Nonlinear Dyn (2014) 78:1989–1998 DOI 10.1007/s11071-014-1578-8
ORIGINAL PAPER
The new result on delayed finance system Xiaoling Chen · Haihong Liu · Chenglin Xu
Received: 3 March 2014 / Accepted: 4 July 2014 / Published online: 22 July 2014 © Springer Science+Business Media Dordrecht 2014
Abstract In this paper, a finance system with time delay is considered. By linearizing the system at the unique equilibrium and analyzing the associated characteristic equation, the asymptotic stability of the unique equilibrium is investigated and Hopf bifurcations are demonstrated. Furthermore, the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem for functional differential equations. Finally, some numerical simulations are carried out for illustrating the theoretical results.
[4–7], the Kaldorian model [8], Goodwin’s accelerator model [9], and other models proposed in various references [10–21]. Many more examples of nonlinear economical models can be found in books on complex economic dynamics [22,23]. References [10–19] have reported a dynamic model of finance, composed of three first-order differential equations. The model describes the time variations of three state variables: the interest rate, x; the investment demand, y; and the price index, z. By choosing an appropriate coordinate system and setting appropriate dimensions for each state variable, references [10–15] offer the simplified finance model as
Keywords Finance system · Delay · Hopf bifurcation · Normal form · Stability · Periodic solution
⎧ ⎨ x˙ = z + (y − a)x, y˙ = 1 − by − x 2 , ⎩ z˙ = −x − cz,
1 Introduction In recent years, there is growing interest in applying nonlinear dynamics to economic model. Examples are the forced van der Pol model [1–3], the IS-LM model X. Chen · H. Liu (B) · C. Xu Department of Mathematics, Yunnan Normal University, Kunming 650092, People’s Republic of China e-mail:
[email protected] X. Chen e-mail:
[email protected] C. Xu e-mail:
[email protected]
(1)
where a ≥ 0 is the savings amount, b ≥ 0 is the cost per investment, and c ≥ 0 is the elasticity of demand of commercial markets. Time delay in a financial system means that one policy from being made to taking effect will have to go through a period of time, and its existence and influence have been known to be not negligible. Though it is still difficult to determine the delay in a financial system accurately, but as an objective existence, it dominates the financial policy and makes it produce effect on the economy to a great extent. Zhen in [21] proposed a delayed fractional order financial system:
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⎧ α ⎨ Dt 1 x(t) = z + (y(t − τ ) − a)x, D α2 y(t) = 1 − by − x 2 (t − τ ), ⎩ tα3 Dt z(t) = −x(t − τ ) − cz,
X. Chen et al.
(2)
where τ ≥ 0 is time delay. Zhen studied its dynamic behaviors, such as single-periodic, multiple-periodic, and chaotic motions. Although some important dynamical behaviors of fractional order system (3) have been discussed extensively in Zhen [21], there are still other meaningful dynamic properties that need to be further understood, which are vital to make clear a dynamical system. Therefore, in the present paper, we consider first order of system (2) with the form ⎧ ⎨ x˙ = z + (y(t − τ ) − a)x, (3) y˙ = 1 − by − x 2 (t − τ ), ⎩ z˙ = −x(t − τ ) − cz, where all of the variables have the same meaning of system (1). Thus, in the present paper, we discuss the stability and the local Hopf bifurcation of system (3) continuously. It has also been observed that time delay can drive the system to occur sustained oscillations, as shown by Hopf bifurcation analysis and limit cycle stability. To the best of our knowledge, few results for system (3) have been obtained in the literature up to now. Therefore, this study might be helpful to the comprehension of finance system. The paper is organized as follows. In Sect. 2, by analyzing the characteristic equation of the linearized system of system (3) at the unique equilibrium, it is found that under suitable conditions on the parameters the unique equilibrium is asymptotically stable when τ is less than a certain critical value and unstable when τ is greater than this critical value. Meanwhile, according to the Hopf bifurcation theorem for functional differential equations (FDEs) [26,27], we find that the system can also undergo a Hopf bifurcation of nonconstant periodic solution at the unique equilibrium when the delay crosses through a sequence of critical values. In Sect. 3, to determine the direction of the Hopf bifurcations and the stability of bifurcated periodic solutions occurring through Hopf bifurcations, an explicit algorithm is given by applying the normal form theory and the center manifold reduction for FDEs developed by Hassard, Kazarinoff, and Wan [24]. To verify our theoretical results, some numerical simulations are also included in Sect. 4.
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2 Stability of the equilibrium and local Hopf bifurcations System (3) is equivalent to system (1) when τ = 0. For system (1), we have known the following results [10–12]. Lemma 2.1 When c−b−abc ≤ 0, i.e., 1+ac− bc > 0, system (3) has an unique equilibrium P0 (0, b1 , 0). Lemma 2.2 When c−b−abc > 0, i.e., 1+ac− bc < 0, system (3) has three equilibriums 1 P0 0, , 0 , b 1 c − b − abc c − b − abc 1 + ac , ,∓ P± ± . c c c c Lemma 2.3 The equilibrium P0 (0, b1 , 0) is stable when 1 + ac − bc > 0 and c + a − b1 > 0. In this section, we investigate the Hopf bifurcation near the unique equilibrium P0 (0, b1 , 0) only. In what follows, we assume that the coefficients in system (3) satisfy the following condition: c 1 > 0 and c + a − > 0. b b In the following, we focus on the existence of local Hopf bifurcation at equilibrium P0 (0, b1 , 0) of system (3). Let x(t) = x(t), y(t) = y(t) − b1 , z(t) = z(t) and still denote x(t), y(t), z(t) by x(t), y(t), z(t), the system (3) is equivalent to the following system: ⎧ ⎨ x˙ = ( b1 − a)x(t) + z(t) + x(t)y(t − τ ), (4) y˙ = − by(t) − x 2 (t − τ ), ⎩ z˙ = − x(t − τ ) − cz(t), (H1) 1 + ac −
and the unique equilibrium P0 (0, b1 , 0) of system (3) is transformed into the zero equilibrium (0, 0, 0) of system (4). It easy to see that characteristic equation of the linearized system of model (4) at the zero equilibrium (0, 0, 0) is (λ + b)(λ2 + a1 λ + e−λτ + a2 ) = 0, where 1 a1 = a + c − , b 1 a2 = a − c. b
(5)
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Lemma 2.4 The root λ = −b of (5) has always negative real parts when τ ≥ 0 with the condition (H1). Next, we only need to consider the distribution of the root of the following transcendental equation: λ2 + a1 λ + e−λτ + a2 = 0.
(6)
Thus, iω(ω > 0) is a root of (6) if and only if ω satisfies the following equation:
−ω2 + a2 + cos ωτ = 0, (7) a1 ω − sin ωτ = 0, which leads to ω4 + (a12 − 2a2 )ω2 + a22 − 1 = 0.
(8)
which leads to
d(Re(λ(τ )) −1 ]τ =τ j dτ λτ (2λ+a1 )e j = Re λ a cos ω0 τ j −2ω0 sin ω0 τ j +i(2ω0 cos ω0 τ j +a1 sin ω0 τ j ) = Re 1 iω0 2ω cos ω0 τ j +a1 sin ω0 τ j = 0 ω0 = 2ω02 − 2a2 + a12 = (a12 − 2a2 )2 − 4(a22 − 1) > 0.
Then we obtain the result of the lemma. Similarly, we can obtain d(Reλ) | − > 0. dτ τ =τk
It is easy to see that if the condition
d(Reλ) |τ =τ + > 0, k dτ
(H2) a12 − 2a2 > 0 and a22 − 1 > 0
Lemma 2.6 For the transcendental equation
holds, then (8) has no positive roots. Hence, all roots of (6) have negative real parts when τ ∈ [0, ∞) under the condition(H2). If (H3) a22 − 1 < 0 holds, then (8) has a unique positive root ω02 . Substituting ω02 into (7), we obtain 1 τj = arccos(ω02 −a2 )+2 jπ , j = 0, 1, 2, .... (9) ω0 If (H4) a12 − 2a2 < 0, a22 −1 > 0 and (a12 −2a2 )2 > 4(a22 − 1) 2 and ω2 . holds, then (8) has two positive roots ω+ − 2 Substituting ω± into (7), we obtain 1 2 arccos(ω± τk± = −a2 )+2kπ , k = 0, 1, 2, .... ω± (10)
Let λ(τ ) = α(τ ) + iω(τ ) be a root of (6) near τ = τ j and α(τ j ) = 0, ω(τ j ) = ω0 , j = 0, 1, 2, . . . ; it is not difficult to verify the following result. Lemma 2.5 The transversalis conditions [ d(Re(λ) dτ ]|τ =τ j > 0( j = 0, 1, 2, )˙ hold. Proof Substituting λ(τ ) into the left-hand side of (6) and taking derivative with respect to τ , we have −1 dλ τ (2λ + a1 )eλτ − , = dτ λ λ
(11)
p(λ, e−λτ1 , . . . , e−λτm ) (0) (0) (0) = λn + p1 λn−1 + · · · + pn−1 λ + pn (1) + [ p1(1) λn−1 + · · · + pn−1 λ + pn(1) ]e−λτ1 + . . . (m) n−1 (m) (m) + · · · + pn−1 λ + pn ]e−λτm + [ p1 λ = 0, as (τ1 , τ2 , . . . , τm ) vary, the sum of orders of the zeros of p(λ, e−λτ1 , . . . , e−λτm ) in the open right half plane can change, and only a zero appears on or crosses the imaginary axis. According to the above analysis and Corollary 2.4 in Ruan and Wei [25], we have the following results. Theorem 2.7 For system (3), assume that (H1) is satisfied. Then the following conclusions hold: (i) If (H2) holds, then the equilibrium P0 (0, b1 , 0) of system (3) is asymptotically stable for all τ ≥ 0. (ii) If (H3) holds, then the equilibrium P0 (0, b1 , 0) of system (3) is asymptotically stable for τ < τ j and unstable for τ > τ j . Furthermore, system (3) undergoes a Hopf bifurcation at the equilibrium P0 (0, b1 , 0) when τ = τ j . (iii) If (H4) holds, then there is a positive integer m such that the equilibrium P0 (0, b1 , 0) is stable − , τm+ ), when τ ∈ [0, τ0+ ) ∪ (τ0− , τ1+ ) ∪ · · · ∪ (τm−1 + − + − and unstable when τ ∈ [τ0 , τ0 ) ∪ (τ1 , τ1 ) ∪ . . . (τm+ , τm− ) ∪ (τm+ , ∞). Furthermore, system (3) undergoes a Hopf bifurcation at the equilibrium P0 (0, b1 , o) when τ = τk± , k = 0, 1, 2, . . . .
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3 Direction and stability of the Hopf bifurcation In the previous section, we studied mainly the stability of the unique equilibrium P0 of system (3) and the existence of Hopf bifurcations at the unique equilibrium P0 . In this section, we shall study the properties of the Hopf bifurcations obtained by Theorem 2.7 and the stability of bifurcated periodic solutions occurring through Hopf bifurcations using the normal form theory and the center manifold and the reduction for retarded functional differential equations (RFDEs) due to Hassard, Kazarinoff, and Wan [24]. Throughout this section, we always assume that system (3) undergoes Hopf bifurcation at the equilibrium P0 for τ = τ j , and then ±iω0 are the corresponding purely imaginary roots of the characteristic equation at the equilibrium P0 . Without loss of generality, we assume that τ ∈ (0, τ0 ). Let x i (t) = xi (τ t)(i = 1, 2, 3) and τ = τ j + μ, x(t) = (x(t), y(t), z(t))T where τ j is defined by (9) and μ ∈ R, drop the bar for simplicity of notation. Then, system (3) can be rewritten as a system of RFDEs in C([−1, 0], R 3 ) of the form ⎧ x(t) ˙ = (τ j + μ)[( b1 − a)x(t) + y(t − 1)x(t) ⎪ ⎪ ⎨ +z(t)], (12) y ˙ (t) = (τ j + μ)[−by(t) − x 2 (t − 1)], ⎪ ⎪ ⎩ z˙ (t) = (τ j + μ)[−x(t − 1) − cz(t)]. Define the linear operator L(μ) : C → R 3 and the nonlinear operator f (·, μ) : C → R 3 by ⎞ ⎛1 ⎞⎛ φ1 (0) b −a 0 1 L μ φ = (τ j + μ) ⎝ 0 −b 0 ⎠ ⎝ φ2 (0) ⎠ (13) φ3 (0) 0 0 −c ⎞ ⎛ ⎞⎛ 0 00 φ1 (−1) (14) +(τ j + μ) ⎝ 0 0 0 ⎠ ⎝ φ2 (−1) ⎠ −1 0 0 φ3 (−1) and
⎛
⎞ φ2 (−1)φ1 (0) ⎠, f (μ, φ) = (τ j + μ) ⎝ −φ12 (−1) 0
(15)
respectively, where φ = (φ1 , φ2 , φ3 )T ∈ C. By the Riesz representation theorem, there exists a 3×3 matrix function η(θ, μ), −1 ≤ θ ≤ 0, whose 0elements are of bounded variation such that L μ φ = −1 dη(θ, μ)φ(θ ) for φ ∈ C([−1, 0], R 3 ). In fact, we can choose η(θ, μ) = (τ j +μ)η0 δ(θ )− (τ j +μ)η−1 δ(θ +1), (16)
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where ⎛1 b
η0 = ⎝ 0 0
−a
⎛ ⎞ 0 1 0 ⎠ , η−1 = ⎝ 0 −1 −c
0 −b 0
0 0 0
⎞ 0 0⎠ 0
For φ ∈ C 1 ([−1, 0], R 3 ), define dφ(θ) dθ
A(μ)φ =
0
−1
,
θ ∈ [−1, 0),
dη(μ, θ )φ(θ ), θ = 0,
(17)
and
R(μ)φ =
0, θ ∈ [−1, 0), f (μ, θ ), θ = 0.
(18)
Then system (12) is equivalent to x˙t = A(μ)xt + R(μ)xt ,
(19)
where xt (θ ) = x(t + θ ). Remark 3.1 Here, it should be pointed out that (19) is the normal form of original system (3). For ψ ∈ C 1 ([0, 1], (R 3 )∗ ), define ∗
A ψ=
− dψ(s) , s ∈ (0, 1], 0 ds dη(t, 0)ψ(−t), s = 0, −1
(20)
and a bilinear inner product ¯ ψ(s), φ(θ ) = ψ(0)φ(0) 0 θ ¯ − θ )dη(θ )φ(ξ )dξ, − ψ(ξ −1 ξ =0
(21) where η(θ ) = η(θ, 0). Then A(0) and A∗ (0) are adjoint operators. In addition, from Sect. 2, we know that ±iω0 τ j are eigenvalues of A(0). Thus, they are also eigenvalues of A∗ (0). Let q(θ ) be the eigenvector of A(0) corresponding to iω0 τ j , and q ∗ (s) is the eigenvector of A∗ (0) corresponding to −iω0 τ j . Let q(θ ) = (1, v1 , v2 )eiω0 τ j θ and q ∗ (s) = G(1, v1∗ , ∗ v2 ) eiω0 τ j s . From the above discussion, it is easy to know that A(0)q(0) = iω0 τ j q(0) and A∗ (0)q ∗ (0) = − iω0 τ j q ∗ (0),
The new result on delayed finance system
that is ⎛1 ⎞ ⎛ 0 1 b −a 0 ⎝0 −b 0 ⎠ q(0) + ⎝ 0 −1 0 0 −c q(−1) = iω0 q(0) and ⎛1 b
⎝0 1 ⎛
−a
0 +⎝0 0
0 −b 0 0 0 0
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0 0 0
Remark 3.2 Actually, (23) is the center manifold of system (3).
⎞
0 0⎠ 0
On the center manifold C0 , we have W (t, θ ) = W (z(t), z(t), θ ) where z2 z¯ 2 + W11 (θ )z z¯ + W02 (θ ) 2 2 z3 (24) + W30 (θ ) + · · · , 6 z and z are local coordinates for center manifold C0 in the direction of q ∗ and q ∗ . Note that W is real if xt is real. We consider only real solutions. For solution xt ∈ C0 of (12), since μ = 0, W (z, z¯ , θ ) = W20 (θ )
⎞ 0 0 ⎠ q(−1)q ∗ (0) −c ⎞
−1 0 ⎠ q(−1)q ∗ (−1) = −iω0 q ∗ (0). 0
z˙ (t) = iω0 τ j z + q ∗ (0) f (0, W (z, z, θ )) + 2{z(t)q(θ )} = iω0 τ j z + q ∗ (0) f 0 ,
Thus, we can easily obtain 1 q(θ ) = (1, 0, iω0 + a − )eiω0 τ j θ b c + iω0 iω0 τ j s q ∗ (s) = G(1, o, 2 )e , c − ω02
that is, z˙ (t) = iω0 τ j z(t) + g(z, z),
since q (s), q(θ ) = q ∗ (0)q(0) − = q ∗ (0)q(0) −
0
θ
q ∗ (ξ − θ )dη(θ )q(ξ )dξ G(1, v ∗1 , v ∗2 )e−iω0 τ j (ξ −θ)
−1 ξ =0 T iω0 τ j ξ
= q ∗ (0)q(0) − q ∗ (0)
0 −1
dξ θ eiω0 τ j θ dη(θ )q(0)
0 00 = q ∗ (0)q(0) + q ∗ (0)τ j 0 0 0 e−iω0 τ j q(0) −1 0 0 ∗ ∗ = G (1 + v1 v 1 + v2 v 2 ) − v ∗2 τ j e−iω0 τ j .
1 −iω τ (1+v1 v ∗1 +v2 v ∗2 )−v ∗2 τ j e 0 j iω τ (1+v 1 v1∗ +v 2 v2∗ )−v2∗ τ j e 0 j
,
xt = W (t, θ ) + 2{z(t)q(θ )} 2 2 = W20 (θ ) z2 + W11 (θ )zz + W02 (θ ) z2 + (1, v1 , v2 )eiω0 τ j θ z + (1, v 1 , v 2 )e−iω0 τ j θ z + . . . (28) It follows together with (15) that g(z, z) = q ∗ (0) f 0 (z, z) = q ∗ (0) f (0, xt )
We may choose G and G as
1
z2 z2 z2 z +g11 zz +g02 +g21 + . . . . (27) 2 2 2 Then it follows from (23) that g(z, z) = g20
−1 ξ =0 0 θ
dη(θ )(1, v1 , v2 ) e
G=
(26)
where
∗
G=
(25)
, (22)
= q ∗ (0)τ10 ⎞ ⎛ (W (1) (0) + z + z)(W (2) (−1) ⎜ +zv1 e−iω0 τ j + zv 1 eiω0 τ j ) ⎟ ⎟ ⎜ ⎠ ⎝ −(W (1) (−1) + z + z)2 0
which assures that q ∗ (s), q(θ ) = 1. By using the same notions as in [24], we first compute the coordinates to describe the center manifold C0 at μ = 0. Let xt be the solution of (12) when μ = 0. Define
= Gτ j (1, v ∗1 , v ∗2 ) ⎛ ⎞ (W (1) (0) + z + z)(W (2) (−1) ⎜ +zv1 e−iω0 τ j + zv 1 eiω0 τ j ) ⎟ ⎜ ⎟ ⎝ −(W (1) (−1) + z + z)2 ⎠ 0
z(t) = q ∗ , xt , W (t, θ ) = xt (θ ) − 2{z(t)q(θ )}.
= Gτ j {2(v1 e−iω0 τ j − v ∗1 )
(23)
+ 2[(v1 e−iω0 τ j ) − v ∗1 ]zz
z2 2
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z2 (1) + [W20 (0)v 1 eiω0 τ j 2 (1) (2) (2) + 2W11 (0)v1 e−iω0 τ j + 2W11 (0) + W20 (0)
From (31) and (33), we get W˙ 20 (θ ) = 2iω0 τ j W20 (θ ) + g20 q(θ ) + g 02 q(θ ).
+ 2(v 1 eiω0 τ j − v ∗1 )
(1)
(1)
− 2v ∗1 W20 (−1) − 4v ∗1 W11 (−1)]
Note that q(θ ) = q(0)eiω0 τ j θ , hence we obtain
z2 z
iω0 τ j θ + W20 (θ ) = ωig020 τ j q(0)e 2iω τ θ + E1e 0 j .
+ . . . }. 2 Comparing the coefficients with (27), we obtain
(35)
Similarly, from (31) and (34), we have W˙ 11 (θ ) = g11 q(θ ) + g 11 q(θ )
g20 = 2Gτ j (v1 e−iω0 τ j − v ∗1 ),
g11 = 2Gτ j [(v1 e−iω0 τ j ) − v ∗1 ],
and
g02 = 4Gτ j (v 1 eiω0 τ j − v ∗1 ),
(1) (1) g21 = Gτ j [W20 (0)v 1 eiω0 τ j + 2W11 (0)v1 e−iω0 τ j (2) (2) (1) + 2W11 (0) + W20 (0) − 2v ∗1 W20 (−1) (1) − 4v ∗1 W11 (−1)].
Since there are W20 (θ ) and W11 (θ ) in g21 , we will need to compute them. From (19) and (23), we have ˙ W˙ = x˙t − z˙ q − zq AW − 2{q ∗ (0) f 0 q(θ )}, θ ∈ [−1, 0) = AW − 2{q ∗ (0) f 0 q(θ )} + f 0 , θ = 0 = AW + H (z, z, θ ),
i g 02 −iω0 τ j θ 3ω0 τ j q(0)e
iω0 τ j θ + W11 (θ ) = − ωig011 τ j q(0)e +E 2 .
i g 11 −iω0 τ j θ ω0 τ j q(0)e
(36)
In what follows, we shall seek appropriate E 1 and E 2 in (35) and (36), respectively. It follows from the definition of A and (31) that
0 −1
dη(θ )W20 (θ ) = 2iω0 τ j W20 (θ ) − H20 (θ )
(37)
dη(θ )W11 (θ ) = −H11 (0),
(38)
and (29)
0 −1
where
where η(θ ) = η(0, θ ). From (29), we have
z2 z2 H (z, z, θ ) = H20 (θ ) + H11 (θ )zz + H02 (θ ) +. . . . 2 2 (30)
⎞ v1 e−iω0 τ j ⎠ H20 (0) = −g20 q(0) − g 02 q(0) + 2τ j ⎝ −1 0 (39)
Substituting the corresponding series into (29) and comparing the coefficients, we obtain (A − 2iω0 τ j )W20 (θ ) = −H20 (θ ) AW11 (θ ) = −H11 (θ ) . . . ,
(31)
Substituting (35) and (39) into (37), we obtain
(33)
0 (2iω0 τ j I − −1 e2iω0 τ j θ dη(θ ))E 1 ⎞ ⎛ v1 e−iω0 τ j ⎠. = 2τ j ⎝ −1 0
(34)
From the definition of A, we have 0 e2iω0 τ j θ dη(θ ) = A(μ)e2iω0 τ j θ = L μ (e2iω0 τ j θ ).
and H11 (θ ) = −g11 q(θ ) − g 11 q(θ ),
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⎞ (v1 eiω0 τ j ) ⎠. H11 (0) = −g11 q(0)−g 11 q(0)+2τ j ⎝ −1 0 (40)
(32)
Comparing the coefficients with (30) gives that H20 (θ ) = −g20 q(θ ) − g 02 q(θ )
and ⎛
from (29), we know that for θ ∈ [−1, 0), H (z, z, θ ) = −q ∗ (0) f 0 q(θ ) − q ∗ (0) f 0 q(θ ) = −g(z, z)q(θ ) − g(z, z)q(θ ).
⎛
−1
(41)
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Therefore, when μ = 0, we have 0 e2iω0 τ j θ dη(θ ) −1 ⎛1 ⎛ ⎞ 0 b −a 0 1 = τj ⎝0 −b 0 ⎠ +τ j ⎝ 0 −1 0 0 −c ⎛1 ⎞ 0 1 b −a = τj ⎝0 −b 0 ⎠ . −c −e−2iω0 τ j 0
bifurcation. There are two kinds of hopf bifurcation. The bifurcation is called supercritical if the bifurcated periodic solution is stable and subcritical if they are unstable. μ2 determines the directions of the Hopf bifurcation: if μ2 > 0(μ2 < 0), then the Hopf bifurcation is supercritical (subcritical).
Therefore, ⎛ 2λ + a − b1 ⎝0 e−2iω0 τ j ⎛ v1 e−iω0 τ j ⎝ =2 1 0
0 2λ + b 0 ⎞
0 0 0
⎞ 0 0 ⎠ e−2iω0 τ j 0
⎞ −1 ⎠ E1 0 2λ + c
⎠,
(42)
(43)
Theorem 3.4 Suppose that (H1) and (H3) hold. If (c1 (0)) < 0 ((c1 (0)) > 0), then the system (3) can undergo a supercritical (subcritical) Hopf bifurcation at the equilibrium P0 (0, b1 , 0) when τ crosses through the critical values τ = τ j . In addition, the bifurcated periodic solutions occurring through Hopf bifurcations are orbitally asymptotically stable on the center manifold if (c1 (0)) < 0 and unstable if (c1 (0)) > 0.
4 Numerical simulations
where λ = iω0 . Similarly, substituting (35) and (40) into (38), we get ⎛ ⎞ 0 (v1 eiω0 τ j ) ⎠. dη(θ )E 2 = − ⎝ −1 (44) −1 0
In this section, we give some numerical simulations for a special case of system (3) to support our analytical results obtained in Sects. 2 and 3. As an example, we consider system (3) with the coefficients a = 1.1, b = 2.7, c = 0.7, that is,
It follows from (35), (36), (42), and (44) that g21 can be expressed. Thus, we can compute the following values:
⎧ ˙ = z + (y(t − τ ) − 1.1)x, ⎨ x(t) ˙ = 1 − 2.7y − x 2 (t − τ ), y(t) ⎩ ˙ z(t) = −x(t − τ ) − 0.7z.
c1 (0) = μ2 =
ω0 τ j (g11 g20 (c1 (0)) − (λ (τ )) , 0 j i
β2 = 2(c1 (0)), T2 =
− 2|g11 |2 −
|g02 |2 g21 3 )+ 2 ,
(45)
(c1 (0))+μ2 (λ0 (τ j )) , ω0
which determine the quantities of bifurcating periodic solutions at the critical value τ j . Specifically, μ2 determines the directions of the Hopf bifurcation: if μ2 > 0(μ2 < 0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for τ > τ j (τ < τ j ); β2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions in the center manifold are stable (unstable) if β2 < 0(β2 > 0); and T2 determines the periodic of the bifurcating periodic solutions: the period increase (decrease) if T2 > 0(T2 < 0). Further, it follows from Lemma 2.5 and (45) that the following results about the direction of the Hopf bifurcation hold. Remark 3.3 The appearance or disappearance of a periodic orbit through a local change in the stability properties of a steady pointed is known as the Hopf
(46)
From Sect. 2, we have c 1 > 0 and c + a − > 0. b b (H3) a22 − 1 < 0, (H1) 1 + ac −
where 1 a2 = (a − )c. b Then we can easily calculate that 1 + ac − bc = 3.71074, c + a − b1 = 1.4296, and a22 − 1 = −0.7391. Clearly, the conditions (H1), (H3), and Lemma 2.5 hold. By computing, we may obtain that a unique positive equilibrium P0 (0, 0.37037, 0), ω0 = 0.69932, τ j = 2.21516 + 8.98475 j, ( j = 0, 1, 2, ...). From Lemma 2.3, we know that the transversal condition is satisfied. Thus, the equilibrium P0 (0, 0.37037, 0) of system (46) is asymptotically stable when τ = 0 (see Fig. 1). On the other hand, the conditions (H1) and (H3) hold. From the second point in Theorem 2.7, we know
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X. Chen et al. y x
z 2.0
2. 10
11
1. 10
11
2. 10
11
1. 10
11
1. 10
11
1.5
50
100
150
200
t
1.0 50
1. 10
0.5 2. 10
100
150
200
t
11
11
50
100
150
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t
Fig. 1 The numerical approximations of system (46) when τ = 0. The positive equilibrium P0 (0, 0.37037, 0) is asymptotically stable y
x 0.370
0.10
0.368 0.05
0.366 50
100
150
200
t 0.364
0.05
0.362 0.10
y
50
100
150
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t
0.3685 0.3690 0.3695 0.3700 0.05
z 0.10
0.05 z 0.00
50
100
150
200
t
0.05
0.10
0.05 0.05
0.00
x
0.05
Fig. 2 The numerical approximations of system (46) when τ < τ0 = 2.21516. The positive equilibrium P0 (0, 0.37037, 0) is asymptotically stable
that the positive equilibrium P0 of system (46) is asymptotically stable when 0 ≤ τ < τ0 = 2.21516 (see Fig. 2), and unstable when τ > τ0 , and system (46) can also undergo a Hopf bifurcation at the positive equilibrium P0 when τ crosses through the critical values τ j = 2.21516 + 8.98475 j, ( j = 0, 1, 2, ...), i.e., a family of periodic solutions bifurcated from P0 (0, 0.37037, 0) (see Fig. 3). By the algorithms derived in Sect. 3, we can obtain μ2 = 0.00008, β2 = −0.00068, andT2 = 0.00028. Therefore, from Theorem 3.1, we know that system
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(46) can undergo a supercritical Hopf bifurcation at the positive equilibrium P0 when τ = τ0 = 2.21516, and the bifurcated periodic solution occurring from the Hopf bifurcation is orbitally asymptotically stable on the center manifold. To summarize, the positive equilibrium point P0 (0, 0.37037, 0) of the system without delay is asymptotically stable. When delay is less than critical values the positive equilibrium P0 (0, 0.37037, 0) is asymptotically stable, as seen from Fig. 2. And P0 is unstable when delay is greater than critical values. Meanwhile,
The new result on delayed finance system
1997 y
x
0.370
0.2 0.365
0.360
0.1
0.355 50
100
150
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t 0.350
0.345
0.1
0.340 0.2
50
100
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t
0.2
z 0.2 0.1 0.1
z 50
100
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0.0
t 0.1
0.1 0.2 0.2
0.37y 0.36 0.35 0.34
0.2
0.1
0.1
0.0
0.2
x
Fig. 3 The numerical approximations of system (46) when τ = 2.312 > τ0 = 2.21516. The positive equilibrium P0 (0, 0.37037, 0) is unstable and a stable periodic solution bifurcates from P0
Hopf bifurcation occurs from the positive equilibrium as seen from Fig. 3.
5 Conclusions In this paper, we considered a financial system with time delay and mainly analyzed its dynamic behaviors by using Hopf bifurcation technique. Specifically, sufficient conditions for existence of Hopf bifurcation are obtained. Moreover, we proved that some families of periodic solutions occur when the delay τ passes through some certain critical values, the equilibrium will lose its stability and Hopf bifurcation will take place. This research might be helpful for further comprehension of finance system. Acknowledgments The author expresses gratitude to the anonymous referee for his/her helpful suggestions and the partial support of Science Foundation (2011FZ086) and the Foundation of Education Commission (2013Z014) of Yunnan Province.
References 1. AC-L. Chian:Nonlinear dynamics and chaos in macroeconomics, Int J Theor Appl Finance, 3 (2000), 3:601. 2. Chian, A.C.-L., Borotto, F.A., Rempel, E.L., Rogers, C.: Attractor merging crisis in chaotic business cycles. Chaos Solitons Fractals 24, 869–875 (2005) 3. Chian, A.C.-L., Rempel, E.L., Rogers, C.: Complex economic dynamics: chaotic saddle, crisis and intermittency. Chaos Solitons Fractals 29, 1194–1218 (2006) 4. Schinasi, G.J.: A nonlinear dynamic model of short run fluctuations. Rev. Econ. Stud. 48(4), 649–656 (1981) 5. Sasakura, K.: On the dynamic behavior of Schinasis business cycle model. J. Macroecon. 16(3), 423–444 (1994) 6. Cesare, L.D., Sportelli, M.: A dynamic IS-LM model with delayed taxation revenues. Chaos Solitons Fractals 25, 233– 244 (2005) 7. Fanti, L., Manfredi, P.: Chaotic business cycles and fiscal policy: an IS-LM model with distributed tax collection lags. Chaos Solitons Fractals 32, 735–744 (2007) 8. Lorenz, H.W.: Nonlinear Economic Dynamics and Chaotic Motion. Springer, New York (1993) 9. Lorenz, H.W., Nusse, H.E.: Chaotic attractors, chaotic saddles, and fractal basin boundaries: goodwin’s nonlinear accelerator model reconsidered. Chaos Solitons Fractals 13, 957–965 (2002)
123
1998 10. Ma, J.H., Chen, Y.S.: Study for the bifurcation topological structure and the global complicated character of a kind of non-linear finance system (I). Appl. Math. Mech. 11(22), 1119–1128 (2001). (in Chinese) 11. Ma, J.H., Chen, Y.S.: Study for the bifurcation topological structure and the global complicated character of a kind of non-linear finance system (II). Appl. Math. Mech. 12(22), 1236–1242 (2001). (in Chinese) 12. Ma, J.H.: The reconstruction technology of complex nonlinear system. Tianjin University Press, Tianjin (2005) 13. Huang, D.S., Li, H.Q.: Theory and method of the nonlinear economics publishing. House of Sichuan University, Chengdu (1993). (in Chinese) 14. Chen, W.C.: Nonlinear dynamics and chaos in a fractionalorder financial system. Chaos Solitons Fractals 36, 1305– 1314 (2008) 15. Chen, W.C.: Dynamics and chaos of a financial system with time-delayed feedbacks. Chaos Solitons Fractals 37, 1198– 1207 (2008) 16. Gao, Q., Ma, J.H.: Chaos and Hopf bifurcation of a finance system. Nonlinear Dyn. 58, 209–216 (2009) 17. Yu, H.J., Cai, G.L., Li, Y.X.: Dynamic analysis and control of a new hyperchaotic finance system. Nonlinear Dyn. 67, 2171–2182 (2012) 18. Ma, J., Bangura, H.: Complexity analysis research of financial and economic system under the condition of three parameters’ change circumstances. Nonlinear Dyn. 70, 2313– 2326 (2012) 19. Cai, G.L., Hu, P., Li, Y.X.: Modified function lag projective synchronization of a financial hyperchaotic system. Nonlinear Dyn. 69, 1457–1464 (2012)
123
X. Chen et al. 20. Pan, I., Korre, A., Das, S., Durucan, S.: Chaos suppression in a fractional order financial system using intelligent regrouping PSO based fractional fuzzy control policy in the presence of fractional Gaussian noise. Nonlinear Dyn. 70, 2445–2461 (2012) 21. Wang, Z., Huang, x, Shi, G.: Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay. Comput. Math. Appl. 62(3), 1531–1539 (2011) 22. T. Puu: Nonlinear economic dynamics. Lecture notes in economics and mathematical systems, vol. 336. Springer, Berlin (1989). 23. Nonlinear dynamics and heterogeneous interacting agents Thomas L, Reitz S, Samanidou E, editors.Lecture notes in economics and mathematical systems, vol. 550. Berlin, Springer (2005). 24. Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge Univ. Press, Cambridge (1981) 25. Wei, J.J., Ruan, S.G.: On the zero of some transcendental functions with applications to stability if delay differential equations with two delays. Dyn. Contin. Discrete Impuls. Syst. Ser. A 10, 863–874 (2003) 26. Hale, J.k: Theory of Functional Differential Equation. Springer, New York (1977) 27. Yan, X.P., Li, W.T.: Hopf bifurcation and global periodic solutions in a delayed predator-prey system. Appl. Math. Comput. 177, 427–445 (2006)