In this paper we investigate the impact of delayed tax revenues on the fiscal
policy out-comes. Choosing the delay as a bifurcation parameter we study the
direction and the stability of the bifurcating periodic solutions. With respect
to the delay we show when the system is stable. Some numerical examples are
finally given for justifying the theoretical results.Comment: 13 pages, 12 figure

Chilarescu, Constantin

Neamtu, Mihaela

Opris, Dumitru

English

arXiv

The paper investigates the impact of delayed tax revenues on the fiscal policy out-comes. Choosing the delay as a bifurcation parameter we study the direction and the stability of the bifurcating periodic solutions. We show when the system is stable with respect to the delay. Some numerical examples are given to confirm the theoretical results

Munich RePEc Personal Archive

MPRAMunich Personal RePEc ArchiveHopf bifurcation in a dynamic IS-LMmodel with time delayMihaela Neamtu and Dumitru Opris and ConstantinChilarescu12. November 2005Online at http://mpra.ub.uni-muenchen.de/13270/MPRA Paper No. 13270, posted 10. February 2009 08:36 UTCarXiv:math/0511322v1  [math.DS]  12 Nov 2005Hopf bifurcation in a dynamic IS-LMmodel with time delayMihaela Neamt¸ua∗, Dumitru Opris¸b, Constantin ChilaˇrescuaaDepartment of Economic Informatics and Statistics,Faculty of Economics, West University of Timis¸oara,str. Pestalozzi, nr. 16A, 300115, Timis¸oara, Romania,E-mails:mihaela.neamtu@fse.uvt.ro, cchilarescu@rectorat.uvt.rob Department of Applied Mathematics, Faculty of Mathematics,West University of Timis¸oara, Bd. V. Parvan, nr. 4, 300223, Timis¸oara, Romania,e-mail: opris@math.uvt.roAbstract. In this paper we investigate the impact of delayed taxrevenues on the fiscal policy out-comes. Choosing the delay as a bi-furcation parameter we study the direction and the stability of thebifurcating periodic solutions. With respect to the delay we showwhen the system is stable. Some numerical examples are finally givenfor justifying the theoretical results.Keywords: delay differential equation, stability, Hopf bifurcation, IS-LM model.2000 AMS Mathematics Subject Classification: 34K20, 34C25, 37G15, 91B64 .1. IntroductionThe differential equations with time delay play an important role for econ-omy, engineering, biology and social sciences, because a lot of phenomena aredescribed with their help. In this paper, we consider a model from economy,of the IS-LM type with time delay and we study how the delay affects the∗Corresponding author1macroeconomic stability. The Hopf bifurcation and normal form theories aretools for establishing the existence and the stability of the periodic solutions.Similar ideas can be found in [5], [9], [10].In [2] Cesare and Sportelli taking into account the papers [6], [7], [8],study a dynamic IS-LM model of the following type:Y˙ (t)=α[I(Y (t), r(t))+g−S(Y (t)−T (Y (t), Y (t−τ))))−T (Y (t), Y (t−τ)))]r˙(t) = β[L(Y (t), r(t))−M(t)]M˙(t) = g − T (Y (t), Y (t−τ))),with Y as income, I as investment, g as government expenditure (constant),S as savings, T as tax revenues, r as rate of interest, L as liquidity, M as realmoney supply and α, β as positive constants. The time delay τ appears infunction T:T (Y (t), Y (t−τ))) = d(1− ε)Y (t) + dεY (t− τ), (1)where d ∈ (0, 1) is a common average tax rate and ε ∈ (0, 1) is the incometax share.Based on the papers [2], [1], [11], we consider the following IS-LM model:Y˙ (t)=α[I(Y (t), r(t))+g−S(Y (t)−T (Y (t), Y (t−τ)))−T (Y (t), Y (t−τ))]r˙(t) = β[L(Y (t), r(t))−M(t)]K˙(t) = I(Y (t− τ), r(t))− δK(t)M˙(t) = g − T (Y (t), Y (t−τ)),(2)with δ > 0 and with the initial conditions:Y (t) = ϕ(t), t ∈ [−τ, 0], r(0) = r1,M(0) = M1, K(0) = K1.In the following analysis we will consider the function T given by (1), theinvestment, the saving and the liquidity of the form:I(Y (t), r(t)) = aY (t)α1r(t)−α2 , a > 0, α1 > 0, α2 > 0,S(Y (t)− T (Y (t), Y (t− τ))) = s(Y (t)− T (Y (t), Y (t− τ))), s ∈ (0, 1),L(r(t)) = mY (t) +γ0r(t)− r2, m > 0, γ0 > 0, r2 > 0.(3)2The paper is organized as follows. In section 2 we investigate the localstability of the equilibrium point associated to system (2). Choosing thedelay as a bifurcation parameter some sufficient conditions for the existenceof Hopf bifurcation are found. In section 3 there is the main aim of the paper,namely the direction, the stability and the period of a limit cycle solution.Section 4 gives some numerical simulations which show the existence and thenature of the periodic solutions. Finally, some conclusions are given.2. The qualitative analysis of system (2).Using functions (1) and (3), system (2) becomes:Y˙ (t)=α[((s−1)(1−ε)d−s)Y (t)+dε(s−1)Y (t−τ)+aY (t)α1r(t)−α2+g]r˙(t) = β[mY (t) +γ0r(t)− r2−M(t)]K˙(t) = aY (t− τ)α1r(t)−α2 − δK(t)M˙(t) = g − d(1− ε)Y (t)− dεY (t− τ),(4)α, β > 0, α1, α2 > 0, m > 0, γ0 > 0, r2 > 0, δ > 0, s ∈ (0, 1), d ∈ (0, 1),ε ∈ (0, 1).System (4) is a system of equations with time delay. The qualitativeanalysis is done using the methods from [3].The equilibrium point of system (4) has the coordinates Y0, r0, K0,M0,where:Y0=gd, r0=[s(1−d)aY 1−α10 ]−1α2 , K0=s(1−d)Y0δ,M0= mY0+γ0r0−r2. (5)Using the translationY = x1 + Y0, r = x2 + r0, K = x3 +K0,M = x4 +M0in (4) and considering the Taylor expansion of the right members from (4)until the third order, we have:x˙(t) = Ax(t) +Bx(t− τ) + F (x(t), x(t− τ)) (6)3whereA=αa1 aαρ01 0 0βm βγ0γ1 0 −β0 aρ01 −δ 0a4 0 0 0, B=αb1 0 0 00 0 0 0aρ10 0 0 0b4 0 0 0(7)where x(t) = (x1(t), x2(t), x3(t), x4(t))T , anda1 = (s− 1)d(1− ε)− s+ aρ10, b1 = (s− 1)dε, a4 = −d(1− ε), b4 = −dερ01 = −α2Yα10 r−(α2+1)0 , ρ10 = α1Yα1−10 r−α20andF (x(t), x(t− τ)) = (F 1(x(t), x(t− τ)), F 2(x(t), x(t− τ)), F 3(x(t), x(t− τ)),(8)F 4(x(t), x(t− τ)))T ,F 1(x(t), x(t− τ)) =αa2ρ20x21(t) + αaρ11x1(t)x2(t) +αa2ρ02x22(t)+αa6(ρ30x31(t) + 3ρ21x1(t)2x2(t) + 3ρ12x1(t)x2(t)2 + ρ03x2(t)3)F 2(x(t), x(t− τ)) =γ02γ2x2(t)2 +γ06γ3x2(t)3F 3(x(t), x(t− τ)) =a2ρ20x1(t− τ)2 + aρ11x1(t− τ)x2(t) +a2ρ02x2(t)2++a6(ρ30x31(t− τ) + 3ρ21x1(t− τ)2x2(t) + 3ρ12x1(t− τ)x2(t)2 + ρ03x2(t)3)F 4(x(t), x(t− τ)) = 0andρ20 = α1(α1 − 1)Yα1−20 r−α20 , ρ11 = −α1α2Yα1−10 r−(α2+1)0ρ02 = α2(α2 + 1)Yα10 r−(α2+2)0 , ρ30 = α1(α1 − 1)(α1 − 2)Yα1−30 r−α20ρ21 = −α1α2(α1 − 1)Yα1−20 r−(α2+1)0 , ρ12 = α1α2(α2 + 1)Yα1−10 r−(α2+2)0ρ03 = −α2(α2 + 1)(α2 + 2)Yα10 r−(α2+3)0 ,γ1 = −1(r0 − r2)2, γ2 =2(r0 − r2)3, γ3 = −6(r0 − r2)4.4The characteristic equation of linear part from (6) is:det(λI − A− Be−λτ ) = (λ+ δ)∆(λ, τ) = 0 (9)where∆(λ, τ) = P (λ) + e−λτQ(λ) (10)P (λ) = λ3 + p2λ2 + p1λ+ p0, Q(λ) = q2λ2 + q1λ+ q0andp2 = −(αa1 + βγ0γ1), p1 = αβ(γ0a1γ1 +maρ01), p0 = αaβa4ρ01q2 = −αb1, q1 = αβγ0b1γ1, q0 = αβb4ρ01.To investigate the local stability of the equilibrium point, we begin byconsidering, as usual, the case without delay (τ = 0 ). In this case thecharacteristic polynomial is:(λ+ δ)(P (λ) +Q(λ)) = 0hence, according to the Hurwitz criterion, the equilibrium point is stable ifand only if:p2 + q2 > 0, (p1 + q1)(p2 + q2) > p0 + q0.When τ > 0, standard results on stability of systems of delay differentialequations postulate that a equilibrium point is asymptotically stable if anonly if all roots of equation (10) have a negative real part. It is well knownthat equation (10) is a transcendental equation which has an infinite numberof complex roots and the some possible roots with positive real part are finitein number.We want to obtain the values τ0 such that the equilibrium point (5)changes from local asymptotic stability to instability or vice versa. We needthe imaginary solutions of equation ∆(λ, τ) = 0. Let λ = ±iω be thesesolutions and without loss of generality we assume ω > 0. We suppose thatP (iω)+Q(iω) 6= 0, for all ω ∈ IR. The previous conditions are equivalent to(p1 + q1)(p2 + q2) 6= p0 + q0.A necessary condition to have ω as a solution of ∆(iω, τ) = 0 is that ωmust be a root of the following equation:f(ω) = ω6 + aFω4 + bFω2 + cF = 0 (11)5where aF = p22 − q22 − 2p1, bF = 2q0q2 − 2p0p2 − q21 + p21, cF = p20 − q20.Let k = −aF3and fD =14[f(k)]4 +19[f′(k)]3 .Using the results from [2], it results:Proposition 1. 1. Let ε >12. Then the following cases can be discerned:(i) If aF ≥ 0 or bF ≤ 0, then equation (11) has only one real positiveroot;(ii) If aF < 0, bF > 0 and fD ≤ 0, then equation (11) has only one realpositive root.2. If ε <12, aF < 0, bF < 0 and fD ≤ 0 then equation (11) has two realpositive roots which are distinct if fD 6= 0.3. If ε =12and:(i) aF < 0, bF = 0 then equation (11) has only one real positive root;(ii)aF < 0, bF > 0, then equation (11) has two real positive roots;(iii)bF < 0, then equation (11) has only one real positive root.Also, we have:Theorem 1. If we suppose that the equilibrium point (Y0, r0, K0,M0) islocally asymptotically stable without time delay, then in conditions of Propo-sition 1 there exists only one stability switch.Theorem 2. If τ0 is a stability switch and fD 6= 0, then a Hopf bifurcationoccurs at τ0, whereτ0 =1ω0arctg(ω0(ω40q2 − ω2(q0 − q1p2 + q2p1) + q0p1 − p0q1)ω40(q1 − q2p2) + ω20(q0p2 − q1p1 + p0q2)− p0q0)and ω0 is a root of (11).3. The normal form for system (4). Cyclical behavior.In this section we describe the direction, stability and the period of thebifurcating periodic solutions of system (4). The method we use is basedon the normal form theory and the center manifold theorem introduced byHassard [4]. Taking into account the previous section, if τ = τ0 then all rootsof equation (9) other than ±iω0 have negative real parts, and any root ofequation (9) of the form λ(τ) = α(τ)± iω(τ) satisfies α(τ0) = 0, ω(τ0) = ω06anddα(τ0)dτ6= 0. For notational convenience let τ = τ0 + µ, µ ∈ IR. Thenµ = 0 is the Hopf bifurcation value for equations (4).Define the space of continuous real-valued functions as C = C([−τ0, 0], IR4).In τ = τ0 + µ, µ ∈ IR, we regard µ as the bifurcation parameter. ForΦ ∈ C we define a linear operator:L(µ)Φ = AΦ(0) +BΦ(−τ)where A and B are given by (7) and a nonlinear operator F (µ,Φ) = F (Φ(0),Φ(−τ)), where F (Φ(0),Φ(−τ)) is given by (8). By the Riesz representa-tion theorem, there exists a matrix whose components are bounded variationfunctions, η(θ, µ) with θ ∈ [−τ0, 0] such that:L(µ)Φ =0∫−τ0dη(θ, µ)φ(θ), θ ∈ [−τ0, 0].For Φ ∈ C1([−τ0, 0], IR4) we define:A(µ)Φ(θ) =dΦ(θ)dθ, θ ∈ [−τ0, 0)0∫−τ0dη(t, µ)φ(t), θ = 0,R(µ)Φ(θ) =0, θ ∈ [−τ0, 0)F (µ,Φ), θ = 0.We can rewrite (6) in the following vector form:u˙t = A(µ)ut +R(µ)ut (12)where u = (u1, u2, u3, u4)T , ut = u(t+ θ) for θ ∈ [−τ0, 0].For Ψ ∈ C1([0, τ0], IR∗4), we define the adjunct operator A∗ of A by:A∗Ψ(s) =−dΨ(s)ds, s ∈ (0, τ0]0∫−τ0dηT (t, 0)ψ(−t), s = 0.7We define the following bilinear form:< Ψ(θ),Φ(θ) >= Ψ¯T (0)Φ(0)−∫ 0−τ0∫ 0ξ=0Ψ¯T (ξ − θ)dη(θ)φ(ξ)dξ,where η(θ) = η(θ, 0).We assume that ±iω0 are eigenvalues of A(0). Thus, they are also eigen-values of A∗. We can easily obtain:Φ(θ) = veλ1θ, θ ∈ [−τ0, 0] (13)where v = (v1, v2, v3, v4)T ,v1 = 1, v2 = −β(a4 + b4e−λ1τ0 −mλ1)λ1(λ1 − βγ0γ1), v4 =a4 + b4e−λ1τ0λ1,v3 =aλ1 + δ[ρ10e−λ1τ0 + ρ01β(a4 + b4e−λ1τ0 −mλ1)λ1(βγ0γ1 − λ1)]is the eigenvector of A(0) corresponding to λ1 = iω0 andΨ(s) = weλ2s, s ∈ [0,∞)where w = (w1, w2, w3, w4),w1 =λ2 − βγ0γ1αaρ011η¯, w2 =1η¯, w3 = 0, w4 = −βλ21η¯η =λ1 − βγ0γ1aαρ01(1+αb1λ1τ0e−λ1τ0−e−λ1τ0 + 1λ21)+v2−βλ1(v4+b4λ1τ0e−λ1τ0−e−λ1τ0 + 1λ21)is the eigenvector of A(0) corresponding to λ2 = −iω0.We can verify that: < Ψ(s),Φ(s) >= 1, < Ψ(s), Φ¯(s) >=< Ψ¯(s),Φ(s) >=0, < Ψ¯(s), Φ¯(s) >= 1.Using the approach of Hassard [4], we next compute the coordinates todescribe the center manifold Ω0 at µ = 0. Let ut = ut(t+ θ), θ ∈ [−τ0, 0) bethe solution of equation (12) when µ = 0 andz(t) =< Ψ, ut >, w(t, θ) = ut(θ)− 2Re{z(t)Φ(θ)}.On the center manifold Ω0, we have:w(t, θ) = w(z(t), z¯(t), θ)8wherew(z, z¯, θ) = w20(θ)z22+ w11(θ)zz¯ + w02(θ)z¯22+ w30(θ)z36+ . . .in which z and z¯ are local coordinates for the center manifold Ω0 in thedirection of Ψ and Ψ¯ and w02(θ) = w¯20(θ). Note that w and ut are real.For solution ut ∈ Ω0 of equation (12), since µ = 0, we have:z˙(t) = λ1z(t) + g(z, z¯) (14)whereg(z, z¯) = Ψ¯(0)F (w(z(t), z¯(t), 0) + 2Re(z(t)Φ(0))) == g20z(t)22+ g11z(t)z¯(t) + g02z¯(t)22+ g21z(t)2z¯(t)2+ . . .whereg20 = F120w¯1 + F220w¯2, g11 = F111w¯1 + F211w¯2, g02 = F102w¯1 + F202w¯2, (15)withF 120 = αa(ρ20 + 2ρ11v2 + ρ02v22), F111 = αa(ρ20 + ρ11(v¯2 + v2) + ρ02v2v¯2),F 220 = γ0γ2v22 , F211 = γ0γ2v2v¯2, F102 = F¯120, F202 = F¯220,andg21 = F121w¯1 + F221w¯2 (16)whereF 121 = αaρ20(2w111(0) + w120(0)) + αaρ11(2w211(0) + w220(0) + 2w111(0)v¯2+w120(0)v2) + αaρ02(2w211(0)v2 + w220(0)v¯2) + αa(ρ30 + 2ρ21v2 + 2ρ12v22++ρ03v22 v¯2 + ρ12v22 + ρ21v¯2)F 221 = γ0γ2(2w211(0)v2 + w220(0)v¯2) + γ0γ3v22 v¯2.The vectors w20(θ), w11(θ) with θ ∈ [−τ, 0] are given by:w20(θ) = −g20λ1veλ1θ −g¯023λ1v¯λ2θe + E1e2λ1θw11(θ) =g11λ1veλ1θ −g¯11λ1v¯eλ2θ + E2(17)9whereE1 = −(A + e−λ1τ0B − 2λ1I)−1F20, E2 = −(A+B)−1F11,where F20 = (F120, F220, F320, 0)T , F11 = (F111, F211, F311, 0)T .Based on the above analysis and calculation, we can see that each gijin (15), (16) are determined by the parameters and delay from system (4).Thus, we can explicitly compute the following quantities:C1(0) =i2ω0(g20g11 − 2|g11|2 −13|g02|2) +g212µ2 = −Re(C1(0))Reλ′(τ0), T2 = −Im(C1(0)) + µ2Imλ′(τ0)ω0, β2 = 2Re(C1(0)).(18)andλ′ =λ(q2λ2 + q1λ+ q0)e−λτ3λ2 + 2p2λ+ p1 + e−λτ (2q2λ+ q1 − τ(q2λ2 + q1λ+ q0))In summary, this leads to the following result:Theorem 3. In formulas (18), µ2 determines the direction of the Hopfbifurcation: if µ2 > 0(< 0), then the Hopf bifurcation is supercritical (sub-critical) and the bifurcating periodic solutions exit for τ > τ0(< τ0); β2 de-termines the stability of the bifurcating periodic solutions: the solutions areorbitally stable (unstable) if β2 < 0(> 0); and T2 determines the period of thebifurcating periodic solutions: the period increases (decreases) if T2 > 0(< 0).4. Numerical example.In this section we find the waveform plots through the formula:X(t+θ)=z(t)Φ(θ)+z¯(t)Φ¯(θ)+12w20(θ)z2(t)+w11(θ)z(t)z¯(t)+12w02(θ)z¯(t)2+X0,where z(t) is the solution of (14), Φ(θ) is given by (13), w20(θ), w11(θ), w02(θ) =w¯20(θ) are given by (17) and X0 = (Y0, r0, K0,M0)T is the equilibrium state.For the numerical simulations we use Maple 9.5. We consider system (4)with a = 0.38, α = 0.96, β = 1, α1 = 0.5, α2 = 0.83, γ0 = 1, d = 0.1, s = 0.3,10r2 = 0.003, δ = 0.2, m = 0.005, g = 50. The equilibrium point is: Y0=500,r0=0.03572181612,K0=675,M0=33.06065092. In what follows we considertwo different shares ǫ of delay tax revenues: ε = 0.3 and ε = 0.8.For ε = 0.3 we obtain: µ2=1.654628706 · 10−8, β2=2.224294680 · 10−9,T2 = 2.092652051 · 10−9, ω0 = 0.6685954740, τ0 = 4.965007916. Then theHopf bifurcation is supercritical, the solutions are orbitally unstable and theperiod of the solution is increasing. The wave plots are given in the followingfigures:Fig.1.Waveplot (t, Y (t)) Fig.2.Waveplot (t, r(t)) Fig.3.Waveplot (t,K(t))t500.00021362500.0000915580500499.9999084560499.9997863840200                        t403.57218347E-2803.5721831E-23.57218236E-2603.57218198E-23.57218161E-23.57218087E-23.57218049E-2203.57218012E-20                        t6.75000122E26.74999939E2806040206.74999878E206.75000061E26.75E2                         Fig.4.Waveplot (t,M(t)) Fig.5.Waveplot (t, I(t)) Fig.6.Waveplot (t, L(t))t33.060184488020 6033.06021533.060211184033.0602035533.0601959233.0601997433.060188290                        100135.00006104135.00004578135.00001526135134.99998474134.99995422134.99993896t50403020                        1033.06066513033.0606613233.0606536933.0606498733.0606460633.0606384333.06063461t50403020                        For ε = 0.8 we obtain: µ2=−9.160756314·10−8, β2=−1.968119398·10−8,T2 = −1.608068638 · 10−8, ω0 = 0.8553440397, τ0 = 3.918246696. Then theHopf bifurcation is subcritical, the solutions are orbitally stable and theperiod of the solution is decreasing. The wave plots are given in the followingfigures:11Fig.7.Waveplot (t, Y (t)) Fig.8.Waveplot (t, r(t)) Fig.9.Waveplot (t,K(t))t60504005.00000092E23020105.00000214E25E24.99999908E24.99999786E2                        3.57218012E-2t60504030201003.5721831E-23.57218198E-23.57218087E-2                        6.75000061E26.75E26.74999939E26050t4010 20 300                         Fig.10.Waveplot (t,M(t)) Fig.11.Waveplot (t, I(t)) Fig.12.Waveplot (t, L(t))33.0597190933.0597114633.0597000133.05968857t605040302010033.05973053                        10134.999954221350t135.0000457830135.0000610420135.00001526134.99993896134.99998474                        33.060661320t33.0606498730201033.0606689533.06063843                        5. Conclusions.From the analysis of the model with continuous time, it results that themodel accepts a limit cycle. The nature of the limit cycle is given by thecoefficients (18) which include the parameters of the model. We establishthe nature of the limit cycle. Because the expressions of the parameter andthe coefficients (18) are difficult to analyze directly, the use of Maple 9.5 wasessential. The paper’s results confirm that a series of economical processes,where a variable with time delay intervenes, have a limit cycle, thus, allowinga prediction concerning the evolution of the model.12References[1] Cai J, Hopf bifurcation in the IS-LM business cycle model with timedelay, Electronic Journal of Differential Equations, 2005(15):1-6.[2] De Cesare L, Sportelli M, A dynamic IS-LMmodel with delayed taxationrevenues, Chaos, Solitons and Fractals, 2005(25):233-44.[3] Hale J. K, Lunel S.M. V, Introduction to Functional DifferentialEquation, Springer-Verlag, New York, Applied Mathematical Sciences,1993(99).[4] Hassard B.D, Kazarinoff N.D, Wan Y.H, Theory and applications ofHopf bifurcation, Cambridge University Press, Cambridge, 1981.[5] Liao X, Li C, Zhou S, Hopf bifurcation and chaos in macroeconomicmodels with policy lag, Chaos, Solitons and Fractals, 2005(25):91-108.[6] Sasakura K, On the dynamic behavior of Schinasi’s business cycle model,J. Macroecon, 1994(16):423-44.[7] Schinasi G.J, A nonlinear dynamic model of short run fluctuations, Rev.Econ. Stud.,1981(48):649-56.[8] Schinasi G.J, Fluctuations in a dynamic intermediate-run IS-LMmodel: applications of the Poicar-Bendixon theorem, J. Econ Theory.,1982(28):369-75.[9] Szydlowski M, Krawiec A, The stability problem in the Kaldor-Kaleckibusiness cycle model, Chaos, Solitons and Fractals, 2005(25):299-305.[10] Takeuchi Y, Yamamura T, Stability analysis of the Kaldor model withtime delays: monetary policy and government budget constraint, Non-linear Analysis, 2004(5):277-308.[11] Torre V, Existence of limit cycles and control in complete Keynesiansystems by theory of bifurcations, Econometrica, 1977(45):1457-66.13

Hopf bifurcation in a dynamic IS-LM model with time delay

Munich Personal RePEc Archive

MPRA
Munich Personal RePEc Archive
Hopf bifurcation in a dynamic IS-LM
model with time delay
Mihaela Neamtu and Dumitru Opris and Constantin
Chilarescu
12. November 2005
Online at http://mpra.ub.uni-muenchen.de/13270/
MPRA Paper No. 13270, posted 10. February 2009 08:36 UTC
ar
X
iv
:m
at
h/
05
11
32
2v
1 
 [m
ath
.D
S]
  1
2 N
ov
 20
05
Hopf bifurcation in a dynamic IS-LM
model with time delay
Mihaela Neamt¸ua∗, Dumitru Opris¸b, Constantin Chilaˇrescua
aDepartment of Economic Informatics and Statistics,
Faculty of Economics, West University of Timis¸oara,
str. Pestalozzi, nr. 16A, 300115, Timis¸oara, Romania,
E-mails:mihaela.neamtu@fse.uvt.ro, cchilarescu@rectorat.uvt.ro
b Department of Applied Mathematics, Faculty of Mathematics,
West University of Timis¸oara, Bd. V. Parvan, nr. 4, 300223, Timis¸oara, Romania,
e-mail: opris@math.uvt.ro
Abstract. In this paper we investigate the impact of delayed tax
revenues on the fiscal policy out-comes. Choosing the delay as a bi-
furcation parameter we study the direction and the stability of the
bifurcating periodic solutions. With respect to the delay we show
when the system is stable. Some numerical examples are finally given
for justifying the theoretical results.
Keywords: delay differential equation, stability, Hopf bifurcation, IS-LM model.
2000 AMS Mathematics Subject Classification: 34K20, 34C25, 37G15, 91B64 .
1. Introduction
The differential equations with time delay play an important role for econ-
omy, engineering, biology and social sciences, because a lot of phenomena are
described with their help. In this paper, we consider a model from economy,
of the IS-LM type with time delay and we study how the delay affects the
∗Corresponding author
1
macroeconomic stability. The Hopf bifurcation and normal form theories are
tools for establishing the existence and the stability of the periodic solutions.
Similar ideas can be found in [5], [9], [10].
In [2] Cesare and Sportelli taking into account the papers [6], [7], [8],
study a dynamic IS-LM model of the following type:
Y˙ (t)=α[I(Y (t), r(t))+g−S(Y (t)−T (Y (t), Y (t−τ))))−T (Y (t), Y (t−τ)))]
r˙(t) = β[L(Y (t), r(t))−M(t)]
M˙(t) = g − T (Y (t), Y (t−τ))),
with Y as income, I as investment, g as government expenditure (constant),
S as savings, T as tax revenues, r as rate of interest, L as liquidity, M as real
money supply and α, β as positive constants. The time delay τ appears in
function T:
T (Y (t), Y (t−τ))) = d(1− ε)Y (t) + dεY (t− τ), (1)
where d ∈ (0, 1) is a common average tax rate and ε ∈ (0, 1) is the income
tax share.
Based on the papers [2], [1], [11], we consider the following IS-LM model:
Y˙ (t)=α[I(Y (t), r(t))+g−S(Y (t)−T (Y (t), Y (t−τ)))−T (Y (t), Y (t−τ))]
r˙(t) = β[L(Y (t), r(t))−M(t)]
K˙(t) = I(Y (t− τ), r(t))− δK(t)
M˙(t) = g − T (Y (t), Y (t−τ)),
(2)
with δ > 0 and with the initial conditions:
Y (t) = ϕ(t), t ∈ [−τ, 0], r(0) = r1,M(0) = M1, K(0) = K1.
In the following analysis we will consider the function T given by (1), the
investment, the saving and the liquidity of the form:
I(Y (t), r(t)) = aY (t)α1r(t)−α2 , a > 0, α1 > 0, α2 > 0,
S(Y (t)− T (Y (t), Y (t− τ))) = s(Y (t)− T (Y (t), Y (t− τ))), s ∈ (0, 1),
L(r(t)) = mY (t) +
γ0
r(t)− r2
, m > 0, γ0 > 0, r2 > 0.
(3)
2
The paper is organized as follows. In section 2 we investigate the local
stability of the equilibrium point associated to system (2). Choosing the
delay as a bifurcation parameter some sufficient conditions for the existence
of Hopf bifurcation are found. In section 3 there is the main aim of the paper,
namely the direction, the stability and the period of a limit cycle solution.
Section 4 gives some numerical simulations which show the existence and the
nature of the periodic solutions. Finally, some conclusions are given.
2. The qualitative analysis of system (2).
Using functions (1) and (3), system (2) becomes:
Y˙ (t)=α[((s−1)(1−ε)d−s)Y (t)+dε(s−1)Y (t−τ)+aY (t)α1r(t)−α2+g]
r˙(t) = β[mY (t) +
γ0
r(t)− r2
−M(t)]
K˙(t) = aY (t− τ)α1r(t)−α2 − δK(t)
M˙(t) = g − d(1− ε)Y (t)− dεY (t− τ),
(4)
α, β > 0, α1, α2 > 0, m > 0, γ0 > 0, r2 > 0, δ > 0, s ∈ (0, 1), d ∈ (0, 1),
ε ∈ (0, 1).
System (4) is a system of equations with time delay. The qualitative
analysis is done using the methods from [3].
The equilibrium point of system (4) has the coordinates Y0, r0, K0,M0,
where:
Y0=
g
d
, r0=[
s(1−d)
a
Y 1−α10 ]
−
1
α2 , K0=
s(1−d)Y0
δ
,M0= mY0+
γ0
r0−r2
. (5)
Using the translation
Y = x1 + Y0, r = x2 + r0, K = x3 +K0,M = x4 +M0
in (4) and considering the Taylor expansion of the right members from (4)
until the third order, we have:
x˙(t) = Ax(t) +Bx(t− τ) + F (x(t), x(t− τ)) (6)
3
where
A=


αa1 aαρ01 0 0
βm βγ0γ1 0 −β
0 aρ01 −δ 0
a4 0 0 0


, B=


αb1 0 0 0
0 0 0 0
aρ10 0 0 0
b4 0 0 0


(7)
where x(t) = (x1(t), x2(t), x3(t), x4(t))
T , and
a1 = (s− 1)d(1− ε)− s+ aρ10, b1 = (s− 1)dε, a4 = −d(1− ε), b4 = −dε
ρ01 = −α2Y
α1
0 r
−(α2+1)
0 , ρ10 = α1Y
α1−1
0 r
−α2
0
and
F (x(t), x(t− τ)) = (F 1(x(t), x(t− τ)), F 2(x(t), x(t− τ)), F 3(x(t), x(t− τ)),
(8)
F 4(x(t), x(t− τ)))T ,
F 1(x(t), x(t− τ)) =
αa
2
ρ20x
2
1(t) + αaρ11x1(t)x2(t) +
αa
2
ρ02x
2
2(t)+
αa
6
(ρ30x
3
1(t) + 3ρ21x1(t)
2x2(t) + 3ρ12x1(t)x2(t)
2 + ρ03x2(t)
3)
F 2(x(t), x(t− τ)) =
γ0
2
γ2x2(t)
2 +
γ0
6
γ3x2(t)
3
F 3(x(t), x(t− τ)) =
a
2
ρ20x1(t− τ)
2 + aρ11x1(t− τ)x2(t) +
a
2
ρ02x2(t)
2+
+
a
6
(ρ30x
3
1(t− τ) + 3ρ21x1(t− τ)
2x2(t) + 3ρ12x1(t− τ)x2(t)
2 + ρ03x2(t)
3)
F 4(x(t), x(t− τ)) = 0
and
ρ20 = α1(α1 − 1)Y
α1−2
0 r
−α2
0 , ρ11 = −α1α2Y
α1−1
0 r
−(α2+1)
0
ρ02 = α2(α2 + 1)Y
α1
0 r
−(α2+2)
0 , ρ30 = α1(α1 − 1)(α1 − 2)Y
α1−3
0 r
−α2
0
ρ21 = −α1α2(α1 − 1)Y
α1−2
0 r
−(α2+1)
0 , ρ12 = α1α2(α2 + 1)Y
α1−1
0 r
−(α2+2)
0
ρ03 = −α2(α2 + 1)(α2 + 2)Y
α1
0 r
−(α2+3)
0 ,
γ1 = −
1
(r0 − r2)2
, γ2 =
2
(r0 − r2)3
, γ3 = −
6
(r0 − r2)4
.
4
The characteristic equation of linear part from (6) is:
det(λI − A− Be−λτ ) = (λ+ δ)∆(λ, τ) = 0 (9)
where
∆(λ, τ) = P (λ) + e−λτQ(λ) (10)
P (λ) = λ3 + p2λ
2 + p1λ+ p0, Q(λ) = q2λ
2 + q1λ+ q0
and
p2 = −(αa1 + βγ0γ1), p1 = αβ(γ0a1γ1 +maρ01), p0 = αaβa4ρ01
q2 = −αb1, q1 = αβγ0b1γ1, q0 = αβb4ρ01.
To investigate the local stability of the equilibrium point, we begin by
considering, as usual, the case without delay (τ = 0 ). In this case the
characteristic polynomial is:
(λ+ δ)(P (λ) +Q(λ)) = 0
hence, according to the Hurwitz criterion, the equilibrium point is stable if
and only if:
p2 + q2 > 0, (p1 + q1)(p2 + q2) > p0 + q0.
When τ > 0, standard results on stability of systems of delay differential
equations postulate that a equilibrium point is asymptotically stable if an
only if all roots of equation (10) have a negative real part. It is well known
that equation (10) is a transcendental equation which has an infinite number
of complex roots and the some possible roots with positive real part are finite
in number.
We want to obtain the values τ0 such that the equilibrium point (5)
changes from local asymptotic stability to instability or vice versa. We need
the imaginary solutions of equation ∆(λ, τ) = 0. Let λ = ±iω be these
solutions and without loss of generality we assume ω > 0. We suppose that
P (iω)+Q(iω) 6= 0, for all ω ∈ IR. The previous conditions are equivalent to
(p1 + q1)(p2 + q2) 6= p0 + q0.
A necessary condition to have ω as a solution of ∆(iω, τ) = 0 is that ω
must be a root of the following equation:
f(ω) = ω6 + aFω
4 + bFω
2 + cF = 0 (11)
5
where aF = p
2
2 − q
2
2 − 2p1, bF = 2q0q2 − 2p0p2 − q
2
1 + p
2
1, cF = p
2
0 − q
2
0.
Let k = −
aF
3
and fD =
1
4
[f(k)]4 +
1
9
[f
′
(k)]3 .
Using the results from [2], it results:
Proposition 1. 1. Let ε >
1
2
. Then the following cases can be discerned:
(i) If aF ≥ 0 or bF ≤ 0, then equation (11) has only one real positive
root;
(ii) If aF < 0, bF > 0 and fD ≤ 0, then equation (11) has only one real
positive root.
2. If ε <
1
2
, aF < 0, bF < 0 and fD ≤ 0 then equation (11) has two real
positive roots which are distinct if fD 6= 0.
3. If ε =
1
2
and:
(i) aF < 0, bF = 0 then equation (11) has only one real positive root;
(ii)aF < 0, bF > 0, then equation (11) has two real positive roots;
(iii)bF < 0, then equation (11) has only one real positive root.
Also, we have:
Theorem 1. If we suppose that the equilibrium point (Y0, r0, K0,M0) is
locally asymptotically stable without time delay, then in conditions of Propo-
sition 1 there exists only one stability switch.
Theorem 2. If τ0 is a stability switch and fD 6= 0, then a Hopf bifurcation
occurs at τ0, where
τ0 =
1
ω0
arctg
(
ω0(ω
4
0q2 − ω
2(q0 − q1p2 + q2p1) + q0p1 − p0q1)
ω40(q1 − q2p2) + ω
2
0(q0p2 − q1p1 + p0q2)− p0q0
)
and ω0 is a root of (11).
3. The normal form for system (4). Cyclical behavior.
In this section we describe the direction, stability and the period of the
bifurcating periodic solutions of system (4). The method we use is based
on the normal form theory and the center manifold theorem introduced by
Hassard [4]. Taking into account the previous section, if τ = τ0 then all roots
of equation (9) other than ±iω0 have negative real parts, and any root of
equation (9) of the form λ(τ) = α(τ)± iω(τ) satisfies α(τ0) = 0, ω(τ0) = ω0
6
and
dα(τ0)
dτ
6= 0. For notational convenience let τ = τ0 + µ, µ ∈ IR. Then
µ = 0 is the Hopf bifurcation value for equations (4).
Define the space of continuous real-valued functions as C = C([−τ0, 0], IR
4).
In τ = τ0 + µ, µ ∈ IR, we regard µ as the bifurcation parameter. For
Φ ∈ C we define a linear operator:
L(µ)Φ = AΦ(0) +BΦ(−τ)
where A and B are given by (7) and a nonlinear operator F (µ,Φ) = F (Φ(0),
Φ(−τ)), where F (Φ(0),Φ(−τ)) is given by (8). By the Riesz representa-
tion theorem, there exists a matrix whose components are bounded variation
functions, η(θ, µ) with θ ∈ [−τ0, 0] such that:
L(µ)Φ =
0∫
−τ0
dη(θ, µ)φ(θ), θ ∈ [−τ0, 0].
For Φ ∈ C1([−τ0, 0], IR
4) we define:
A(µ)Φ(θ) =


dΦ(θ)
dθ
, θ ∈ [−τ0, 0)
0∫
−τ0
dη(t, µ)φ(t), θ = 0,
R(µ)Φ(θ) =


0, θ ∈ [−τ0, 0)
F (µ,Φ), θ = 0.
We can rewrite (6) in the following vector form:
u˙t = A(µ)ut +R(µ)ut (12)
where u = (u1, u2, u3, u4)
T , ut = u(t+ θ) for θ ∈ [−τ0, 0].
For Ψ ∈ C1([0, τ0], IR
∗4), we define the adjunct operator A∗ of A by:
A∗Ψ(s) =


−
dΨ(s)
ds
, s ∈ (0, τ0]
0∫
−τ0
dηT (t, 0)ψ(−t), s = 0.
7
We define the following bilinear form:
< Ψ(θ),Φ(θ) >= Ψ¯T (0)Φ(0)−
∫ 0
−τ0
∫ 0
ξ=0
Ψ¯T (ξ − θ)dη(θ)φ(ξ)dξ,
where η(θ) = η(θ, 0).
We assume that ±iω0 are eigenvalues of A(0). Thus, they are also eigen-
values of A∗. We can easily obtain:
Φ(θ) = veλ1θ, θ ∈ [−τ0, 0] (13)
where v = (v1, v2, v3, v4)
T ,
v1 = 1, v2 = −
β(a4 + b4e
−λ1τ0 −mλ1)
λ1(λ1 − βγ0γ1)
, v4 =
a4 + b4e
−λ1τ0
λ1
,
v3 =
a
λ1 + δ
[ρ10e
−λ1τ0 + ρ01
β(a4 + b4e
−λ1τ0 −mλ1)
λ1(βγ0γ1 − λ1)
]
is the eigenvector of A(0) corresponding to λ1 = iω0 and
Ψ(s) = weλ2s, s ∈ [0,∞)
where w = (w1, w2, w3, w4),
w1 =
λ2 − βγ0γ1
αaρ01
1
η¯
, w2 =
1
η¯
, w3 = 0, w4 = −
β
λ2
1
η¯
η =
λ1 − βγ0γ1
aαρ01
(1+αb1
λ1τ0e
−λ1τ0−e−λ1τ0 + 1
λ21
)+v2−
β
λ1
(v4+b4
λ1τ0e
−λ1τ0−e−λ1τ0 + 1
λ21
)
is the eigenvector of A(0) corresponding to λ2 = −iω0.
We can verify that: < Ψ(s),Φ(s) >= 1, < Ψ(s), Φ¯(s) >=< Ψ¯(s),Φ(s) >=
0, < Ψ¯(s), Φ¯(s) >= 1.
Using the approach of Hassard [4], we next compute the coordinates to
describe the center manifold Ω0 at µ = 0. Let ut = ut(t+ θ), θ ∈ [−τ0, 0) be
the solution of equation (12) when µ = 0 and
z(t) =< Ψ, ut >, w(t, θ) = ut(θ)− 2Re{z(t)Φ(θ)}.
On the center manifold Ω0, we have:
w(t, θ) = w(z(t), z¯(t), θ)
8
where
w(z, z¯, θ) = w20(θ)
z2
2
+ w11(θ)zz¯ + w02(θ)
z¯2
2
+ w30(θ)
z3
6
+ . . .
in which z and z¯ are local coordinates for the center manifold Ω0 in the
direction of Ψ and Ψ¯ and w02(θ) = w¯20(θ). Note that w and ut are real.
For solution ut ∈ Ω0 of equation (12), since µ = 0, we have:
z˙(t) = λ1z(t) + g(z, z¯) (14)
where
g(z, z¯) = Ψ¯(0)F (w(z(t), z¯(t), 0) + 2Re(z(t)Φ(0))) =
= g20
z(t)2
2
+ g11z(t)z¯(t) + g02
z¯(t)2
2
+ g21
z(t)2z¯(t)
2
+ . . .
where
g20 = F
1
20w¯1 + F
2
20w¯2, g11 = F
1
11w¯1 + F
2
11w¯2, g02 = F
1
02w¯1 + F
2
02w¯2, (15)
with
F 120 = αa(ρ20 + 2ρ11v2 + ρ02v
2
2), F
1
11 = αa(ρ20 + ρ11(v¯2 + v2) + ρ02v2v¯2),
F 220 = γ0γ2v
2
2 , F
2
11 = γ0γ2v2v¯2, F
1
02 = F¯
1
20, F
2
02 = F¯
2
20,
and
g21 = F
1
21w¯1 + F
2
21w¯2 (16)
where
F 121 = αaρ20(2w
1
11(0) + w
1
20(0)) + αaρ11(2w
2
11(0) + w
2
20(0) + 2w
1
11(0)v¯2+
w120(0)v2) + αaρ02(2w
2
11(0)v2 + w
2
20(0)v¯2) + αa(ρ30 + 2ρ21v2 + 2ρ12v
2
2+
+ρ03v
2
2 v¯2 + ρ12v
2
2 + ρ21v¯2)
F 221 = γ0γ2(2w
2
11(0)v2 + w
2
20(0)v¯2) + γ0γ3v
2
2 v¯2.
The vectors w20(θ), w11(θ) with θ ∈ [−τ, 0] are given by:
w20(θ) = −
g20
λ1
veλ1θ −
g¯02
3λ1
v¯λ2θe + E1e
2λ1θ
w11(θ) =
g11
λ1
veλ1θ −
g¯11
λ1
v¯eλ2θ + E2
(17)
9
where
E1 = −(A + e
−λ1τ0B − 2λ1I)
−1F20, E2 = −(A+B)
−1F11,
where F20 = (F
1
20, F
2
20, F
3
20, 0)
T , F11 = (F
1
11, F
2
11, F
3
11, 0)
T .
Based on the above analysis and calculation, we can see that each gij
in (15), (16) are determined by the parameters and delay from system (4).
Thus, we can explicitly compute the following quantities:
C1(0) =
i
2ω0
(g20g11 − 2|g11|
2 −
1
3
|g02|
2) +
g21
2
µ2 = −
Re(C1(0))
Reλ′(τ0)
, T2 = −
Im(C1(0)) + µ2Imλ
′(τ0)
ω0
, β2 = 2Re(C1(0)).
(18)
and
λ′ =
λ(q2λ
2 + q1λ+ q0)e
−λτ
3λ2 + 2p2λ+ p1 + e−λτ (2q2λ+ q1 − τ(q2λ2 + q1λ+ q0))
In summary, this leads to the following result:
Theorem 3. In formulas (18), µ2 determines the direction of the Hopf
bifurcation: if µ2 > 0(< 0), then the Hopf bifurcation is supercritical (sub-
critical) and the bifurcating periodic solutions exit for τ > τ0(< τ0); β2 de-
termines the stability of the bifurcating periodic solutions: the solutions are
orbitally stable (unstable) if β2 < 0(> 0); and T2 determines the period of the
bifurcating periodic solutions: the period increases (decreases) if T2 > 0(< 0).
4. Numerical example.
In this section we find the waveform plots through the formula:
X(t+θ)=z(t)Φ(θ)+z¯(t)Φ¯(θ)+
1
2
w20(θ)z
2(t)+w11(θ)z(t)z¯(t)+
1
2
w02(θ)z¯(t)
2+X0,
where z(t) is the solution of (14), Φ(θ) is given by (13), w20(θ), w11(θ), w02(θ) =
w¯20(θ) are given by (17) and X0 = (Y0, r0, K0,M0)
T is the equilibrium state.
For the numerical simulations we use Maple 9.5. We consider system (4)
with a = 0.38, α = 0.96, β = 1, α1 = 0.5, α2 = 0.83, γ0 = 1, d = 0.1, s = 0.3,
10
r2 = 0.003, δ = 0.2, m = 0.005, g = 50. The equilibrium point is: Y0=500,
r0=0.03572181612,K0=675,M0=33.06065092. In what follows we consider
two different shares ǫ of delay tax revenues: ε = 0.3 and ε = 0.8.
For ε = 0.3 we obtain: µ2=1.654628706 · 10
−8, β2=2.224294680 · 10
−9,
T2 = 2.092652051 · 10
−9, ω0 = 0.6685954740, τ0 = 4.965007916. Then the
Hopf bifurcation is supercritical, the solutions are orbitally unstable and the
period of the solution is increasing. The wave plots are given in the following
figures:
Fig.1.Waveplot (t, Y (t)) Fig.2.Waveplot (t, r(t)) Fig.3.Waveplot (t,K(t))
t
500.00021362
500.00009155
80
500
499.99990845
60
499.99978638
40200
                        
t
40
3.57218347E-2
80
3.5721831E-2
3.57218236E-2
60
3.57218198E-2
3.57218161E-2
3.57218087E-2
3.57218049E-2
20
3.57218012E-2
0
                        
t
6.75000122E2
6.74999939E2
80604020
6.74999878E2
0
6.75000061E2
6.75E2
 
                        
Fig.4.Waveplot (t,M(t)) Fig.5.Waveplot (t, I(t)) Fig.6.Waveplot (t, L(t))
t
33.06018448
8020 60
33.060215
33.06021118
40
33.06020355
33.06019592
33.06019974
33.06018829
0
                        
100
135.00006104
135.00004578
135.00001526
135
134.99998474
134.99995422
134.99993896
t
50403020
                        
10
33.06066513
0
33.06066132
33.06065369
33.06064987
33.06064606
33.06063843
33.06063461
t
50403020
                        
For ε = 0.8 we obtain: µ2=−9.160756314·10
−8, β2=−1.968119398·10
−8,
T2 = −1.608068638 · 10
−8, ω0 = 0.8553440397, τ0 = 3.918246696. Then the
Hopf bifurcation is subcritical, the solutions are orbitally stable and the
period of the solution is decreasing. The wave plots are given in the following
figures:
11
Fig.7.Waveplot (t, Y (t)) Fig.8.Waveplot (t, r(t)) Fig.9.Waveplot (t,K(t))
t
6050400
5.00000092E2
302010
5.00000214E2
5E2
4.99999908E2
4.99999786E2
                        
3.57218012E-2
t
6050403020100
3.5721831E-2
3.57218198E-2
3.57218087E-2
                        
6.75000061E2
6.75E2
6.74999939E2
6050
t
4010 20 300
 
                        
Fig.10.Waveplot (t,M(t)) Fig.11.Waveplot (t, I(t)) Fig.12.Waveplot (t, L(t))
33.05971909
33.05971146
33.05970001
33.05968857
t
6050403020100
33.05973053
                        
10
134.99995422
135
0
t
135.00004578
30
135.00006104
20
135.00001526
134.99993896
134.99998474
                        
33.06066132
0
t
33.06064987
302010
33.06066895
33.06063843
                        
5. Conclusions.
From the analysis of the model with continuous time, it results that the
model accepts a limit cycle. The nature of the limit cycle is given by the
coefficients (18) which include the parameters of the model. We establish
the nature of the limit cycle. Because the expressions of the parameter and
the coefficients (18) are difficult to analyze directly, the use of Maple 9.5 was
essential. The paper’s results confirm that a series of economical processes,
where a variable with time delay intervenes, have a limit cycle, thus, allowing
a prediction concerning the evolution of the model.
12
References
[1] Cai J, Hopf bifurcation in the IS-LM business cycle model with time
delay, Electronic Journal of Differential Equations, 2005(15):1-6.
[2] De Cesare L, Sportelli M, A dynamic IS-LMmodel with delayed taxation
revenues, Chaos, Solitons and Fractals, 2005(25):233-44.
[3] Hale J. K, Lunel S.M. V, Introduction to Functional Differential
Equation, Springer-Verlag, New York, Applied Mathematical Sciences,
1993(99).
[4] Hassard B.D, Kazarinoff N.D, Wan Y.H, Theory and applications of
Hopf bifurcation, Cambridge University Press, Cambridge, 1981.
[5] Liao X, Li C, Zhou S, Hopf bifurcation and chaos in macroeconomic
models with policy lag, Chaos, Solitons and Fractals, 2005(25):91-108.
[6] Sasakura K, On the dynamic behavior of Schinasi’s business cycle model,
J. Macroecon, 1994(16):423-44.
[7] Schinasi G.J, A nonlinear dynamic model of short run fluctuations, Rev.
Econ. Stud.,1981(48):649-56.
[8] Schinasi G.J, Fluctuations in a dynamic intermediate-run IS-LM
model: applications of the Poicar-Bendixon theorem, J. Econ Theory.,
1982(28):369-75.
[9] Szydlowski M, Krawiec A, The stability problem in the Kaldor-Kalecki
business cycle model, Chaos, Solitons and Fractals, 2005(25):299-305.
[10] Takeuchi Y, Yamamura T, Stability analysis of the Kaldor model with
time delays: monetary policy and government budget constraint, Non-
linear Analysis, 2004(5):277-308.
[11] Torre V, Existence of limit cycles and control in complete Keynesian
systems by theory of bifurcations, Econometrica, 1977(45):1457-66.
13


http://mpra.ub.uni-muenchen.de/13270/1/MPRA_paper_13270.pdf

Hopf bifurcation in a dynamic IS-LM model with time delay

Abstract

Similar works

Full text

Available Versions

Munich RePEc Personal Archive

Munich Personal RePEc Archive