# Analysis of the Stability and Hopf Bifurcation of Currency Supply Delay in an Opened Kaldorian Business Cycle Model

Authors: Liming Zhao; Zhipei Zhao
Journal: Mathematical Problems in Engineering
 (2016)

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## Abstract

First of all, we establish a three-dimension open Kaldorian business cycle model under the condition of the fixed exchange rate. Secondly, with regard to the model, we discuss the existence of equilibrium point and the stability of the system near it with a time delay in currency supply as the bifurcating parameters of the system. Thirdly, we discuss the existence of Hopf bifurcation and investigate the stability of periodic solution generated by the Hopf bifurcation; then the direction of the Hopf bifurcation is given. Finally, a numerical simulation is given to confirm the theoretical results. This paper plays an important role in theoretical researching on the model of business cycle, and it is crucial for decision-maker to formulate the macroeconomic policies with the conclusions of this paper.


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## Body

## 1. Introduction

Economy is a pivotal factor in the national development, and it not only closely links to the daily life of singular people but also plays an important role in maintaining the stability of society. Because of the complexity and the instability of the economic system operation, any tiny external condition changes may give rise to instability of the whole system. In recent years, more and more economists put their eyes on the field of nonlinear dynamic system and use it to build relevant economic models and do some research on the operation law of the economic system [1–3]. As is well known, a time delay always exists in most of economic systems. Therefore, as a branch of nonlinear dynamics, the delayed nonlinear dynamics have been used widely by economists. Kaldor used ordinary differential equations to research the business cycle model, and he proposed that nonlinear investment function and saving function were an important factor to generate periodical fluctuation in economic system [4]. His research had given a new direction in business cycle research. Kalecki [5] proposed a new viewpoint in researching the business cycle: the capital for operation is related to the time. In other words, original part of the investment capital is the profit saving in a period of time previously; the investment decision will influence the capital stock [6], and a period or a time delay always exists in capital investment for production in economic operational process. After summing up the idea of Kaldor and Kalecki, Krawiec and Szydlowski [7] bring about a time delay in Kaldor model and get the Kaldor-Kalecki model which shows that(1)Y′t=αIYt,Kt-SYt,Kt,K′t=IYt-τ,Kt-dKt.Reference [8], in allusion to the model above, did some research on the influences of the time delay on the stability of the system and analyzed the generation condition of the Hopf bifurcation. Kaddar and Talibi Alaoui [9] thought that investment decision would exert an influence on both the gross value of product and capital stock. They modified the second equation in (1) and transfer the investment function I(Y(t-τ),K(t)) into I(Y(t-τ),K(t-τ)); that is,(2)Y′t=αIYt,Kt-SYt,Kt,K′t=IYt-τ,Kt-τ-dKt.Reference [10] puts the time delay in Kaldor-Kalecki model and did some research on its dynamic properties. Wang and Wu found that the time delay would destroy the stability of the system and concluded that the variation of capital stock caused by investment decision was a factor of leading the economic system to an instable status.Most of researches on Kaldor-Kalecki model, from the viewpoint of the functional differential equation or the viewpoint of the ordinary differential equation, concentrate on the enclosed type of economic system. Therefore, we will focus on an open Kaldorian business model following the research method and conclusion of the Kaldor-Kalecki model [11, 12]. Kaldorian business cycle theory is based on the initial thoughts of Kaldor and was enriched by Lorenz [13], Gandolfo [14], and Agliari et al. [15]. According to [16], the Kaldorian model can be written in the following form:(3)dYtdt=αc1-ε-1Yt+γ-β1δYt-d1Kt+β1β2Mt+cT0+C0+G,dKtdt=γ-β1δYt-d1Kt+β1β2Mt,dMtdt=δ1Y+βδYt-β2M-r0,where Y is actual gross domestic product, C is actual consumption expenditure, K is physical capital stock, T is actual income tax, G is government expenditure, M is nominal currency supply, α is the market adjustment coefficient, and β is the degree of capital movement.Furthermore, we denotem=c(1-ε)-1, n=γ-β1δ, b=β1β2, and a=cT0+C0+G; then the model can be transformed to(4)dYtdt=αmYt+nYt-d1Kt+bMt+a,dKtdt=nYt-d1Kt+bMt,dMtdt=δ1Yt+βδYt-β2Mt-r0.Considering that the right-hand sides of (4) should be analytical, we bring about the transform Y→Y, and the system is C1(0,+∞) class which is shown as(5)dYtdt=α2mYt+n-d1KtY-1t+bMtY-1t+aY-1t,dKtdt=nYt-d1Kt+bMt,dMtdt=δ1Y2t+βδYt-β2Mt-r0+c.The time delay of money supply is a crucial factor which influences the effects of monetary policy. Monetary policy changes normally take a certain amount of time to have an effect on the economy. The time delay could span anywhere from nine months up to two years. Fiscal policy and its effects on output have a shorter time delay. When monetary policy attempts to stimulate the economy by lowering interest rates, it may take up to 18 months for evidence of any improvement in economic conditions to show up. However, there are few papers considering the macroeconomic model with currency supply delay. In this paper, we propose that there is a finite delay between the rate and the currency supply in a frame work. We study how the delay affects the macroeconomic stability. By using the theory of normal and center manifold theorem, an explicit algorithm for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions is derived. To the best of our knowledge, this paper is the first one to deal with Hopf bifurcation about the dynamic Kaldor-Kalecki model with discrete delay in currency supply. Considering currency supply delay to macroeconomic model is very important to implement the monetary policy and the government’s stabilization policy.This paper is organized as follows. In Section2, we investigate the local stability of the equilibrium point of system (6). By choosing the delay as the bifurcation parameter, some sufficient conditions for the existence of Hopf bifurcation are found. In Section 3, the formulas determining the direction and the stability of the bifurcating periodic solutions are obtained by the normal form theory and center manifold theorem introduced by Hassard et al. [17]. In Section 4, some numerical simulations consequences are given. Finally, some conclusions are made.


## 2. Equilibrium Stability and Existence of the Hopf Bifurcation

Bring the time delay in currency supply; system (5) becomes(6)dYtdt=α2mYt+n-d1KtY-1t+bMt-τY-1t+aY-1t,dKtdt=nYt-d1Kt+bMt,dMtdt=δ1Y2t+βδYt-β2Mt-r0+c.At first, we will find the equilibrium of system (6) with τ=0, and the equilibrium point is equal to the solutions of the equation sets:(7)αmYt+n-d1KtY-1t+bMt-τY-1t+aY-1t=0,nYt-d1Kt+bMt=0,δ1Y2t+βδYt-β2Mt-r0+c=0.Then, the equilibrium of system (6) is E(Y∗,K∗,M∗), where(8)Y∗=Y0=-2am=2cT0+2C0+2G1-c1-ε,K∗=bβδmY0+βmnβ2Y0-bβmr0-abδ1+bcm2mββ2d1,M∗=βδmY0-βmr0-aδ1+cm2mββ2.Through the linear transformation,(9)x1t=Yt-Y∗,x2t=Kt-K∗,x3t=Mt-M∗.The equilibrium pointE(Y∗,K∗,M∗) of system (6) can be translated into the original point, and the system is transformed as(10)dx1tdt=α2Y∗m+nx1-d1x2+mx12+bx3t-τ2x1+2Y0,dx2tdt=-nx1+d1x2+bx3,dx3tdt=βδ+2Y∗δ1x1-ββ2mx3+δ1x12.Considering Taylor expansion of the right members from (10) until the third order, we derive that(11)x˙t=At+Bxt-τ+fxt,where f(x(t)) is the nonlinear part of system (10) and its expression is(12)fxt=12Bx,x+16Cx,x,x+ox4,where B(x,y) and C(x,y,u) are the bilinear and trilinear functions about the variation x,y,u shown as(13)Bix,y=∑j,k=12∂2Fiξ,0∂ξj∂ξkξ=0xjyk,i=1,2,Cix,y,u=∑j,k=13∂3Fiξ,0∂ξj∂ξk∂ξlξ=0xjykul,i=1,2,3.According to the expressions above, we denote that(14)A=12Y02αY02Y0m+n-αY02m+2a-2Y03αd102Y02n-2Y02d12Y02b4Y03δ1+2Y02βδ0-2Y02ββ2,B=12Y000αb000000,fxt,xt-τ=f1xt,xt-τ03δ1x12, where(15)fxt,xt-τ=32-αmY02+α2Y0m+nY03x13t+αbY03x12tx3t-τ-αd1Y03x12tx2t+32αmY0-α2Y0m+nY02x12t-αbY02x1tx3t-τ+αd1Y02x1tx2t.Furthermore, the characteristic equation of the linear part of system (10) is(16)λ3+Aλ2+Bλ+C+Dλ+Ee-λτ=0,where(17)A=-αm2+ββ2+d1-αn2Y0+aαY02,B=-αβmβ22-12αmd1+ββ2d1-αβnβ22Y0+aαββ2Y02+aαd1Y02,C=-αβmβ2d2+αbd1δ1+αbβδd12Y0+aαββ2d1Y02,D=-αbδ1-αbβδ2Y0,E=-αbd1δ1-αbβδd12Y0.When τ=0, according to the Routh-Hurwitz criteria, if the system meets(H1)(18)A>0,AB+D-C+E>0,then all roots of (16) are negative, and at this time the equilibrium E(Y∗,K∗, M∗) is stable; when τ≠0, according to the Hopf bifurcating condition, we assume that (16) has a pair of conjugate pure imaginary characteristic roots, and let λ=iω, and substitute it into (16); we get(19)-iω3-Aω2+iBω+C+iDω+Ee-iτω=0.Separate the real and imaginary parts of (19), and we get(20)-Aω2+C+Ecos⁡ωτ+Dωsin⁡ωτ=0,-ω3+Bω+Dωcos⁡ωτ-Esin⁡ωτ=0;then(21)-Aω2+C=-Ecos⁡ωτ-ωDsin⁡ωτ,-ω3+Bω=-ωDcos⁡ωτ+Esin⁡ωτ.Square both sides of (21) and add them all:(22)ω6+A2-2Bω4+B2-D2-2ACω2+C2-E2=0.For simplicity, we denote that(23)ω6+pω4+qω2+r=0,where p=A2-2B, q=B2-D2-2AC, and r=C2-E2.Letz=ω2; we obtain that(24)hz=z3+pz2+qz+r=0;then, we discuss the roots of (24).(H2) When r⩾0 and p2-3q2<0, (24) has no positive roots.(H3) When p2q2+18pqr-27r2-4p3r-4q3<0, (24) has a positive root and a pair of conjugate pure imaginary roots; that is, (24) has a positive root.(H4) When p2-3q=0, pq-9r=0, and p<0, (24) has three equal positive roots: zk=-p/3, k=1,2,3, and ω2=-p/3. That is, (24) has a unique positive root: ω=-p/3.(H5) When p<pq-9r/p2-3q<0 and p2q2-18pqr+27r2+4p3r+4q3=0, (24) has positive roots. That is, z1=-p+pq-9r/p2-3q and z2=z3=-1/2pq-9r/p2-3q. And (24) has two solutions: ω1,2=±-p+pq-9r/p2-3q.(H6) When p2q2+18pqr-27r2-4p3r-4q3>0, (24) has three positive roots; that is, z1=-p-2p2-3qcos⁡θ/3/3 and z2,3=-p-2p2-3qsin⁡(θ/3±π/6)/3, where θ=arccos⁡T, T=2(p2-3q)p-3(pq-9r)/2(p2-3q)3, and (p2-3q>0,-1<T<1).With regard to anyωj above, we denote(25)τjk=1ωjθj+2kπ,j=1,2,3,k=0,1,2,….Thus, according to (21),(26)cos⁡θj=Dωj4+AEωj2-BDωj2-CED2ωj2+E2,sin⁡θj=ωjADωj2-Eωj2+BE-CDD2ωj2+E2and we will discuss transversality condition; we have the following lemmas.Lemma 1.
Assume that (16) has roots which is shown as λ(τ)=α(τ)+iω(τ), where α(τ) and ω(τ) meet the conditions α(τjk)=0,ω(τjk)=ωj; then sign⁡dRe⁡λ(τjk)⁡/dτjk=sign⁡(h′(τjk)).Proof.
Substituteλ(τ)=α(τ)+iω(τ) into the characteristic equation (16), and differentiate it with respect to τ; we have(27)3λ2dλdτ+2Aλdλdτ+Bdλdτ+Ddλdτe-λτ+Dλ+Ee-λτ-dλdττ-λ=0and then(28)dλdτ=Dλ2+Eλe-λτ3λ2+2Aλ+B+D-Dλτ-Eτe-λτwith respect to (16). Consider(29)Dλ2+Eλe-λτ=-λ4+Aλ3+Bλ2+Cλ;substitute it into the equation above; we have(30)Re⁡dλτjkdτjk=ωj2S3ωj4+2ωj2A2-2B+B2-2AC-D2=ωj2S3ωj4+2ωj2p+q=ωj2h′zjS,where(31)S=-3ωj2+B-DEτjksin⁡ωjτjk+D-Eτjkcos⁡ωjτjk2+2Aωj-D-Eτjksin⁡ωjτjk-DEτjkcos⁡ωjτjk2>0.Thus, we have(32)sign⁡dReλτjkdτjk=sign⁡h′τjk.
According to Lemma1, we know that if system (10) meets hypothesis (H3) or (H4), then α′(τk)>0; and if system (10) meets hypothesis (H5), then α′(τ1k)>0 and α′(τ2k)=0; if system (10) meets hypothesis (H6), then α′(τ1k)>0, α′(τ2k)<0, and α′(τ3k)>0, where τjk,j=1,2,3;k=0,1,2,…,   defined by (25).With regard to Lemma1 we can conclude the following.Lemma 2.
With respect to (6),(1)
if system (6) meets (H1) and (H2), for any τ∈[0,+∞), then the real parts of the roots of (16) are negative,(2)
if system (6) meets (H1) and (H3) or (H1) and (H4), then there exists a series of {τk}k≥0; when τ∈[0,τ0), all roots of (16) have negative part; and when τ=τk, (16) has a pair of conjugate pure imaginary roots; and when τ∈(τk,τk+1], (16) has k+1 pairs of characteristic roots with positive real part,(3)
if system (6) meets conditions (H1) and (H5), then there exists a series of {τ1k}k≥0; when all of characteristic roots are negative and when τ=τ1k, (16) has a pair of conjugate pure imaginary roots ±iω0,(4)
if system (6) meets conditions (H1) and (H6), there exists a series of {τjk}k≥0, k=0,1,2,…,j=1,2,3, and if τ10<τ20<τ30, then when τ∈[0,τ10), all the characteristic roots of (16) have negative real part; when τ∈(τ10,τ20], (16) has a pair of characteristic roots with real part.Theorem 3.
If system (6) meets hypothesis (H1), one can conclude the following:(1)
If system (6) meets hypothesis (H2), for all τ∈[0,+∞), the system is stable near the equilibrium E∗(Y∗,K∗,M∗).(2)
If system (6) meets hypothesis (H3) or (H4), for all τ∈[0,τ0), the system is stable near the equilibrium E∗(Y∗,K∗,M∗); and when τ=τ0, system (6) undergoes Hopf bifurcation at E∗(Y∗,K∗,M∗).(3)
If system (6) meets hypothesis (H5) or (H6), one denotes τ0=min1⩽j⩽3{τj0}; system (6) is stable near the equilibrium E∗(Y∗,K∗,M∗) for all τ∈[0,τ0), and when τ=τ0, system (6) undergoes Hopf bifurcation at E∗(Y∗,K∗,M∗); that is, system (6) has a branch of periodic solutions bifurcation the zero solution near τ=τ0.


## 3. Stability of Bifurcating Periodic Solutions

In this section, formulation for determining the direction of Hopf bifurcation periodic solutions of system (6) at τ0 will be presented following the normal form method and center manifold theorem introduced by Hassard et al. in [17].Denoteτ as the criteria value of any τ=τk(k=1,2,3,…) and assume that the pure imaginary characteristic roots of system (6) are ±iω with respect to τ. For convenience, through linear transformation t→t/τ, we normalize the time delay based on system (10), and we get(33)dX1tdt=ατ2Y∗m+nX1-d1X2+mX12+bX3t-12x1+2Y0,dX2tdt=τ-nX1+d1X2+bX3,dX3tdt=τβδ+2Y∗δ1X1-ββ2mX3+δ1X12.Assume thatτ=τ0+μ; when μ=0, the system undergoes Hopf bifurcation, and (33) can be transformed to the equation on C([-1,0],R3); that is,(34)Z˙t=LμZt+fZt,μ.For φ=(φ1,φ2,φ3)T∈C, we obtain(35)Lμφ=τ0+μαm2+12nY0φ10-αd1φ20+αbφ-112nY0φ10-d1φ20+bφ30d12+12βδY0φ10-ββ2φ30,fφ,μ=f1φ,μ03δφ120,where(36)fxt,xt-τ=32-αmY02+α2Y0m+nY03φ130+αbY03φ120φ3-1-αd1Y03φ120φ20+32αmY0-α2Y0m+nY02φ120-αbY02φ10φ3-1+αd1Y02φ10φ20,and we easily know that the characteristic equation of L is(37)det⁡λI-∫-10eλθdθημ,θ=0.According to Riesz theorem, we take the function of bounded variation; that is, η(θ)=τAδ(θ)+τBδ(θ+1). Consider(38)Lμφ=∫-10eλθdηθφθ,φ∈C.Then, we can define infinitely small generating element A(μ)=A with respect to (34); we have(39)Aφθ=dφθdθ;θ∈-1,0Lμφ=∫-10dηθφθ;θ=0,where φ∈C([-1,0],C3) and(40)Rφθ=0;θ∈-1,0fμ,φ;θ=0;thus, (34) is equal to abstract ordinary differential equation:(41)x˙tt=Axt+Rxt.Assume that±iω0 are the solutions to the characteristic equation of system (37) when τ=τ0, and ±iω0τ0 are the characteristic value of A, and the corresponding characteristic vector is p(θ)=p(0)eiω0τ0θ, where p(0)=p1(0),p2(0),p3(0)T meets the characteristic equation (iω0τ0I-∫-10‍eiω0τ0θdη(θ))p(0)=0. That is to say that (iω0τ0I-A-Be-iω0τ0)p(0)=0, and we have(42)iω0τ0-αm+n2Y0Y0αd1αbe-iω0τ0-n2Y0iω0τ0+d1-b-d1-βδ2Y00iω0τ0+ββ2p10p20p30=0,where(43)p10=1,p20=iω0τ0n+2Y0bd1+bβδ+nββ22Y0-ω02τ02+iω0τ0ββ2+id1ω0τ0+d1ββ2,p30=2iY0τ0ω0d1+iβδτ0ω0+2d12Y0+d1βδ2Y0-ω02τ02+iω0τ0ββ2+iω0τ0d1+ββ2d1,∀ψ∈C([0,1],C3); denote A∗ as the formally adjoint operator of A:(44)A∗ψs=-dψsds,s∈0,1∫-10ψ-sdηs,s=0and assume that qs=Dq0eiω0τ0s is the characteristic vector with respect to the characteristic value-iω0τ0 of A∗, where q(0)=(q1(0),q2(0),q3(0))T meets the characteristic equation D-iω0τ0I-AT-BTeiω0τ0q0=0. Consider(45)-iω0τ0-αm+n2Y0-n2Y0-d1-βδ2Y0αd1-iω0τ0+d10-αbeiω0τ0-b-iω0τ0+ββ2q10q20q30=0.By calculating, q1(0)=1,q2(0)=αd1/iω0τ0-d1,q30=-αbiω0τ0-d1eiω0τ0+d1/iω0τ0-d1iω0τ0-ββ2; thus, q(s)=D(q10,q20,q30)Teiω0τ0 according to bilinear form q,p=q¯0p0-∫-10‍∫0θ‍q¯ξ-θdηθpξdξ; that is,(46)qs,pθ=D¯1,q¯10,q¯201,p10,p20T-∫-10∫0θq¯10,q¯20,q¯30e-iω0τ0ξ-θdηθp10,p20,p30eiω0τ0ξ=D¯1+q¯10p10+q¯20p20-αbq10p30e-iω0τ0,where D¯-1=q¯1(0)p1(0)+q¯2(0)p2(0)+q¯3(0)p3(0)-αbq¯1(0)p3(0)e-iω0τ0.According to the solution of (33) at τ=τ0, we define z(t)=q,xt and denote(47)Wt,θ=xtθ-ztpθ-z¯tp¯θ;thus, on center manifold W(t,θ)=W(z,z¯,θ), where W(z,z¯,θ) can be expressed into power series form with variations z and z¯,(48)Wz,z¯,θ=W20θz22+W11θzz¯+W02θz¯22+⋯.We can obtain the form of (34) on center manifold which is shown as follows:(49)z˙t=iω0τ0zt+q¯0fτ0,zpθ+z¯p¯θ+Wz,z¯,θ≜iω0τ0zt+q¯0f0z,z¯,and we denote(50)f0z,z¯=fz2z22+fzz¯zz¯+fz¯2z¯22+fz2z¯z2z¯2+⋯;for simplicity, we denote (49) as the form shown as(51)z˙t=Iω0τ0zt+gz,z¯and denote(52)gz,z¯=g20z22+g11zz¯+g02z¯22+g21z2z¯2+⋯,where (51) is the norm form of (34) on center manifold.In order to obtain the direction and stability of Hopf bifurcation, we need to calculate regulating coefficientsg20, g11, g02, and g21 and compare with formulas (48), (50), and (51); we have(53)g20=q¯0fz2,g11=q¯0fzz¯,g02=q¯0fz¯2,g21=q¯0fz2z¯.With regard to (47),(54)xtθ=ztpθ+z¯tp¯θ+Wz,z¯,θ=p10p20p30eiω0τ0θzt+p¯10p¯20p¯30eiω0τ0θz¯t+W201θW202θW203θz22+W111θW112θW113θzz¯+W021θW022θW023θz¯22+⋯.Furthermore, we obtain that(55)xt=p0zt+p¯0z¯t+W200z22+W110zz¯+W020z¯22+⋯,xt-1=p0zte-iω0τ0+p¯0z¯teiω0τ0+W20-1z22+W11-1zz¯+W02-1z¯22+⋯.Substitutex(t) into formula f0 and compare the coefficient with formula (50); and(56)fz2=fz21,0,3δ1p¯120T,fzz¯=fzz¯1,0,6δ1p10p¯10T,fz¯2=fz¯21,0,3δ1p¯120T,fz2z¯=τ0fz2z¯1,0,6δ1p10W1110+3δ1p¯10W2010T,where(57)fz21=αd1S2p10p20-αbS2p10p30-3αn2S2p120+3αap120S3,fzz¯1=αd1S2p10p¯20+αd1S2p¯10p20+6aαS3p10p¯10-3αnS2p10p¯10-αbS2p10p¯30-αbS2p¯10p30,fz¯21=αd1p¯10p¯20S2-αbp¯10p¯320S2-3αnp¯1202S2+3αap¯120S3,fz2z¯1=-2αd1p10p¯10p20S3+2αbp10p¯10p30S3+3αap¯10W2010S3-9αap120p¯10S4-αd1p120p¯20S3-3αnp¯10W20102S2+αd1p¯10W20202S2+αd1W2010p¯202S2+9αnp120p¯102S3-αbp¯10W20302S2+αbp120p¯30S3-αbW2010p¯202S2+αd1W1110p20S2-αbp10W113-1S2-αbW1110p30S2-3αnp1W1110S2+6αap1W1110S3+αd1p1W1120S2.Substitutefz2,fzz¯,fz¯2 into formula (53); then(58)g20=q¯10fz21+3δ1q¯30p¯120,g11=q¯10fzz¯1+6δ1q¯30p10p¯10,g02=q¯10fz¯21+3δ1q¯30p¯120.However, there are some unknown terms,W20(0),W11(0),W11(-1), in formula fz2z¯, and thus in order to obtain g21 we must calculate W20(θ),W11(θ) firstly, and with respect to formula (52) we know that W˙=x˙t-z˙p-z¯p¯; that is,(59)W˙=AW-gpθ-g¯p¯θ,θ∈-1,0,AW-gp0-g¯p¯0+f0,θ=0.On the other hand, the system can be transferred intoW˙=Wzz˙+Wz¯z¯˙, the center manifold near the origin point:(60)W˙=Wzz˙+Wz¯z¯˙=W20θz+W11θz¯z˙+W11θz+W20θz¯z¯˙=W20θz+W11θz¯iω0τ0z+gz,z¯+W11θz+W02θz¯iω0τ0z+g¯z,z¯.Compared with the coefficients ofz2/2 and zz¯ in formulas (59) and (60),(61)2iω0τ0I-AW20θ=-g20pθ-g¯02p¯θ,θ∈-1,0,-g20p0-g¯02p¯0+fz2,θ=0,(62)-AW11θ=-g11pθ-g¯11p¯θ,θ∈-1,0,-g11p0-g¯11p¯0+fzz¯,θ=0.According to formulas (37) and (61), we have(63)W20θ=2iω0τ0W20θ+g20pθ+g¯02p¯θ.Substitutep(θ)=p(0)eiω0τ0θ into formula above; then(64)W20θ=ig20ω0τ0p0eiω0τ0θ+ig¯023ω0τ0p¯0e-iω0τ0θ+E1e2iω0τ0θ.Furthermore, we have(65)∫-10dηθW20θ=2iω0τ0W200+g20p0+g¯02p¯0-fz2.Substitute formula (64) into (65) with respect to (iω0τ0I-∫-10‍eiω0τ0θdη(θ))p(0)=0; we have(66)2iω0τ0I-∫-10e2iω0τ0θdηθE1=fz2.Denote(67)2iω0τ0I-∫-10e2iω0τ0θdηθ-1=a11a12a13a21a22a23a31a32a33;then,(68)E1=1S1a11fz21+3a13δ1p¯12a21fz21+3a23δ1p¯12a31fz21+3a33δ1p¯12=E11E12E13,where(69)
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.In a similar way, with regard to formulas (37) and (62), we have W˙11(θ)=g11p(θ)+g¯11p¯(θ) and we get(70)W11θ=-ig11ω0τ0p0eiω0τ0θ+ig¯11ω0τ0p¯0e-iω0τ0θ+E2.According to (62), when θ=0,(71)∫-10dη0,θW11θ=g11p0+g¯11p¯0-fzz¯.Substitute (70) into (71) and we have -∫-10‍dηθE2=fzz¯, where(72)E2=1S2-2ββ2Y0αfzz¯1+6b1-2Y0δ1p10p¯10-4Y0bδ1+2bβδ+2βnβ2Y0αd1fzz¯1+6b2Y0m-2Y0n+nδ1p10p¯10d1-4Y0δ1+2βδY0αfzz¯1+12Y0mδ1p10p¯10,S2=4Y02bδ1+2Y0bβδ+2Y0βmβ2-2Y0bδ1-bβδ; with regard to (64) and (70), we have(73)W20θ=ig20ω0τ0p10eiω0θ+ig¯023ω0τ0p¯10eiω0θ+E1e2iω0θ,W11θ=-ig11ω0τ0p10eiω0θ+ig¯11ω0τ0p¯10eiω0θ+E2e2iω0θ. And we acquired g21 shown as follows from the formula above:(74)g21=q¯10τ0fz2z¯1+6δ1q¯10p10W1110+3δ1q¯10p¯10W2010.Now, we have acquired all values ofg20,g11,g02,g21; thus we get normal form on center manifolds (51) of (34). From the certain equation, we can get the parameters of Hopf bifurcation such as C1(0),μ2, and β2 which are shown as(75)C10=i2ω0g20g11-2g112-g0223+g212,(76)μ2=-Re⁡C10Re⁡λ′τ0,β2=2Re⁡C10,τ2=-Im⁡C10+μ2ω′0ω0,P=2πω01+τ2ε2+Oε4.From the parametersμ2andβ2, the direction of Hopf bifurcation at τ=τ0 and the stability of periodic solution generated by the Hopf bifurcation can be presented, where μ2 decides the direction of the Hopf bifurcation and the parameter β2 gives the stability of the periodic solution which is shown as (77)xt,με=2εRe⁡q0eiω0t+2ε2Re⁡q0g202iω0e2iω0t-g11iω0-g026iω0e2iω0t+e2Re⁡W200e2iω0t+W110+Oε3,0⩽t⩽Pε=2πω01+τ2ε2+⋯.


## 4. Numerical Simulation

In order to reflect the basic situation of the macroeconomic business cycle system with currency supply delay and confirm the theoretical analysis above, we do some numerical simulation to system (6).We take parametersα,β,β2,m,n,d1,a,b,c,δ,δ1,r0 with the value shown as follows:(78)α=1,β=10,m=-0.66,n=15,d1=0.3,a=108,b=1,δ1=-0.3,δ=10,β2=1,r0=3. Then system (6) can be transformed into(79)dYtdt=-0.33Yt+7.5-0.15Kt-0.5Mt-τ-54Yt,dKtdt=15Yt-0.3Kt+Mt,dMtdt=-0.3Y2t+100Yt-10Mt-10.Through calculating, the system has a unique equilibrium pointE∗(163.6363,1046.3,122), and we let τ=0; according to Theorem 3, when the system meets hypothesis (H1), the equilibrium is stable as shown in Figures 1–4.Figure 1
History of gross domestic productY when τ=0.Figure 2
History of physical capital stockK when τ=0.Figure 3
History of currency supplyM when τ=0.Figure 4
Phase diagram of economic growth system whenτ=0.We conclude that the gross domestic productY and physical capital stock K and the currency supply M undergo some oscillation at the initial time and then converge into the equilibrium point E∗(163.6363,1046.3,122) rapidly.Furthermore, we consider system (79); (24) has only one positive solution, z0+=0.1615. According to (25), we obtain τk=22.045+4.9762kπ. Through the calculation by (75), we obtain that C1(0)=-2.7964+4.3725i; μ2>0; β2=-5.5928<0, and according to Theorem 3, Hopf bifurcation is generated when τ>τ0, and the bifurcating periodical solution is asymptotically stable. Figures 5–7 show the time histories of the variations, gross domestic product Y, physical capital stock K, and currency supply M when τ=22.045. Figure 8 shows the phase diagram of economic growth system. We can see clearly that there is a limit cycle near equilibrium E∗(163.6363,1046.3,122).Figure 5
History of gross domestic productY when τ=22.045.Figure 6
History of physical capital stockK when τ=22.045.Figure 7
History of currency supplyM when τ=22.045.Figure 8
Phase diagram of economic growth system whenτ=22.045.


## 5. Conclusion

Nonlinear dynamic finance system model provides rich dynamical behaviors. Whether from the viewpoint of nonlinear system or from implementation of macroeconomic policy, the analysis of the researchers is useful in solving problems of both theoretical and practical importance.In this paper, based on the research on open Kaldorian business cycle model in [11, 12, 16], a three-dimension business cycle model under the fixed exchange rate condition is established. Firstly, we discuss the existence of equilibrium point and investigate the stability of the system near the equilibrium point with the time delay in currency supply. We find that the system presents different stable status under the different hypothesis, which is shown in Theorem 3. Secondly, we investigate the existence condition of Hopf bifurcation and discuss the stability of periodic solution which is generated by Hopf bifurcation; then, the direction of Hopf bifurcation is given. Furthermore, a numerical simulation is given to confirm the theoretical results.We conclude that the system will maintain a stable status under the condition that the parameterτ is in a certain range. Accompanied with the increase of the time delay, the system will lose its stability gradually. Once the parameter τ is out of the certain range, the system will generate periodical fluctuation; that is, the business cycle is generated. In other words, when the time delay increases to the critical value, the system will undergo Hopf bifurcation and a bifurcation periodical solution will be generated. From our research, the time delay in the money supply in the Kaldorian business cycle model can change the stability of the whole system. The money supply is one of the crucial factors of the business cycle forming. This paper plays an important role in theoretical researching on the model of business cycle with time delay in currency supply, and it is crucial for decision-maker to formulate the macroeconomic policies with the conclusions of this paper.



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*Source: 8056894-2016-08-15.xml*