+ , . . . , xn+ ), (x1− , x2− , . . . , xn− ) and the origin, where xj+ > 0 and xj− < 0 for j = 1, 2, . . . , n. Remark. The conclusions tell us that the line An =
n
i=1 Fi (0)
is a pitchfork bifurcation curve of system (2.1).
Proof. The assumption (H1 ) implies that the origin (0, 0, . . . , 0) is an equilibrium of system (2.1). Since Fi (x) ∈ C 2 (i = 1, 2, . . . , n) are all monotonically increasing functions, then Fi (x) 0. Furthermore, by the assumption xF i (x) < 0 (i = 1, 2, . . . , n) for x = 0 we have Fi (0) = maxx∈R Fi (x) (i = 1, 2, . . . , n). (x1∗ , x2∗ , . . . , xn∗ ) is an equilibrium of system (2.1) if and only if (x1∗ , x2∗ , . . . , xn∗ ) satisfies the a1 x1 = F1 xn , a2 x2 = F2 (x1 ), . . . , an xn = Fn (xn−1 ).
(5.1)
x1∗
This leads to the condition that satisfies $ $ $ $ % %%% 1 1 1 1 1 x − F1 Fn Fn−1 · · · F3 F2 (x) · · · = 0. a1 an an−1 a3 a2 Let 1 h(x) = x − F1 a1
$
1 Fn an
$
1 an−1
$ Fn−1
1 · · · F3 a3
$
(5.2)
% %%% 1 F2 (x) · · · . a2
Under the inequality An −
n
Fi (0) > 0,
i=1
by the definition of h(x) we have $ $ $ %%% 1 1 1 1 h (x) = 1 − F Fn Fn−1 · · · F2 (x) · · · A n 1 an a a2 $ $ n−1 $ $ %%% % 1 1 1 1 · Fn Fn−1 Fn−2 · · · F2 ((x)) · · · · · · · · F3 F2 (x) · F2 (x). an−1 an−2 a2 a2
(5.3)
J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
From (H1 ) we know that Fi (0) = 0 for i = 1, 2, . . . , n. Hence n j =1 Fj (0) h (0) = 1 − . An
2247
(5.4)
Observing Fi (0) = maxx∈R Fi (x) (i = 1, 2, . . . , n), we have h (x) > 0, for x ∈ R, and hence x = 0 is the unique zero of h(x). Thus the origin is the unique equilibrium of (2.1). Under the inequality An −
n
Fi (0) < 0,
(5.5)
i=1
from (5.4) it follows that h (0) < 0. Hence, there exists a neighborhood, N (0), of x =0 such that h (x) < 0, for x ∈ N (0). From h(0) = 0 we have h(x) < 0 when x ∈ N (0) and x > 0. On the other hand, from |Fi (x)| L (i = 1, 2, . . . , n), we have h(x) → +∞ when x → +∞. There exists x1+ > 0 such that h(x1+ ) = 0, and h(x) < 0 when x ∈ (0, x1+ ). So h (x1+ )0. By (H4 ), we know that Fi (x) (i = 1, 2, . . . , n) are monotonically decreasing for x > 0. Thus, from (H1 ), $ $ $ %%% 1 1 1 1 h (x) = 1 − F1 Fn Fn−1 · · · F2 (x) · · · An an a a2 $ $ n−1 $ $ %%% % 1 1 1 1 · Fn Fn−1 Fn−2 · · · F2 ((x)) · · · · · · · · F3 F2 (x) · F2 (x) > 0, an−1 an−2 a2 a2 when x > x1+ . This implies that x1+ is a unique positive zero of the function h(x). Let xi+ =
1 + Fi (xi−1 ) ai
(i = 2, . . . , n),
then (x1+ , x2+ , . . . , xn+ ) is a positive equilibrium to system (2.1), and (2.1) has no other positive equilibrium. Similarly, there exists x1− < 0 such that x1− is the unique negative zero of the function h(x). Hence (x1− , x2− , . . . , xn− ) − ) < 0 (i = 2, . . . , n), and (2.1) has no other negative is an equilibrium of system (2.1), where xi− = (1/ai )Fi (xi−1 equilibrium. The proof is complete. Corollary 5.2. Suppose that (H1 ), (H4 ) and (5.5) are satisfied. Then 1 + F (x + )F (x + ) · · · Fn (xn−1 ) < 1, a1 a2 · · · an 1 n 2 1
(5.6)
1 − F (x − )F (x − ) · · · Fn (xn−1 ) < 1, a1 a2 · · · an 1 n 2 1
(5.7)
where (x1+ , x2+ , . . . , xn+ ) and (x1− , x2− , . . . , xn− ) are defined as in Theorem 5.1. Proof. By the proof of Theorem 5.1, we know that h (x1+ ) 0 and h(x) < 0 when x ∈ (0, x1+ ). Hence, from h(x1+ ) = 0, there exists an x0 ∈ (0, x1+ ) such that h (x0 ) > 0. By the monotonically increasing property of Fi (x) and decreasing property of Fi (x) for i = 1, 2, . . . , n, we have h (x) > h (x0 ) > 0 when x > x0 . Then h (x11 ) > 0, that + ) < 1. The proof of (5.7) is similar, and we shall omit it. is (1/a1 a2 · · · an )F1 (xn+ )F2 (x1+ ) · · · Fn (xn−1 On the stability of the equilibria, we have the following results. Theorem 5.3. Suppose that (H1 ), (H2 ), (H4 ) are satisfied. (i) If (H3 ) holds, then the unique equilibrium, the origin, of system (2.1) is asymptotically stable for all 0. (ii) If (5.5) holds, then the zero solution of system (2.1) is unstable for all 0; and the equilibria (x1+ , x2+ , . . . , xn+ ) and (x1− , x2− , . . . , xn− ) are both asymptotically stable for all 0.
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Proof. (i) From Theorem 5.1 we know that the origin is the unique equilibrium of system (2.1). Then the conclusion follows from (1) of Theorem 3.3. (ii) Clearly, (5.5) implies that Eq. (2.3) with = 0 has at least one positive real root. By the increasing property of Fi (x) and (5.5), we have that Fi (0) > 0. Hence, A2n < ( ni=1 Fi (0))2 is satisfied. Applying Lemma 3.1, Eq. (2.3) with > 0 has purely imaginary roots if and only if = j . Applying Lemma 3.2, Eq. (2.3) has at least one positive real root when ∈ [0, 0 ], and at least three roots with positive real parts when > 0 . Therefore, the zero solution of system (2.1) is unstable for all 0. From Theorem 5.1 we know that (x1+ , x2+ , . . . , xn+ ) and (x1− , x2− , . . . , xn− ) are equilibria of system (2.1). Noticing that the linearization of system (2.1) around the equilibrium (x1+ , x2+ , . . . , xn+ ) is given by ⎧ y˙ (t) = −a y (t) + F (x + )y (t − ), 1 1 1 n 1 n ⎪ ⎪ ⎪ ⎨ y˙2 (t) = −a2 y2 (t) + F (x + )y1 (t), 2 1 .. ⎪ ⎪ . ⎪ ⎩ + )yn−1 (t), y˙n (t) = −an yn (t) + Fn (xn−1
(5.8)
and its characteristic equation is n + A1 n−1 + A2 n−2 + A3 n−2 + · · · + An −
n
+ Fi (xi−1 )e− = 0,
(5.9)
i=1
with x0+ = xn+ . Applying (5.6), similar to the proof of (1) in Theorem 3.3, we can obtain the second conclusion. We shall omit the detail. 6. Examples In Section 4, we have derived an algorithm for determining the direction of the Hopf bifurcations and stability of the bifurcating periodic solutions. Even though the formula is quite long, one can compute the important quality such as Re c1 (0) for some special transfer functions. For example, consider the following network model: u˙ i (t) = −aui (t) + bi F (ui−1 (t − i )),
(6.1)
i (mod n),
where a > 0 and bi (i = 1, 2, . . . , n) are constants. In view of the properties of the function tanh(x), we make the following assumptions: (P1 ) F ∈ C 3 , xF (x) > 0 for x = 0, F (0) = 1, F (0) = 0, and F (0) = 0. Denote 1 C a 1 = na, 2 = 3n 3 C a n
1 Cn a 3 3 1 C a , = 3 n 2 2 Cn a 5 5 Cn a
1 Cn1 a 3 3 2 Cn a Cn a 2 . .. n = .. . 2n−1 2n−1 2n−2 Cn a 2n−2 Cn a n n = a − bi n−1 . i=1
1 Cn2 a 2 Cn4 a 4
0 1 , . . . , Cn3 a 3
0
0
1 .. .
0 .. .
Cn2n−3 a 2n−3
Cn2n−4 a 2n−4
··· 0 .. ··· . n n n · · · Cn a − bi
···
0
i=1
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We make the following assumptions on 1 , 2 , . . . , n : 0, 2 > 0, . . . , n−1 > 0. (P2 ) 1 > (P3 ) a n > ni=1 bi . Applying Theorem 3.3 we have the following. Theorem 6.1. Suppose that (P1 ) is satisfied. (1) If a 2n > ( ni=1 bi )2 , (P2 ) and (P3 ) hold, then the zero solution of system (6.1) is asymptotically stable for all 0. (2) If a 2n < ( ni=1 bi )2 , (P2 ) and (P3 ) hold, then the zero solution of system (6.1) is asymptotically stable for ∈ [0, 0 ), and unstable when > 0 . Furthermore, system (6.1) undergoes a Hopf bifurcation at the origin when = j (j = 0, 1, . . .). (3) If a 2n > ( ni=1 bi )2 , and either (P2 ) or n > 0 is not satisfied, then the zero solution of system (6.1) is unstable for all 0. Here ⎧ 2 n−2 n−2 +a n 2 2 (−1)m n0 +(−1)m−1 n−2 0 Cn a +···−0 Cn a ⎨ 1 (arccos + 2j ), n = 2m, n 0 i=1 bi j = n−2 ⎩ 1 (−1)m n0 +(−1)m−1 0 Cn2 a 2 +···−30 Cnn−3 a n−3 +0 Cnn−1 a n−1 + 2j ), n = 2m + 1, 0 (arcsin − ni=1 bi √ j = 0, 1, . . . , and 0 = z0 , z0 is the unique positive zero of the function n 2 n n−1 n−2 h(z) = z + B2 z + B4 z + · · · + B2n−2 z + B2n − bi , ⎡
B2 = a 2 ⎣n2 + ⎡
⎤
⎡
j (−1)i+1 2Cni Cn ⎦ , B4
i+j =2,i=j
B6 = a 6 ⎣(Cn3 )2 + ⎡
= a 4 ⎣(Cn2 )2 +
⎤ j (−1)i+2 2Cni Cn ⎦ ,
i+j =4,i=j
⎤ (−1)i+3 2Cni Cn ⎦ , . . . , j
i+j =6,i=j
B2n−2 = a 2n−2 ⎣(Cnn−1 )2 +
i=1
⎤ j (−1)i+n 2Cni Cn ⎦ ,
B2n = a 2n , i, j = 0, 1, . . . , n.
i+j =2n−2,i=j
Applying Theorems 5.1 and 5.3, we have the following. Theorem 6.2. Suppose that xF (x) > 0 and xF (x) < 0 for x = 0, F (x) > 0 and F (0) = 1, bi > 0 (i = 1, 2, . . . , n) are satisfied. (i) If a n > nj=1 bj , then the model (6.1) has only one equilibrium at the origin, and the origin is asymptotically stable for all 0. (ii) If a n < nj=1 bj , then the model (6.1) has just three equilibria (x1+ , x2+ , . . . , xn+ ), (x1− , x2− , . . . , xn− ) and the origin, and the origin is unstable, the other two equilibria are both asymptotically stable for all 0, where xj+ > 0 and xj− < 0 solve % % $ $ b1 bn−1 bn bn b2 ± ± ± ± ± x1 = F = ) and xn± = F (xn−1 ), ···F F (x1 ) · · · , . . . , xn−1 F (xn−2 a a a a a respectively. The conclusions of Theorems 6.1 and 6.2 are illustrated in Fig. 1.
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J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
b b = an G3
G2
a
G1 b = −an
Fig. 1. The bifurcation set. Here we set b = nj=1 bj . The curves b = ±a n divide the right half plane into three regions: G1 is a conditional stable region; G2 is an absolutely stable region; and G3 is an unstable region. b = a n is a pitchfork bifurcation curve.
Now, let the conditions of (2) in Theorem 6.1 be satisfied. From F (0) = 0 and the formulae on gij derived in Section 4, we have g11 = g20 = g02 = 0, and
g21 = ∗ F (0)
(a + i0 )(n + ∗ a − i∗ 0 )
1+
(n + ∗ a)2 + 2∗ 20
b22 a 2 + 20
+
b22 b32 (a 2 + 20 )2
+ ··· +
b22 b32 · · · bn2
.
(a 2 + 20 )n−1
It follows that Re c1 (0) = Re
g21 2
= ∗ F (0)
[a(n + ∗ a) + ∗ 20 ] (n + ∗ a)2 + 2∗ 20
1+
b22 a 2 + 20
+
b22 b32 (a 2 + 20 )2
+ ··· +
b22 b32 · · · bn2 (a 2 + 20 )n−1
.
Hence, from 2 =2 Re c1 (0) and 2 =−Rec1 (0)/Re (∗ ), and applying Lemma 3.2, we have that sign 2 =sign F (0) and sign 2 = −sign F (0). Consequently, we have the following. Theorem 6.3. Suppose that the conditions of (2) in Theorem 6.1 are satisfied. If F (0) > 0 (< 0), then, at each bifurcation value j , the direction of the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions on the center manifold are orbitally asymptotically stable (unstable). Particularly, the stability of the bifurcating periodic solutions in the phase space and on the center manifold are coincidence at the first bifurcation value 0 . To illustrate the analytical results found, let us consider the following special case of Eq. (6.1): ⎧ ⎨ u˙1 (t) = −au1 (t) + b1 tanh(u3 (t − 1 )), (6.2) u˙ (t) = −au2 (t) + b2 tanh(u1 (t − 2 )), ⎩ 2 u˙3 (t) = −au3 (t) + b3 tanh(u1 (t − 3 )). √ . Let a = 1, b1 = −2 2, b2 = b3 = 1, then 0 = 0.782, and 0 = 1. Under this group of data, from Theorem 6.3, . we have that the origin is asymptotically stable when ∈ [0, 0.782), and unstable when > 0 = 0.782, the Hopf bifurcation is supercritical at 0 , as well as the bifurcating periodic solutions are asymptotically stable. . . In Fig. 2, = 0.6 < 0 = 0.782, we show that the origin is asymptotically stable. In Fig. 3, = 1 > 0 = 0.782, we . show that the origin is unstable, and a periodic solution bifurcates from the origin for > 0 = 0.782, as well as the bifurcating periodic solution is asymptotically stable.
J. Wei, C. Zhang / Nonlinear Analysis: Real World Applications 9 (2008) 2234 – 2252
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0.1 0.05 0 −0.05 −0.1 0.2
0.1
0 −0.1 −0.2
−0.2
−0.1
0.1
0
0.2
√ Fig. 2. A Matlab simulation of the asymptotically stable origin to system (6.2) with a = 1, b1 = −2 2, b2 = b3 = 1, 1 = 2 = 3 = 0.2, and . = 3i=1 i = 0.6 < 0 = 0.782.
0.5 0.4 0.3 0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 1 0.5 0
−0.5 −1
−1
−0.5
0
0.5
1
√ Fig. 3 3. A Matlab .simulation of a periodic solution to system (6.2) with a = 1, b1 = −2 2, b2 = b3 = 1. 1 = 2 = 0.3, 3 = 0.4, and i=1 i = 1 > 0 = 0.782.
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