Analysis of the stability and Hopf bifurcation of money supply delay in complex macroeconomic models

Nonlinear DynDOI 10.1007/s11071-013-1143-x

O R I G I NA L PA P E R

Analysis of the stability and Hopf bifurcation of moneysupply delay in complex macroeconomic models

Junhai Ma · Hongliang Tu

Received: 6 July 2013 / Accepted: 29 October 2013© Springer Science+Business Media Dordrecht 2013

Abstract Considering the macroeconomic model ofmoney supply, this paper carries out the correspondingextension of the complex dynamics to macroeconomicmodel with time delays. By setting the parameters, wediscuss the effect of delay variation on system stabilityand Hopf bifurcation. Results of analysis show that thestability of time-delay systems has important signifi-cance with the length of time delay. When time delayis short, the stable point of the system is still in a stableregion; when time delay is long, the equilibrium pointof the system will go into chaos, and the Hopf bifurca-tion will appear in certain conditions. In this paper, us-ing the normal form theory and center manifold theo-rem, the periodic solutions of the system are obtained,and the related numerical analysis are also given; thispaper has important innovation-theoretical value andacts as important actual application in macroeconomicsystem.

Keywords Stability · Hopf bifurcation · Dynamics ·Macro-economy system · Delay

J. Ma · H. Tu (B)College of Management and Economics, TianjinUniversity, Tianjin 300072, Chinae-mail: [email protected]

J. Ma (B)e-mail: [email protected]

H. TuSchool of Management, China University of Miningand Technology, Xuzhou 221116, China

1 Introduction

Economic dynamics have recently become very popu-lar in mainstream economics. Its influence has beenquite pervasive and has affected both macroeco-nomics. Researchers are striving to explain the centralfeatures of economic data: irregular microeconomicfluctuations, erratic macroeconomic fluctuations, ir-regular growth, structural changes, and overlappingwaves of economic development. Especially, eco-nomic dynamics seem to devote new interest to de-lay differential equations. This is because some eco-nomic phenomena cannot be described exhaustivelywith pure (linear or non-linear) differential equations.Differential equations with time delay play an impor-tant role in economy, engineering, biology and socialsciences, because great deal of problems may be de-scribed with their help. In this paper, we consider aneconomical model of a four-dimensional dynamic fi-nance system and study how the saving rate and thetime delay affect the stability of the dynamic financesystem.

We would like to mention that Hopf bifurcationsin a dynamics with delay have already been investi-gated by many researchers [1–11]. However, these re-searches are aimed at population dynamics model, andtwo- or three-dimensional business cycle model withdelays. In [12–16], the researchers have investigatedthe IS-LM model with taxation delay and shown thattax collection time delays create a wide variety of dy-namic behaviors.

mailto:[email protected]

mailto:[email protected]

J. Ma, H. Tu

References [5–7, 11] have reported a dynamicmodel of finance which is composed of three first-order differential equations. The model describes thetime variation of three variables: the interest rate, x,the investment demand, y, and the prize index, z. Thefactors that affect changes in x mainly come from con-tradictions in the investment market and structural ad-justments from the prices of goods. The changing rateof y is in proportion to the rate of investment, and inproportion to an inversion with the cost of investmentand interest rates. Changes in z, on the one hand, arecontrolled by a contradiction between supply and de-mand in commercial markets, and on the other hand,are influenced by inflation rates. By choosing an ap-propriate coordinates and setting proper dimensionsfor every state variable, [5–7, 11] offer the simplifiedfinance model as

⎧⎨

⎩

x(t) = z + (y − a)x,

y(t) = 1 − by − x2,

z(t) = −x − cz,

(1.1)

where a ≥ 0 is the saving amount, b ≥ 0 is the costper investment, and c ≥ 0 is the elasticity of demandof commercial markets.

As far as we know, there is little literature on dy-namic economic system considering the change ofprice index. As the price index closely contacts withinflation, so studying economic system including thechange of price index has an important theoretical aswell as practical value. Based on the IS-LM model (see[14–18]) and the model of (1.1) (see [5–7, 11]), webuild up a four-dimensional dynamic macro-economysystem as follows:

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

Y (t) = α1[I (Y (t), r(t)) + g − S(YD(t)) − T (t)]+ α2[r(t) + cP (t) − 1]Y(t),

r(t) = β1P + β2[L(Y (t), r(t)) − M(t)],P (t) = 1 − r(t) − cP (t) + γ1[g − νY (t)],M(t) = g − νY (t)

(1.2)

with Y as income, I as investment, g as governmentexpenditure (constant), S as savings, T as tax rev-enues, r as the rate of interest, L as liquidity, M asreal money supply and α, β as positive constants, withtime delay τ appearing in real money supply M .

In the following analysis we will consider the in-vestment, the liquidity, the saving and the tax in theform⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

I (Y (t)r(t)) = aY (t)α1r(t)−α2 ,

a > 0, α1 > 0, α2 > 0,

L(Y (t), r(t)) = mY(t) + r1r(t)−r2

,

m > 0, r1 > 0, r2 > 0,

S(YD(t)) = s(1 − ε)Y (t), s ∈ (0,1),

T (t) = εY (t), ε > 0.

(1.3)

The paper is organized as follows. In Sect. 2, weinvestigate the local stability of the equilibrium pointto system (1.2). Choosing the delay as a bifurcationparameter some sufficient conditions for the existenceof Hopf bifurcation are found. In Sect. 3, the formu-las determining the direction and the stability of thebifurcating periodic solutions are obtained by the nor-mal form theory and center manifold theorem intro-duced by Hassard et al. [17]. Section 4 is devoted tonumerical simulations of a specified version of themodel, in which we show the existence and the natureof the period solutions. Finally, conclusions are madein Sect. 5.

2 Qualitative analysis of system (2.3)

Using the functions (1.3), system (1.2) becomes⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Y (t) = α1[aY (t)κ1r(t)−κ2 + g − s(1 − ε)Y (t)

− νY (t)] + α2[r(t) + cP (t) − 1]Y(t),

r(t) = β1P(t) + β2[mY(t) + r1r(t)−r2

− M(t − τ)],P (t) = 1 − r(t) − cP (t) + γ1[g − νY (t)],M(t) = g − νY (t),

(2.1)

where α1 > 0, α2 > 0, a > 0, κ1 > 0, κ2 > 0, β1 > 0,β2 > 0, m > 0, r1 > 0, r2 > 0, ν ∈ (0,1), s ∈ (0,1).

System (2.1) is a system of equations with moneysupply delay, and the equilibrium point of system (2.1)is as follows:

Y0 = g

ν, r0 =

[aY

κ1−10

s(1 − ν)

] 1κ2

,

P0 = 1 − r0

c, M0 = mY0 + r1

r0 − r2+ β1P0

β2.

(2.2)

Analysis of the stability and Hopf bifurcation of money supply delay in complex macroeconomic models

In (2.1) and considering the Taylor expansion of theright-side members from (2.1) up to the third order, wecan derive that

x(t) = A(t) + Bx(t − τ) + f(x(t)

)(2.3)

where

A =

⎛

⎜⎜⎝

a11 a12 α2cY0 0β2m a22 β1 0−γ1ν −1 −c 0−ν 0 0 0

⎞

⎟⎟⎠ ,

B =

⎛

⎜⎜⎝

0 0 0 00 0 0 −β2

0 0 0 00 0 0 0

⎞

⎟⎟⎠ ,

f(x(t)

) =

⎛

⎜⎜⎝

f1(x(t))

f2(x(t))

00

⎞

⎟⎟⎠ ,

a11 = α1

[

aY

α1−10 α1

rα20

− s(1 − ν) − ν

]

,

a12 = α2Y0 − α1aα2Y

α10

r1+α20

, a22 = − β2r1

(r0 − r2)2,

f1(x(t)

) = b11x21 + b12x1x2 + α2cx1x3 + b13x

22

+ b14x31 + b15x

32 + b16x1x

22 + b17x

21x2

+ h.o.t.,

f2(x(t)

) = c11x22 + c12x

32 + h.o.t.,

b11 = aα1Y

α1−20 α1(α1 − 1)

rα20

,

b12 = α2 − aα1Y

α1−10 α1α2

r1+α20

,

b13 = aα1Y

α10 α2(α2 + 1)

2rα2+20

,

b14 = aα1Y

α1−30 α1(α1 − 1)(α1 − 2)

3!rα20

,

b15 = aα1Y

α10 α2(α2 + 1)(−α2 − 2)

3!rα2+30

,

b16 = aα1Y

α1−10 α1α2(α2 + 1)

2r2+α20

,

b17 = aα1Y

α1−20 α1α2(1 − α1)

2r1+α20

,

c11 = β2r1

(r0 − r2)3, c12 = − β2r1

(r0 − r2)4,

x(t) = (Y(t) − Y0, r(t) − r0,K(t)

− K0,M(t) − M0)T

.

The characteristic equation of the linearized system(2.3) is

det

⎛

⎜⎜⎝

λ − a11 −a12 −α2cY0 0−β2m λ − a22 −β1 β2e

−λτ

γ1ν 1 λ + c 0ν 0 0 λ

⎞

⎟⎟⎠ = 0.

(2.4)

Furthermore, the following four-degree exponentialpolynomial equation is obtained:

λ4 + d3λ3 + d2λ

2 + d1λ + d11(λ + c)e−λτ

+ d12e−λτ = 0 (2.5)

where

d1 = c(a11a22 − β2ma12) + α2cY0(β2m − γ1νa22)

+ β1(a12γ1ν − a11),

d11 = −β2νa12, d12 = β1να2cY0,

d2 = a11a22 − β2ma12 − c(a11 + a22)

+ α2cY0γ1ν + β1, d3 = c − a11 − a22.

Let λ = iω0, τ = τ0, and substituting this into (2.5),for the sake of simplicity, denote ω0, τ0 by ω, τ ; then(2.5) becomes

ω4 − d3ω3i − d2ω

2 + d1ωi + [(d11c + d12) − d11ωi

]

× [cos(ωτ) − i sin(ωτ)

] = 0. (2.6)

Separating the real and imaginary parts, it is easy toget

ω4 − d2ω2 + b11ω sin(ωτ)

+ (d11c + d12) cos(ωτ) = 0,

− d3ω3 + d1ω + d11ω cos(ωτ)

− (d11c + d12) sin(ωτ) = 0.

(2.7)

By a simple calculation, the following equationscan be obtained:

cos(ωτ)

= (d2ω2 − ω4)(d11c + d12) + (d3ω

3 − d1ω)d11ω

(d11c + d12)2 + d211ω

2,

sin(ωτ)

= (d2ω2 − ω4)d11ω + (d3ω

3 − d1ω)(d11c + d12)

(d11c + d12)2 + d211ω

2.

(2.8)

J. Ma, H. Tu

Since sin2(ωτ) + cos2(ωτ) = 1, we have

d11ω10 + e4ω

8 + e3ω6 + e2ω

4 + e1ω2

− (d11c + d12)4 = 0, (2.9)

where

e1 = (d2

1 − 2d211

)(d11c + d12)

2,

e2 = [d2(d11c + d12) − d1d11

]2

+ d1(d11c + d12)[d2d11 − d3(d11c + d12)

] − d411,

e3 = [d2d11 − d3(d11c + d12)

]2 − d1d11(d11c + d12)

+ 2[d2(d11c + d12) − d1d11

]

× [d3d11 − (d11c + d12)

],

e4 = [d3d11 − (d11c + d12)

]2

− d11[d2d11 − d3(d11c + d12)

].

By denoting ς = ω2, (2.9) becomes

d11ς5 + e4ς

4 + e3ς3 + e2ς

2 + e1ς

− (d11c + d12)4 = 0. (2.10)

Let

ϕ(ς) = d11ς5 + e4ς

4 + e3ς3 + e2ς

2 + e1ς

− (d11c + d12)4. (2.11)

Since Lim ϕ(ς)ς→∞ = +∞, we conclude that if d11c+

d12 �= 0, then (2.10) has at least one positive real root.It is easy to use computer to calculate the roots of(2.10) when α1, α2, a, κ1, κ2, β1, β2, m, r1, r2, s, g, ν

of the system (2.1) are given.Without loss of generality, assume that it has five

positive roots, defined by ς1, . . . , ς5, respectively.Then (2.9) has five positive roots ωi = √

ςi, i =1, . . . ,5.

In view of (2.8), we have

τ ki = 1

ωi

arcos

{(d2ω

2 − ω4)(d11c + d12) + (d3ω3 − d1ω)d11ω

(d11c + d12)2 + d211ω

2+ 2kπ

}

. (2.12)

where i = 1, . . . ,5, k = 0,1,2,3, . . .. Then ±iωi is apair of purely imaginary roots of (2.7) with τ k

i .Define

τ0 = τ 0i0

= min

i = 1, . . . ,5τ 0i , ω0 = ωk0 . (2.13)

Now when τ = 0, (2.5) becomes

λ4 + d3λ3 + d2λ

2 + (d1 + d11)λ + d11c + d12 = 0.

(2.14)

According to well-known Routh–Hurwitz criteria,all roots of (2.14) have a negative real part if and onlyif the following four conditions are satisfied:

(1) d3 > 0,

(2) d2d3 − (d1 + d11) > 0,

(3) (d1 + d11)[d2d3 − (d1 + d11)

]

− d23 (d11c + d12) > 0,

(4) d3(d11c + d12) > 0.

(2.15)

Taking the derivative of τ in (2.5), it is easy to ob-tain

dλ

dτ= λ(d11λ + d11c + d12)e

−λτ

4λ3 + 3d3λ2 + 2d2λ + d1 − τ(d11λ + d11c + d12)e−λτ. (2.16)

For the sake of simplicity, denote ω0 τ0 respectivelyby ω, τ ; then

Re

[(dλ

dτ

)∣∣∣∣λ=iω0,τ=τ0

]

= k1k3 + k2k4

k23 + k2

4

, (2.17)

where

k1 = −d11ω2 cos(ωτ) + (d11c + d12)ω sin(ωτ),

k2 = d11ω2 sin(ωτ) + (d11c + d12)ω cos(ωτ),

k3 = −3d3ω2 + d11

[1 + cos(ωτ)

]

− τ[(d11c + d12) cos(ωτ) + d11ω sin(ωτ)

],

k4 = −4ω3 + 2d2ω − d11 sin(ωτ)

− τ[d11ω cos(ωτ) − (d11c + d12) sin(ωτ)

].

Clearly, if k23 + k2

4 �= 0 holds, then


sign

[

Re

(dλ

dτ

)∣∣∣∣τ=τ0

]

= sign

[

Re

(dλ

dτ

)−1∣∣∣∣τ=τ0

]

.

(2.18)

Up to now we can employ a result from Ruan andWei [19] to analyze (2.5), which is stated as follows:

Lemma 2.1 Consider the exponential polynomial

P(λ, e−λτ1 , .., e−λτm

)

= λn + p(0)1 λn−1 + · · · + p

(0)n−1λ + p(0)

n

+ [p

(1)1 λn−1 + · · · + p

(1)n−1λ + p(1)

n

]e−ωτ1 + · · ·

+ [p

(m)1 λn−1 + · · · + p

(m)n−1λ + p(m)

n

]e−ωτm

(2.19)

where τi ≥ 0 (i = 1,2, . . . ,m) and p(i)j (i = 1,2, . . . ,

m; j = 1,2, . . . , n) are constants. As (τ1, τ2, . . . , τm)

vary, the sum of the order of the zero of P(λ, e−λτ1 , ..,

e−λτm) on the open right half-plane can change only ifa zero appears on or cross the imaginary axis.

Theorem 2.1 Making the following assumptions:

(P1) If (2.15) holds, (2.14) has four roots with neg-ative real parts, system (2.3) is stable near theequilibrium;

(P2) Re( dλdτ

) �= 0;

then the following results hold:

(I) For Eq. (2.3), its zero solution is asymptoticallystable for τ ∈ [0, τ0);

(II) Equation (2.3) undergoes a Hopf bifurcation atthe origin when τ = τ0.

That is, system (2.3) has a branch of bifurcating peri-odic solutions from the zero solution near τ = τ0.

It is implying that the government can stabilize in-trinsically unstable economy if the monetary policydelay is sufficiently short, but the system becomes lo-cally unstable when the monetary policy delay is toolong.

3 Stability of bifurcating periodic solutions

In this section, formulas for determining the directionof Hopf bifurcation periodic solutions of system (2.3)at τ0 are presented by employing the normal formmethod and center manifold theorem introduced byHassard et al. [17].

For convenience, let t = sτ , xi(t) = ui(tτ ) and τ =τ0 + μ, μ ∈ R. Then system (2.3) is equivalent to thesystem:

xt = τ[Lμxt +�(μ,xt )

](3.1)

where Ck[−1,0] = {ϑ |ϑ : [−1,0] → R4, each com-ponent of ϑ has k order continuous derivative},xt (θ) = x(t + θ) = (x1(t + θ), x2(+θ), x3(t + θ),

x4(t + θ))T ∈ C,Lμ is a one-parameter family ofbounded linear operators in C → R4, given by

Lμφ = τAφ(0) + τBφ(−1), (3.2)

where φ(θ) = (φ1(θ),φ2(θ),φ3(θ),φ4(θ))T ∈C[−1,0] and f (x(t)) = τ�(μ,xt ),� : R × C → R4.

By the Riesz representation theorem, there exists amatrix whose components are bounded variation func-tions η(θ,μ) in [0,1] → R4, such that

Lμφ =∫ 0

−1dη(θ,0)φ(θ), φ ∈ C[−1,0]. (3.3)

In fact, we can choose

η(θ,μ) = τAδ(θ) + τBδ(θ + 1), (3.4)

where δ(θ) is a Dirac function, such that (3.2) is satis-fied.

For φ ∈ C1[0,1], define

Φ(μ)ψ ={

dφ(θ)dθ

, −1 ≤ θ < 0,∫ 0−1 dη(θ,μ)φ(θ), θ = 0,

Ψ (μ)φ =

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

⎛

⎜⎜⎝

0000

⎞

⎟⎟⎠ , −1 ≤ θ < 0,

�(μ,φ), θ = 0.

(3.5)

In order to conveniently study Hopf bifurcation, wetransform system (3.1) into an operator equation of theform

xt = Φxt + Ψ xt (3.6)

where xt = x(t + θ) = (x1(t + θ), x2(+θ),

x3(t + θ), x4(t + θ))T , θ ∈ (−1,0].The adjoint operator Φ∗ of Φ is defined by

Φ∗(μ)ψ ={− dψ(κ)

dκ, 0 < κ ≤ 1,

∫ 0−1 dηT (κ,μ)φ(−κ), κ = 0,

(3.7)

where ηT is the transport of the matrix η.The domains of Φ and Φ∗ are C1[−1,0] and

C1[0,1], respectively. In order to normalize the eigen-vector of operator Φ and adjoint operator Φ∗, the fol-lowing bilinear form is needed to be introduced:

J. Ma, H. Tu

〈ψ,φ〉 = ψ(0) · φ(0)

−∫ 0

θ=−1

∫ θ

ξ=0ψT (ξ − θ) dη(θ)φ(ξ) dξ.

(3.8)

Here η(θ) = η(θ,0), C2 is complex plane. And for c

and d in C2, c · d = ∑4i=1 cidi , where ci and di are

components of c and d , respectively.Then, as usual, we normalize p and p∗ by the con-

dition

〈ψ∗,Φφ〉 = ⟨Φ∗ψ,φ

⟩,

⟨p∗,p

⟩ = 1,⟨p∗, p

⟩ = 0,(3.9)

for (φ,ψ) ∈ D(A) × D(A∗).From discussion in Sect. 2 and transformation

t = sτ , it follows that iω0τ0 is the eigenvalue of Φ(0)

and other eigenvalues have strictly negative real parts.Thus they are also eigenvalues of Φ∗. Next we cal-culate the eigenvector p of Φ belonging to the eigen-value iω0τ0 and the eigenvector p∗ of Φ∗ belongingto the eigenvalue −iω0τ0.

Let

p(θ) =

⎛

⎜⎜⎝

1p1

p2

p3

⎞

⎟⎟⎠ eiτ0ω0θ , −1 < θ ≤ 0. (3.10)

From the above discussion, it can be easily shownthat

Φp(0) = iτ0ω0p(0). (3.11)

Furthermore, we can obtain

p1 = −γ1ν − (ω0i + c)ω0i − a11 + a12γ1ν

α2cY0a12(c + iω0),

p2 = ω0i − a11 + a12γ1ν

α2cY0a12(c + iω0), p3 = − ν

iω0.

(3.12)

Similarly, suppose that the eigenvector q∗ of Φ∗ is

p∗(κ) = 1

ρ

⎛

⎜⎜⎝

1p∗

1p∗

2p∗

3

⎞

⎟⎟⎠ eiτ0ω0κ , 0 ≤ κ < 1, (3.13)

where ρ is a coefficient which will be determined later.Then, the following relationship is obtained:

Φ∗p∗(0) = −iτ0ω0p∗(0). (3.14)

Therefore, we obtain

q∗1 = α2cY0 + a12(ω0i − c)

(−ω0i + c)(ω0i + a22) − β1,

q∗2 = a12 + (ω0i + a22)q

∗1 , q∗

3 = β2

ω0ieiω0τ0q∗

1 .

(3.15)

As 〈p∗,p〉 = 1 and recalling the definition of bilin-ear form (3.8), we can deduce that⟨p∗,p

⟩ = p∗(0) · p(0)

−∫ 0

θ=−1

∫ θ

ξ=0p∗T (ξ − θ) dη(θ)p(ξ) dξ

= 1

ρ

(1, p∗

1, p∗2, p∗

3

)

−∫ 0

θ=−1

∫ θ

ξ=0

1

ρ

(1, p∗

1, p∗2, p∗

3

)

× e−iω0τ0(ξ−θ) dη(θ)

⎛

⎜⎜⎝

1p1

p2

p3

⎞

⎟⎟⎠ eiτ0ω0ξ dξ

= 1

ρ

[(1 + p1p

∗1 + p2p

∗2 + p3p

∗3

)

− τ0β2p3p∗1

]. (3.16)

Consequently, we have

ρ = 1 + p1p∗1 + p2p

∗2 + p3p

∗3 − τ0β2p3p

∗1 . (3.17)

Using the same method it is easy to prove that〈p∗, p〉 = 0.

Next, we study the stability of bifurcating periodicsolutions. The bifurcating period solution Z(t,μ(ε))

has an amplitude O(ε) and a non-zero Floquet expo-nent β(ε) with β(0) = 0. Under the hypotheses, μ, β

are given by{

μ = μ2ζ2 + μ4ζ

4 + · · · ,β = β2ζ

2 + β4ζ4 + · · · . (3.18)

The sign of μ2 indicates the direction of bifurcationwhile that of β2 determines the stability Z(t,μ(ε)). Inthe following, we will show how to derive the coeffi-cients in these expansions.

We first construct the coordinates to describe a cen-ter manifold Ω0 near μ = 0, which is a local invariant,attracting a two-dimensional manifold.

Let

z(t) = ⟨p∗, xt

⟩,

W(t, θ) = xt − 2 Re[z(t)p(θ)

],

(3.19)

where xt is a solution of (3.1). On the manifold Ω0:W(t, θ) = W(z(t), z(t), θ), where

W(z(t), z(t), θ

) = W20(θ)z2

2+ W11zz

+ W02z2

2+ · · · . (3.20)


In fact, z, z are local coordinates of the center man-ifold Ω0 in the directions of q and q∗, respectively.

The existence of center manifold Ω0 enables us toreduce to (3.1) an ordinary differential equation in asingle complex variable on Ω0.

For the solution xt ∈ Ω0 of (3.1), since μ = 0,

z(t) = ⟨p∗, xt

⟩ = ⟨p∗,Φxt + Ψ xt

⟩

= ⟨q∗,Φxt

⟩ + ⟨p∗,Ψ xt

⟩ = ⟨Φ∗p∗, xt

⟩ + ⟨p∗,Ψ xt

⟩

= iω0τ0z + p∗(0) ·�(0,W(t,0)

+ 2 Re[z(t)p(0)

]). (3.21)

Rewrite (3.21) as

z(t) = iω0τ0z + g(z, z), (3.22)

where

g(z, z) = g20(θ)z2

2+ g11zz + g02

z2

2+ g21z

2z + · · · .(3.23)

In the following, the motivation is to expand g inpowers of z and z and then obtain, from the coeffi-cients of this expansion, the values of μ2 and β2 usingalgorithm presented by Hassard et al. [17].

According to (3.6) and (3.21), we have

W = xt − zp − ˙zp= Φxt + Ψ xt − [

iω0τ0z + p∗(0) ·�(z, z)]p

− [−iω0τ0z + p∗(0) · �(z, z)]p

= ΦW − 2 Re[p∗(0) ·�(z, z)p(θ)

] + Rxt

=

⎧⎪⎪⎨

⎪⎪⎩

ΦW − 2 Re[p∗(0) ·�(z, z)p(θ)],−1 ≤ θ < 0,

ΦW − 2 Re[p∗(0) ·�(z, z)p(θ)] +�,

θ = 0.

(3.24)

Let

W = ΦW + H(z, z, θ), (3.25)

where

H(z, z, θ) = H20(θ)z2

2+ H11zz + H02

z2

2+ · · · .

(3.26)

Taking the derivative of W with respect to t in(3.20), we have

W = Wzz + Wz˙z. (3.27)

Substituting (3.20) and (3.22) into (3.27), we obtain

W = (W20z + W11z + · · ·)(iτ0ω0z + g)

+ (W11z + W02z + · · ·)(−iτ0ω0z + g). (3.28)

Then substituting (3.20) and (3.26) into (3.25), thefollowing results is obtained:

W = (ΦW20 + H20)Z2

2+ (ΦW11 + H11)zz

+ (ΦW02 + H02)z2

2+ · · · . (3.29)

Comparing the coefficients of (3.28) and (3.29), wehave

(Φ − 2iτ0ω0)W20(θ) = −H20(θ),

ΦW11(θ) = −H11(θ).(3.30)

Combing (3.21) and (3.22), we can see that

g(z, z) = q∗(0) ·�(z, z)

= τ0

ρ

(1, p∗

1, p∗2, p∗

3

)�(z, z) (3.31)

and, as is known,

xt (θ) = (x1t (θ), x2t (θ), x3t (θ), x4t (θ)

)T

= W(t, θ) + zq(θ) + zq(θ),

x1t (θ) = z + z + W(1)20 (0)

z2

2+ W

(1)11 (0)zz

+ W(1)02 (0)

z2

2+ O

(∣∣(z, z)

∣∣3)

x2t (θ) = zp1 + zp1 + W(2)20 (0)

z2

2+ W

(2)11 (0)zz

+ W(2)02 (0)

z2

2+ O

(∣∣(z, z)

∣∣3)

,

x3t (θ) = zp2 + zp2 + W(3)20 (0)

z2

2+ W

(3)11 (0)zz

+ W(3)02 (0)

z2

2+ O

(∣∣(z, z)

∣∣3)

.

Thus, it follows that

g(z, z) = g20(θ)z2

2+ g11zz + g02

z2

2+ g21z

2z + · · · ,

g20 = 2τ0

ρ

[b11 + b12p1 + b13p

21 + α2cp2 + c11p

21p

∗1

],

g11 = τ0

ρ

[2b11 + b12(p1 + p1) + 2b13|p1|2

+ α2c(p2 + p2) + 2c11|p1|2p∗1

],

g02 = 2τ0

ρ

[b11 + b12p1 + b13p

21 + α2cp2 + c11p

21p

∗1

],

g21 = 2τ0

ρ

{

b11

(

W(1)11 (0) + W

(1)20 (0)

2

)

+ b12

(

W(2)11 (0) + W

(1)20 (0)

2p1

J. Ma, H. Tu

+ W(1)11 (0)p1

)

(3.32)

+ b13

(

p1W(2)11 (0) + W

(2)20 (0)

2p1

)

+ 3b14

+ 3b15p21p1 + b16

(2p1p1 + p2

1

)

+ b17(2p1 + p1)

+ α2c

[

W(3)11 (0) + W

(3)20 (0)

2p1

+ W(1)20 (0)

2p2 + W

(1)11 (0)p2

]

+[

c11

(

p1W(2)11 (0) + W

(2)20 (0)

2

)

+ 3c12p21p1

]

p∗1

}

.

In the following, we focus on the computation ofW20(θ) and W11(θ).

Relations (3.24) and (3.27) imply that

H(z, z, θ)

= −2 Re[q∗(0) ·�(z, z)q(θ)

] + Rxt

= −gp(θ) − gp(θ) + Rxt

= −(

g20(θ)z2

2+ g11zz + g02

z2

2+ · · ·

)

p(θ)

−(

g20(θ)z2

2+ g11zz + g02

z2

2+ · · ·

)

p(θ)

+ Rxt . (3.33)

Comparing the coefficients of (3.26) with (3.33),we obtain

H20(θ) = −g20p(θ) − g02p(θ), −1 ≤ θ < 0,

H11(θ) = −g11p(θ) − g11p(θ), −1 ≤ θ < 0.(3.34)

Substituting (3.34) into (3.30), it follows that

W20(θ) = 2iτ0ω0W20(θ) + g20p(θ) + g02p(θ),

W11(θ) = g11p(θ) + g11p(θ).(3.35)

It is easy to obtain the solution of (3.35):

W20(θ) = ig20

τ0ω0p(0)eiτ0ω0θ

+ ig02

3τ0ω0p(0)e−iτ0ω0θ + E1e

2iτ0ω0θ (3.36)

and

W11(θ) = g11

iτ0ω0p(0)eiτ0ω0θ

+ ig11

τ0ω0p(0)e−iτ0ω0θ + E2, (3.37)

where E1 = (E(1)1 ,E

(2)1 ,E

(3)1 ,E

(4)1 )T ∈ R4, E2 =

(E(1)2 ,E

(2)2 ,E

(3)2 ,E

(4)2 )T ∈ R4.

Next we focus on the computation of E1 and E2.From (3.30), we have

ΦW20(0) = 2iτ0ω0W20(0) − H20(0) (3.38)

and

ΦW11(0) = −H11(0). (3.39)

From the definition of Φ in (3.5), we obtain∫ 0

−1dη(θ)W20(θ) = 2iτ0ω0W20(0) − H20(0) (3.40)

and∫ 0

−1dη(θ)W11(θ) = −H11(0). (3.41)

From (3.1), (3.33) and (3.34), we have

H20(0) = −g20q(0) − g02q(0) + 2τ0

⎛

⎜⎜⎝

b11 + b12p1 + b13p21 + α2cp2

c11p21

00

⎞

⎟⎟⎠ (3.42)

and

H11(θ) = −g11q(θ) − g11q(θ) + τ0

⎛

⎜⎜⎝

2b11 + b12(p1 + p1) + 2b13|p1|2 + α2c(p2 + p2)

2c11|p1|200

⎞

⎟⎟⎠ . (3.43)

Note that


{

iτ0ω0I −∫ 0

−1eiτ0ω0θ dη(θ)

}

p(0) = 0, and

{

−iτ0ω0I −∫ 0

−1e−iτ0ω0θ dη(θ)

}

p(0) = 0.

(3.44)

Substituting (3.36) and (3.42) into (3.40), we obtain

(

2iτ0ω0I −∫ 0

−1e2iτ0ω0θ dη(θ)

)

E1 = 2τ0

⎛

⎜⎜⎝

b11 + b12p1 + b13p21 + α2cp2

c11p21

00

⎞

⎟⎟⎠ . (3.45)

Consequently, we obtain

E1 = 2τ0

⎛

⎜⎜⎝

2iω0 − a11 −a12 0 0−β2m 2iω0 − a22 0 β2e

−2iω0τ0

γ1ν 1 2iω0 + c 0ε 0 0 2iω0

⎞

⎟⎟⎠

−1 ⎛

⎜⎜⎝

b11 + b12p1 + b13p21 + α2cp2

c11p21

00

⎞

⎟⎟⎠ . (3.46)

Similarly, substituting (3.37) and (3.43) into (3.41), we get

E2 = τ0

⎛

⎜⎜⎝

−a11 −a12 0 0−β2m −a22 0 β2

γ1ν 1 c 0ν 0 0 0

⎞

⎟⎟⎠

−1 ⎛

⎜⎜⎝

2b11 + b12(p1 + p1) + 2b13|p1|2 + α2c(p2 + p2)

2c11|p1|200

⎞

⎟⎟⎠ . (3.47)

Finally, Eq. (3.33) can be obtained. The following

parameters can be calculated:

C1(0) = i

2ω0

(

g20g11 − 2|g11|2 − 1

3|g02|2

)

+ g21

2;

μ2 = −ReC1(0)

Reλ′(τ0); β2 = 2 ReC1(0).

(3.48)

Above all, the following result is established:

Theorem 3.1 Under the conditions of Theorem 2.1:

(I) μ = 0 is Hopf bifurcation of system (3.1).

(II) The direction of Hopf bifurcation is determined

by the sign of μ2: if μ2 > 0, the Hopf bifurcation

is supercritical; if μ2 < 0, the Hopf bifurcation

is subcritical.

(III) The stability of bifurcation periodic solutions is

determined by β2: if β2 > 0, they are unstable; if

β2 < 0, they are stable.

4 Numerical simulation

In this section, some numerical results of simulatingsystem (2.3) are presented at justifying the theoremobtained above.

We consider system (2.1) with a = 0.37, α1 = 0.95,α2 = 0.65, κ1 = 0.39, κ2 = 0.92, s = 0.25, β1 = 0.9,β2 = 0.92, m = 0.0035, r1 = 1, r2 = 0.0075, c = 0.96,γ1 = 0.01, ν = 0.15, g = 30.

The equilibrium point of system (2.1) is as follows:

Y0 = 200, r0 = 0.0544,

P0 = 0.9849, M0 = 22.9568.(4.1)

So, the system (2.3) becomes

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

Y (t) = 0.95[0.37(x1(t) + Y0)0.39(x2(t) + r0)

−0.92

+ 30 − 0.3625(x1(t) + Y0)]+ 0.65[x2(t) + 0.96x3(t)][x1(t) + Y0],

x2(t) = 0.9(x3(t) + P0) + 0.92[0.0035(x1(t) + Y0)

+ 1x2(t)+r0−0.0075 − (x4(t − τ) + M0)],

x3(t) = −x2(t) − 0.96x3(t) − 0.0015x1(t),

x4(t) = −0.15x1(t).

(4.2)

J. Ma, H. Tu

Fig. 1 τ = 1.7 < τ0

Fig. 2 τ = 2.1001 = τ0

In the following, we consider the system (4.2);Eq. (2.10) has only one positive solution ς+

0 = 0.1615.According to (2.12), we obtain τ k = 2.1001 +4.9762kπ (k = 0,1,2,3, . . .).

First, we choose τ = 1.7 < τ0: then the correspond-ing waveform plots are in Fig. 1(a) and phase plots inFig. 1(b); by Theorem 2.1, we know that the zero so-lutions of system (4.2) are asymptotically stable.

Second, we choose τ = 2.1001 = τ0. The state ofthe system (4.2) changes from an equilibrium to a cy-cle, the variables x2, x3 are still stable, but the vari-ables x1, x4 are periodic, as shown by the waveformplots in Fig. 2(a) and phase plots in Fig. 2(b).

Finally, we choose τ = 2.2 > τ0: the correspond-ing waveform plots are in Fig. 3(a) and phase plots inFig. 3(b). It is easy to see that (a) and (b) in Fig. 3undergo a Hopf bifurcation, the variables x2, x3 arestill stable, but the variables x1, x4 become larger andlarger.

By computation through the method provided inSect. 3, we can derive μ2 = 1.6939061832 > 0. Henceby Theorem 3.1, we know that the bifurcating point issupercritical.

Correspondingly β2 = −0.3794932121 < 0, and sothe bifurcating solutions under the condition of β2 =−0.3794932121 < 0, are stable.


Fig. 3 τ = 2.2 > τ0

5 Conclusions

Non-linear dynamic finance system model providesrich dynamical behaviors. Whether from the viewpointof non-linear system or from implement of macroeco-nomic policy, the system analysis is useful in solvingproblems of both theoretical and practical value.

In this paper, first of all, based on the IS-LM modelstudied in Refs. [14–16] and the model of (1.1) re-searched in Refs. [5–7, 11], a four-dimensional dy-namic macro-economy system has been established.Then, we proposed a money supply delay in themacro-economy system. Choosing the delay as a bi-furcation parameter, the stability of the system and theconditions for Hopf bifurcation were derived. Further-more, the direction and the stability of the bifurcatingperiodic solutions were obtained by the normal formtheory and the center manifold theorem. This researchhas an important theoretical as well as practical value.

Acknowledgements This work was supported by The Na-tional Nature Science Foundation of China (grant No. 61273231)and supported by Doctoral Fund of Ministry of Education ofChina (grant No. 20130032110073).

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Analysis of the stability and Hopf bifurcation of money supply delay in complex macroeconomic models