Makovínyiová, Katarína

Zimka, Rudolf

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 11Katarína Makovínyiová; Rudolf ZimkaOn the existence of Hopf bifurcation in an open economy modelIn: Marek Fila and Angela Handlovičová and Karol Mikula and Milan Medveď and Pavol Quittner and Daniel Ševčovič(eds.): Proceedings of Equadiff 11, International Conference on Differential Equations. Czecho-Slovak series,Bratislava, July 25-29, 2005, [Part 2] Minisymposia and contributed talks. Comenius University Press, Bratislava,2007. Presented in electronic form on the Internet. pp. 229--236.Persistent URL: http://dml.cz/dmlcz/700416Terms of use:© Comenius University, 2007Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documentsstrictly for personal use. Each copy of any part of this document must contain these Terms of use.This paper has been digitized, optimized for electronic delivery and stamped with digitalsignature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czProceedings of Equadiff-112005, pp. 229–236ISBN 978-80-227-2624-5ON THE EXISTENCE OF HOPF BIFURCATION IN AN OPENECONOMY MODELKATARÍNA MAKOVÍNYIOVÁ∗ AND RUDOLF ZIMKA†Abstract. In the paper a four dimensional open economy model describing the development ofoutput, exchange rate, interest rate and money supply is analyzed. Sufficient conditions for the existenceof equilibrium stability and the existence of business cycles are found. Formulae for the calculation ofbifurcation coefficients are derived.Key words. dynamical model, equilibrium, stability, bifurcation, cyclesAMS subject classifications. 37Gxx1. Introduction. In economic theory there are many macroeconomic models de-scribing the development of output Y in an economy. The majority of them are basedon the well-known IS-LM model. One of them is the Schinasi’s model (see [5]) describingthe development of output Y , interest rate R and money supply Ls in a closed economy.In [8] the Schinasi’s model was extended to an open economy model of the kindẎ = α[I(Y, R) + G + X(ρ)− S(Y D, R)− T (Y )−M(Y, ρ)]ρ̇ = β [M(Y, ρ) + Cx(R)−X(ρ)− Cm(R)]Ṙ = γ [L(Y,R)− LS ]L̇S = G + M(Y, ρ) + Cx(R)− T (Y )−X(ρ)− Cm(R),(1.1)where Y – output, ρ – exchange rate, R – interest rate, LS – money supply, I – invest-ments, S – savings, G – government expenditures, T – tax collections, X –– export, M – import, Cx – capital export, Cm – capital import, L – money demand, α,β, γ – positive parameters, t – time andY D = Y − T (Y ), Ẏ = dYdt, ρ̇ =dρdt, Ṙ =dRdt, L̇S =dLSdt.The economic properties of the functions in (1.1) are expressed by the following partialderivatives:∂I(Y,R)∂Y> 0,∂I(Y, R)∂R< 0,∂S(Y D, R)∂Y> 0,∂S(Y D, R)∂R> 0,∂T (Y )∂Y> 0,∂X(ρ)∂ρ> 0,∂M(Y, ρ)∂Y> 0,∂M(Y, ρ)∂ρ< 0,∂Cx(R)∂R< 0,∂Cm(R)∂R> 0,∂L(Y, R)∂Y> 0,∂L(Y, R)∂R< 0.(1.2)∗Department of Mathematics and Descriptive Geometry, Faculty of Wood Sciences and Technology,Technical university Zvolen, Zvolen, Slovakia (kmakovin@vsld.tuzvo.sk).†Department of Quantitative Methods and Informatics, Faculty of Economics, Matej Bel University,Banská Bystrica, Slovakia (rudolf.zimka@umb.sk).229230 K. Makov́ınyiová and R. ZimkaThe development of Y, ρ,R and LS in the model (1.1) in a neighborhood of its equi-librium was studied under fixed exchange rate regime in [6] and under flexible exchangerate regime in [2] from the point of equilibrium stability and the existence of businesscycles. In [4] this model was analyzed under assumption that both ρ and R are flexible.In the paper [4] the assertion on the existence of business cycles was formulated underassumption of the existence of the Liapunov bifurcation constant a. In the present papera formula for the calculation of this Liapunov bifurcation constant a is found supposingthat the functions I(Y, R) and S(Y D, R) are nonlinear with respect to Y and linear withrespect to R, and other functions in (1.1) are linear with respect to all their variables ex-cept government expenditures G, which are constant. In the end of the paper an exampleillustrating the achieved results is presented.2. Analysis of the model (1.1). Using denotationB(Y, R) = I(Y, R)− S(Y D, R)C(R) = Cm(R)− Cx(R)D(Y ) = G− T (Y )F (Y, ρ) = X(ρ)−M(Y, ρ),the model (1.1) has the formẎ = α [B(Y, R) + D(Y ) + F (Y, ρ)]ρ̇ = −β [F (Y, ρ) + C(R)]Ṙ = γ [L(Y, R)− LS ]L̇S = D(Y )− F (Y, ρ)− C(R),(2.1)where on the base of (1.2)∂B(Y,R)∂R< 0,∂C(R)∂R> 0,∂D(Y )∂Y< 0,∂F (Y, ρ)∂Y< 0,∂F (Y, ρ)∂ρ> 0,and the sign of ∂B(Y,R)∂Y can be positive or negative or zero according to the properties ofI(Y, R) and S(Y D, R).Assume:1. The functions I(Y,R), S(Y D, R), T (Y ), C(R), D(Y ), F (Y, ρ), L(Y,R) are of thekindI(Y,R) = i0 + i1√Y − i3R, C(R) = c0 + c3R,S(Y D, R) = s0 + s1(Y D)2 + s3R, D(Y ) = d0 − d1Y,Y D = Y − T (Y ), F (Y, ρ) = f0 − f1Y + f2ρ,T (Y ) = t0 + t1Y, L(Y, R) = l0 + l1Y − l3R,with positive coefficients i1, i3, s1, s3, t1, c3, d1, f1, f2, l1, l3.2. The model (2.1) has an isolated equilibrium E∗ = (Y ∗, ρ∗, R∗, L∗S), Y∗ > 0,ρ∗ > 0, R∗ > 0, L∗S > 0.Remark 1. A sufficient condition for the existence of an isolated equilibrium E∗ of themodel (2.1) was presented in [4].Consider an isolated equilibrium E∗ = (Y ∗, ρ∗, R∗, L∗S) of (2.1). After the transfor-mationY1 = Y − Y ∗, ρ1 = ρ− ρ∗, R1 = R−R∗, LS1 = LS − L∗S ,Equadiff-11. On the existence of Hopf bifurcation in an open economy model 231the equilibrium E∗ goes into the origin E∗1 = (Y∗1 = 0, ρ∗1 = 0, R∗1 = 0, L∗S1 = 0) and themodel (2.1) takes the formẎ1 = α [B(Y1 + Y ∗, R1 + R∗) + D(Y1 + Y ∗) + F (Y1 + Y ∗, ρ1 + ρ∗)]ρ̇1 = −β [F (Y1 + Y ∗, ρ1 + ρ∗) + C(R1 + R∗)]Ṙ1 = γ [L(Y1 + Y ∗, R1 + R∗)− LS1 − LS∗1]L̇S1 = D(Y1 + Y∗)− F (Y1 + Y ∗, ρ1 + ρ∗)− C(R1 + R∗).(2.2)Performing Taylor expansion of the functions on the right-hand side in (2.2) at the equi-librium E∗1 we get the modelẎ1 = α [(b1 − d1 − f1)Y1 + f2ρ1 − b3R1] + α[4∑k=2akYk1 + O(|Y1|5)]ρ̇1 = β (f1Y1 − f2ρ1 − c3R1)Ṙ1 = γ (l1Y1 − l3R1 − LS1)L̇S1 = (f1 − d1)Y1 − f2ρ1 − c3R1,(2.3)where b1 =∂B(Y ∗,R∗)∂Y , b3 = i3 + s3, ak =∂kB(Y ∗,R∗)∂Y k, k = 2, 3, 4.The linear approximation matrix of (2.3) isA(α, β, γ) =α(b1 − d1 − f1) αf2 −αb3 0βf1 −βf2 −βc3 0γl1 0 −γl3 −γf1 − d1 −f2 −c3 0 . (2.4)The characteristic equation of A(α, β, γ) isλ4 + a1(α, β, γ)λ3 + a2(α, β, γ)λ2 + a3(α, β, γ)λ + a4(α, β, γ) = 0, (2.5)wherea1 = α(−b1 + d1 + f1) + βf2 + γl3,a2 = −γc3 + βγf2l3 + α(βf2(−b1 + d1) + γ(b3l1 + l3(−b1 + d1 + f1))),a3 = αγ(βf2(l1(b3 + c3) + l3(−b1 + d1))− c3(−b1 + d1 + f1)− b3(f1 − d1)),a4 = αβγd1f2(b3 + c3).As we are interested in equilibrium stability and in the existence of cycles in the model(2.3) it is suitable to find such values of parameters α, β, γ at which the equation (2.5) hasa pair of purely imaginary eigenvalues λ1 = iω, λ2 = −iω, and the rest two eigenvaluesλ3, λ4 are negative or have negative real parts. We shall call such values of parametersα, β, γ as critical values of the model (2.3) and denote them α0, β0, γ0. Mentioned typesof eigenvalues are ensured by the Liu’s conditions ([3]):a1 > 0, a2 > 0, a3 > 0, a4 > 0, (2.6)∆3 = (a1a2 − a3)a3 − a 21 a4 = 0. (2.7)232 K. Makov́ınyiová and R. ZimkaThe conditions (2.6) are satisfied if− b1 + d1 ≥ 0,α > α∗ =c3l3(−b1 + d1 + f1) + b3l1,(−b1 + d1 + f1)c3 + (f1 − d1)b3 < 0.(2.8)The relation (2.7) can be expressed in the form[a(1)(β, γ)β + a(2)γ]α2 +[b(1)(β, γ)β + b(2)(γ)γ]α + c(1)(β, γ)β + c(2)γ2 = 0,where in the power of (2.8) there isa(2) = (−b1 + d1 + f1)[b3(f1 − d1)+c3(−b1 + d1 + f1)][−b3l1 − l3(−b1 + d1 + f1)] > 0,c(2) = c3l3[b3(f1 − d1) + c3(−b1 + d1 + f1)] < 0.Take γ at any positive level and denote it γ0. Then a(2)γ0 > 0 and c(2)γ 20 < 0. DenoteA(β) = a(1)(β, γ0)β + a(2)γ0,B(β) = b(1)(β, γ0)β + b(2)(γ0)γ0,C(β) = c(1)(β, γ0)β + c(2)γ 20 .Take β such small that A(β) > 0 and C(β) < 0. Denote this β as β0. Then the equationA(β0)α2 + B(β0)α + C(β0) = 0 has two rootsα1,2 =−B(β0)±√(B(β0))2 − 4A(β0)C(β0)2A(β0)={α1 < 0α2 > 0,and the function ∆3(α) = A(β0)α2 + B(β0)α + C(β0) has the following form (Fig. 2.1):Fig. 2.1.Denote α2 = α0. The triple (α0, β0, γ0) is the critical triple of the model (2.3). Thusfor given specific values of β0, γ0 it is always possible to find such a value α0 that thecondition (2.7) is satisfied. From Figure 1 we get in a small neighborhood of α0:If α > α0 then ∆3(α) = (a1a2 − a3)a3 − a 21 a4 > 0.If α < α0 then ∆3(α) = (a1a2 − a3)a3 − a 21 a4 < 0. (2.9)If α = α0 then ∆3(α) = (a1a2 − a3)a3 − a 21 a4 = 0.Equadiff-11. On the existence of Hopf bifurcation in an open economy model 233Consider a critical triple (α0, β0, γ0) of the model (2.3). Further we shall investigatethe behavior of Y1, ρ1, R1 and LS1 around the equilibrium E∗1 with respect to parameterα, α ∈ (α0 − ε, α0 + ε), ε > 0, and fixed parameters β = β0, γ = γ0.Performed considerations enable us to answer the question about the stability of theequilibrium E∗1 of the model (2.3) in a small neighborhood of the critical value α0.Theorem 1. Let the conditions (2.8) be satisfied and let (α0, β0, γ0) be such a criticaltriple of the model (2.3) that A(β0) > 0 and C(β0) < 0. Then:1. If α0 ≤ α∗ then at every α > α∗ the equilibrium E∗1 is asymptotically stable.2. If α0 > α∗ then at every α > α0 the equilibrium E∗1 is asymptotically stable andat every α∗ < α < α0 the equilibrium E∗1 is unstable.Proof. The conditions (2.8) guarantee that a1 > 0, a2 > 0, a3 > 0, and a4 > 0 whattogether with (2.9) at α > α∗ ≥ α0 means that the Routh-Hurwitz necessary and sufficientconditions for all the roots of the equation (2.5) to have negative real parts (∆1(α) >0,∆2(α) > 0,∆3(α) > 0,∆4(α) > 0) are satisfied. If α∗ < α < α0 then ∆3(α) < 0 whatmeans that the Routh-Hurwitz conditions are not satisfied.To gain the bifurcation equation of the model (2.3) it is suitable to transform (2.3)to its partial normal form on invariant surface. After the shift of α0 into the origin byrelation α1 = α− α0 the model (2.3) takes the formẎ1 = α0 [(b1 − d1 − f1)Y1 + f2ρ1 − b3R1] + (b1 − d1 − f1)Y1α1+ f2ρ1α1 − b3R1α1 +4∑k=2α0akYk1 +4∑k=2α1akYk1 + O(|Y1|5)ρ̇1 = β0 (f1Y1 − f2ρ1 − c3R1)Ṙ1 = γ0 (l1Y1 − l3R1 − LS1)L̇S1 = (f1 − d1)Y1 − f2ρ1 − c3R1.(2.10)Consider the matrix M which transfers the matrix A(α0, β0, γ0) into its Jordan formJ. Then the transformation x = My, x = (Y1, ρ1, R1, LS1)T , y = (Y2, ρ2, R2, LS2)T takesthe model (2.10) into the modelẎ2 = λ1Y2 + F1(Y2, ρ2, R2, LS2, α1)ρ̇2 = λ2ρ2 + F2(Y2, ρ2, R2, LS2, α1)Ṙ2 = λ3R2 + F3(Y2, ρ2, R2, LS2, α1)L̇S2 = λ4LS2 + F4(Y2, ρ2, R2, LS2, α1),(2.11)where ρ2 = Y 2, F2 = F 1, and F3, F4 are real (the symbol ” - ” means complex conjugateexpression in the whole article; for the sake of simplicity we suppose that λ3, λ4 are realand not equal).Theorem 2. There exists a polynomial transformationY2 = Y3 + h1(Y3, ρ3, α1)ρ2 = ρ3 + h2(Y3, ρ3, α1)R2 = R3 + h3(Y3, ρ3, α1)LS2 = LS3 + h4(Y3, ρ3, α1),(2.12)234 K. Makov́ınyiová and R. Zimkawhere hj(Y3, ρ3, α1), j = 1, 2, 3, 4, are nonlinear polynomials with constant coefficients ofthe kindhj(Y3, ρ3, α1) =∑m1,m2,m3ν(m1,m2,m3)Y m13 ρm23 αm31 , j = 1, 2, 3, 4, h2 = h1,with the propertyhj(√α1Y3,√α1ρ3, α1) =∑m1,m2,m3ν(m1,m2,m3)(√α1)kY m13 ρm23 , k ≤ 4,which transforms the model (2.11) into its partial normal form on invariant surfaceẎ3 = λ1Y3+δ1Y3α1+δ2Y 23 ρ3+U◦(Y3, ρ3, R3, LS3, α1)+U∗(Y3, ρ3, R3, LS3, α1)ρ̇3 = λ2ρ3 + δ1ρ3α1 + δ2Y3ρ 23 + U◦ + U∗Ṙ3 = λ3R3 + V ◦(Y3, ρ3, R3, LS3, α1) + V ∗(Y3, ρ3, R3, LS3, α1)L̇S3 = λ4LS3 + W◦(Y3, ρ3, R3, LS3, α1) + W ∗(Y3, ρ3, R3, LS3, α1),(2.13)where U◦(Y3, ρ3, 0, 0, α1) = V ◦(Y3, ρ3, 0, 0, α1) = W ◦(Y3, ρ3, 0, 0, α1) = 0 andU∗(√α1Y3,√α1ρ3,√α1R3,√α1LS3,α1)=V ∗(√α1Y3,√α1ρ3,√α1R3,√α1LS3,α1)= W ∗(√α1Y3,√α1ρ3,√α1R3,√α1LS3, α1) = O(√α1)5.The resonant terms δ1 and δ2 in the model (2.13) are determined by the formulaeδ1 =∂2F1∂α1∂Y2,δ2 =12λ2∂2F1∂Y 22∂2F1∂Y2∂ρ2+16λ1∂2F1∂ρ 22∂2F2∂Y 22+1λ1∂2F1∂Y2∂ρ2∂2F2∂Y2∂ρ2− 1λ3∂2F1∂Y2∂R2∂2F3∂Y2∂ρ2+12(2λ1 − λ3)∂2F1∂ρ2∂R2∂2F3∂Y 22− 1λ4∂2F1∂Y2∂Ls2∂2F4∂Y2∂ρ2+12(2λ1 − λ4)∂2F1∂ρ2∂Ls2∂2F4∂Y 22+12∂3F1∂Y 22 ∂ρ2,where all derivatives in δ1 and δ2 are calculated at E∗1 and α1 = 0.Proof. Differentiating (2.12) in the power of (2.11) and (2.13) we get the equations forthe determination of the individual terms of the polynomials hj , j = 1, 2, 3, 4, and theresonant terms δ1, δ2 by standard “step by step” procedure. As the whole record of thisprocedure is rather long we are omitting it.The model (2.13) takes in polar coordinates Y3 = reiϕ, ρ3 = re−iϕ the formṙ = r(ar2 + bα1) + Ũ◦(r, ϕ,R3, LS3, α1) + Ũ∗(r, ϕ,R3, LS3, α1)ϕ̇ = ω + cα1 + dr2 +1r[Φ◦(r, ϕ,R3, LS3, α1) + Φ∗(r, ϕ,R3, LS3, α1)]Ṙ3 = λ3R3 + Ṽ ◦1 (r, ϕ,R3, LS3, α1) + Ṽ∗1 (r, ϕ,R3, LS3, α1)L̇S3 = λ4LS3 + W̃◦1 (r, ϕ,R3, LS3, α1) + W̃∗1 (r, ϕ,R3, LS3, α1),(2.14)where a = Re δ2, b = Re δ1.Equadiff-11. On the existence of Hopf bifurcation in an open economy model 235The behavior of solutions of the model (2.14) around its equilibrium for small param-eters α1 depends on the signs of the constants a, b. It is known that to every constantsolution of the bifurcation equation ar2+bα1 = 0 a periodic solution of (2.14) corresponds(see for example [1]).The following theorem ([4]) gives a sufficient condition for the negativeness of thecoefficient b.Theorem 3. Let the assumptions of Theorem 1 be satisfied. Let in addition α0 > α∗and the critical value β0 be such that c3 − β0f2l3 > 0. Then the value of the coefficient bis negative.Analyzing the model (2.14) and taking into account all transformations which havebeen done to get (2.14) we can formulate on the base of Poincaré-Andronov-Hopf bifur-cation theorem (see for example [7]) the following statement.Theorem 4. Let the assumptions of Theorem 1 be satisfied. Let in addition α0 > α∗and the critical value β0 be such that c3 − β0f2l3 > 0. Then:1. If a > 0 then the equilibrium E∗1 of the model (2.3) is unstable also at the criticaltriple (α0, β0, γ0) and to every α > α0 there exists an unstable limit cycle.2. If a < 0 then the equilibrium E∗1 of the model (2.3) is asymptotically stable alsoat the critical triple (α0, β0, γ0) and to every α∗ < α < α0 there exists a stablelimit cycle.2.1. Numerical example. Take the functions I, S, T , C, D, F , L from the model(1.1) in the formI = 1.8 + 0.6√Y − 0.3R, C = 0.024 + 0.2R,S = −0.8 + 0.01(Y D)2 + 0.3R, D = 2.7− 0.3Y,Y D = 0.7Y + 0.2, F = −0.6− 0.05Y + 0.03ρ,T = −0.2 + 0.3Y, L = 2 + 0.8Y − 0.2R.Then (1.1) has the following formẎ = α[0.6√Y − 0.01(0.7Y + 0.2)2 − 0.35Y + 0.03ρ− 0.6R + 4.7]ρ̇ = −β (−0.05Y + 0.03ρ + 0.2R− 0.576)Ṙ = γ (0.8Y − 0.2R− LS + 2)L̇S = −0.25Y − 0.03ρ− 0.2R + 3.276.(2.15)The equilibrium of (2.15) isE∗ = (Y ∗ = 9, ρ∗ = 1.2541667, R∗ = 4.941875, L∗S = 8.211625),where the values of Y and LS are taken in 102 milliards. After the transformation of E∗into the origin and Taylor expansion the model (2.15) has the formẎ1 = α(−0.341Y1 + 0.03ρ1 − 0.6R1)+ α[− 69190000Y 21 +16480Y 31 −193312Y 41 + O(|Y1|5)]ρ̇1 = β (0.05Y1 − 0.03ρ1 − 0.2R1)Ṙ1 = γ (0.8Y1 − 0.2R1 − LS1)L̇S1 = − 0.25Y1 − 0.03ρ1 − 0.2R1.(2.16)236 K. Makov́ınyiová and R. ZimkaThe matrix of linear approximation of this model isA(α, β, γ) =−0.341α 0.03α −0.6α 00.05β −0.03β −0.2β 00.8γ 0 −0.2γ −γ−0.25 −0.03 −0.2 0 .Take β0 = 1, γ0 = 1. Then α0 =5(26610277753+462357√46690885481)933779562749.= 0.6774. The eigenval-ues of the matrix A(α0, β0, γ0) areλ1.= 0.3886 i, λ2.= −0.3886 i, λ3.= −0.3748, λ4.= −0.0862.The formulae for the calculation of the resonant constants δ1 and δ2, which are introducedin Theorem 2, give the valuesδ1.= −0.3180− 0.3537 i, δ2.= 0.0020− 0.0024 i.The bifurcation coefficients area = Re δ2.= 0.0020, b = Re δ1.= −0.3180,and the partial normal form of the model (2.16) on invariant surface for critical triple(α0, β0, γ0) = (0.6774, 1, 1) in polar coordinates isṙ = r(0.0020r2 − 0.3180α1) + Ũ◦(r, ϕ,R3, LS3, α1) + Ũ∗(r, ϕ,R3, LS3, α1)ϕ̇ = 0.3886− 0.3537α1 − 0.0024r2 +1r[Φ◦ + Φ∗]Ṙ3 = −0.3748R3 + Ṽ ◦1 (r, ϕ,R3, LS3, α1) + Ṽ ∗1 (r, ϕ,R3, LS3, α1)L̇S3 = −0.0862LS3 + W̃ ◦1 (r, ϕ,R3, LS3, α1) + W̃ ∗1 (r, ϕ,R3, LS3, α1).The value of α∗ from (2.8) is α∗ .= 0.3648. On the base of the values a and b we getthe following result: To every value of parameter α > α0.= 0.6774 the equilibrium of themodel (2.15) is asymptotically stable and there exists an unstable limit cycle. To everyvalue of parameter α ∈ (0.3648, 0.6774) this equilibrium is unstable.REFERENCES[1] Yu. N. Bibikov, Multi-frequency non-linear oscillations and their bifurcations, The Publishing Houseof the Saint Petersburg University, Saint Petersburg, 1991, (in Russian).[2] P. Fellnerová and R. Zimka, On Stability and Fluctuations in Open Economy Model under Flex-ible Exchange Rate Regime, Prace Naukowe Akademii Ekonomicznej we Wroclawiu, 1036 (2004),339–347.[3] W. M. Liu, Criterion of Hopf Bifurcations without using Eigenvalues, Journal of MathematicalAnalysis and Applications, 182 (1994), 250–256.[4] K. Makov́ınyiová and R. Zimka, On Fluctuations in an Open Economy Model, Proceeding of the7th International conference “Applications of Mathematics and Statistics in Economy”, Universityof Economics Prague, Faculty of Informatics and Statistics, Professional Publishing, Prague (2004),144–152.[5] G. J. Schinasi, Fluctuations in a Dynamic, Intermediate-Run IS-LM Model: Applications of thePoincaré-Bendixon Theorem, Journal of Economics Theory, 28 (1982), 369–375.[6] L. Striežovská and R. Zimka, Cycles in an Open Economy Model under Fixed Exchange RateRegime with Pure Money Financing, Operations Research and Decisions, Quarterly 3–4, Wroclaw(2002), 133–141.[7] S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, Springer-Verlag,Berlin, 1990.[8] E. Zimková and R. Zimka, The Stability of an Open Economy Model, Mundus Symbolicus 10,Economic University, Prague (2002), 187–193.

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On the existence of Hopf bifurcation in an open economy model

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On the existence of Hopf bifurcation in an open economy model

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