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

Transcription

1 arxiv:mah/511322v1 [mah.ds] 12 Nov 25 Hopf bifurcaion in a dynamic IS-LM model wih ime delay Mihaela Neamţu a, Dumiru Opriş b, Consanin Chilǎrescu a a Deparmen of Economic Informaics and Saisics, Faculy of Economics, Wes Universiy of Timişoara, sr. Pesalozzi, nr. 16A, 3115, Timişoara, Romania, s:mihaela.neamu@fse.uv.ro, cchilarescu@recora.uv.ro b Deparmen of Applied Mahemaics, Faculy of Mahemaics, Wes Universiy of Timişoara, Bd. V. Parvan, nr. 4, 3223, Timişoara, Romania, opris@mah.uv.ro Absrac. In his paper we invesigae he impac of delayed ax revenues on he fiscal policy ou-comes. Choosing he delay as a bifurcaion parameer we sudy he direcion and he sabiliy of he bifurcaing periodic soluions. Wih respec o he delay we show when he sysem is sable. Some numerical examples are finally given for jusifying he heoreical resuls. Keywords: delay differenial equaion, sabiliy, Hopf bifurcaion, IS-LM model. 2 AMS Mahemaics Subjec Classificaion: 34K2, 34C25, 37G15, 91B Inroducion The differenial equaions wih ime delay play an imporan role for economy, engineering, biology and social sciences, because a lo of phenomena are described wih heir help. In his paper, we consider a model from economy, of he IS-LM ype wih ime delay and we sudy how he delay affecs he Corresponding auhor 1

2 macroeconomic sabiliy. The Hopf bifurcaion and normal form heories are ools for esablishing he exisence and he sabiliy of he periodic soluions. Similar ideas can be found in [5], [9], [1]. In [2] Cesare and Sporelli aking ino accoun he papers [6], [7], [8], sudy a dynamic IS-LM model of he following ype: Ẏ()=α[I(Y(),r())+g S(Y() T(Y(),Y( τ)))) T(Y(),Y( τ)))] ṙ() = β[l(y(),r()) M()] Ṁ() = g T(Y(),Y( τ))), wih Y as income, I as invesmen, g as governmen expendiure (consan), S as savings, T as ax revenues, r as rae of ineres, L as liquidiy, M as real money supply and α,β as posiive consans. The ime delay τ appears in funcion T: T(Y(),Y( τ))) = d(1 ε)y()+dεy( τ), (1) where d (,1) is a common average ax rae and ε (,1) is he income ax share. Based on he papers [2], [1], [11], we consider he following IS-LM model: Ẏ()=α[I(Y(),r())+g S(Y() T(Y(),Y( τ))) T(Y(),Y( τ))] ṙ() = β[l(y(),r()) M()] K() = I(Y( τ),r()) δk() Ṁ() = g T(Y(),Y( τ)), (2) wih δ > and wih he iniial condiions: Y() = ϕ(), [ τ,],r() = r 1,M() = M 1,K() = K 1. In he following analysis we will consider he funcion T given by (1), he invesmen, he saving and he liquidiy of he form: I(Y(),r()) = ay() α 1 r() α 2, a >,α 1 >,α 2 >, S(Y() T(Y(),Y( τ))) = s(y() T(Y(),Y( τ))), s (,1), γ L(r()) = my()+, m >,γ >,r 2 >. r() r 2 (3) 2

3 The paper is organized as follows. In secion 2 we invesigae he local sabiliy of he equilibrium poin associaed o sysem (2). Choosing he delay as a bifurcaion parameer some sufficien condiions for he exisence of Hopf bifurcaionarefound. In secion 3 here ishe mainaimof he paper, namely he direcion, he sabiliy and he period of a limi cycle soluion. Secion 4 gives some numerical simulaions which show he exisence and he naure of he periodic soluions. Finally, some conclusions are given. 2. The qualiaive analysis of sysem (2). Using funcions (1) and (3), sysem (2) becomes: Ẏ()=α[((s 1)(1 ε)d s)y()+dε(s 1)Y( τ)+ay() α 1 r() α 2 +g] γ ṙ() = β[my()+ M()] r() r 2 K() = ay( τ) α 1 r() α 2 δk() Ṁ() = g d(1 ε)y() dεy( τ), (4) α,β >, α 1,α 2 >, m >, γ >, r 2 >, δ >, s (,1), d (,1), ε (,1). Sysem (4) is a sysem of equaions wih ime delay. The qualiaive analysis is done using he mehods from [3]. The equilibrium poin of sysem (4) has he coordinaes Y,r,K,M, where: Y = g d,r =[ s(1 d) Y 1 α 1 ] 1 a Using he ranslaion α 2,K = s(1 d)y δ,m = my + γ r r 2. (5) Y = x 1 +Y,r = x 2 +r,k = x 3 +K,M = x 4 +M in (4) and considering he Taylor expansion of he righ members from (4) unil he hird order, we have: ẋ() = Ax()+Bx( τ)+f(x(),x( τ)) (6) 3

4 where αa 1 aαρ 1 αb 1 βm βγ γ 1 β A=,B= aρ 1 δ aρ 1 a 4 b 4 (7) where x() = (x 1 (),x 2 (),x 3 (),x 4 ()) T, and a 1 = (s 1)d(1 ε) s+aρ 1,b 1 = (s 1)dε,a 4 = d(1 ε),b 4 = dε ρ 1 = α 2 Y α 1 r (α 2+1),ρ 1 = α 1 Y α 1 1 r α 2 and F(x(),x( τ)) = (F 1 (x(),x( τ)),f 2 (x(),x( τ)),f 3 (x(),x( τ)), (8) F 4 (x(),x( τ))) T, F 1 (x(),x( τ)) = αa 2 ρ 2x 2 1 ()+αaρ 11x 1 ()x 2 ()+ αa 2 ρ 2x 2 2 ()+ αa 6 (ρ 3x 3 1 ()+3ρ 21x 1 () 2 x 2 ()+3ρ 12 x 1 ()x 2 () 2 +ρ 3 x 2 () 3 ) F 2 (x(),x( τ)) = γ 2 γ 2x 2 () 2 + γ 6 γ 3x 2 () 3 F 3 (x(),x( τ)) = a 2 ρ 2x 1 ( τ) 2 +aρ 11 x 1 ( τ)x 2 ()+ a 2 ρ 2x 2 () a 6 (ρ 3x 3 1 ( τ)+3ρ 21x 1 ( τ) 2 x 2 ()+3ρ 12 x 1 ( τ)x 2 () 2 +ρ 3 x 2 () 3 ) F 4 (x(),x( τ)) = and ρ 2 = α 1 (α 1 1)Y α 1 2 r α 2, ρ 11 = α 1 α 2 Y α 1 1 r (α 2+1) ρ 2 = α 2 (α 2 +1)Y α 1 r (α 2+2), ρ 3 = α 1 (α 1 1)(α 1 2)Y α 1 3 r α 2 ρ 21 = α 1 α 2 (α 1 1)Y α 1 2 r (α 2+1), ρ 12 = α 1 α 2 (α 2 +1)Y α 1 1 r (α 2+2) ρ 3 = α 2 (α 2 +1)(α 2 +2)Y α 1 r (α 2+3), γ 1 = (r r 2 ) 2,γ 2 = (r r 2 ) 3,γ 3 = (r r 2 ) 4. 4

5 The characerisic equaion of linear par from (6) is: de(λi A Be λτ ) = (λ+δ) (λ,τ) = (9) where (λ,τ) = P(λ)+e λτ Q(λ) (1) and P(λ) = λ 3 +p 2 λ 2 +p 1 λ+p, Q(λ) = q 2 λ 2 +q 1 λ+q p 2 = (αa 1 +βγ γ 1 ), p 1 = αβ(γ a 1 γ 1 +maρ 1 ), p = αaβa 4 ρ 1 q 2 = αb 1, q 1 = αβγ b 1 γ 1, q = αβb 4 ρ 1. To invesigae he local sabiliy of he equilibrium poin, we begin by considering, as usual, he case wihou delay (τ = ). In his case he characerisic polynomial is: (λ+δ)(p(λ)+q(λ)) = hence, according o he Hurwiz crierion, he equilibrium poin is sable if and only if: p 2 +q 2 >, (p 1 +q 1 )(p 2 +q 2 ) > p +q. When τ >, sandard resuls on sabiliy of sysems of delay differenial equaions posulae ha a equilibrium poin is asympoically sable if an only if all roos of equaion (1) have a negaive real par. I is well known ha equaion (1) is a ranscendenal equaion which has an infinie number of complex roos and he some possible roos wih posiive real par are finie in number. We wan o obain he values τ such ha he equilibrium poin (5) changes from local asympoic sabiliy o insabiliy or vice versa. We need he imaginary soluions of equaion (λ, τ) =. Le λ = ±iω be hese soluions and wihou loss of generaliy we assume ω >. We suppose ha P(iω)+Q(iω), for all ω IR. The previous condiions are equivalen o (p 1 +q 1 )(p 2 +q 2 ) p +q. A necessary condiion o have ω as a soluion of (iω,τ) = is ha ω mus be a roo of he following equaion: f(ω) = ω 6 +a F ω 4 +b F ω 2 +c F = (11) 5

6 where a F = p 2 2 q2 2 2p 1, b F = 2q q 2 2p p 2 q 2 1 +p2 1, c F = p 2 q2. Le k = a F 3 and f D = 1 4 [f(k)] [f (k)] 3. Using he resuls from [2], i resuls: Proposiion Le ε > 1. Then he following cases can be discerned: 2 (i) If a F or b F, hen equaion (11) has only one real posiive roo; (ii) If a F <, b F > and f D, hen equaion (11) has only one real posiive roo. 2. If ε < 1 2, a F <, b F < and f D hen equaion (11) has wo real posiive roos which are disinc if f D. 3. If ε = 1 2 and: (i) a F <, b F = hen equaion (11) has only one real posiive roo; (ii)a F <, b F >, hen equaion (11) has wo real posiive roos; (iii)b F <, hen equaion (11) has only one real posiive roo. Also, we have: Theorem 1. If we suppose ha he equilibrium poin (Y,r,K,M ) is locally asympoically sable wihou ime delay, hen in condiions of Proposiion 1 here exiss only one sabiliy swich. Theorem 2. If τ is a sabiliy swich and f D, hen a Hopf bifurcaion occurs a τ, where τ = 1 ( ω (ω 4 arcg q 2 ω 2 ) (q q 1 p 2 +q 2 p 1 )+q p 1 p q 1 ) ω ω(q 4 1 q 2 p 2 )+ω(q 2 p 2 q 1 p 1 +p q 2 ) p q and ω is a roo of (11). 3. The normal form for sysem (4). Cyclical behavior. In his secion we describe he direcion, sabiliy and he period of he bifurcaing periodic soluions of sysem (4). The mehod we use is based on he normal form heory and he cener manifold heorem inroduced by Hassard [4]. Taking ino accoun he previous secion, if τ = τ hen all roos of equaion (9) oher han ±iω have negaive real pars, and any roo of equaion (9) of he form λ(τ) = α(τ)±iω(τ) saisfies α(τ ) =, ω(τ ) = ω 6

7 and dα(τ ). For noaional convenience le τ = τ + µ,µ IR. Then dτ µ = is he Hopf bifurcaion value for equaions (4). Definehespaceofconinuousreal-valuedfuncionsasC = C([ τ,],ir 4 ). In τ = τ + µ,µ IR, we regard µ as he bifurcaion parameer. For Φ C we define a linear operaor: L(µ)Φ = AΦ()+BΦ( τ) where A and B are given by (7) and a nonlinear operaor F(µ,Φ) = F(Φ(), Φ( τ)), where F(Φ(),Φ( τ)) is given by (8). By he Riesz represenaion heorem, here exiss a marix whose componens are bounded variaion funcions, η(θ,µ) wih θ [ τ,] such ha: L(µ)Φ = τ dη(θ,µ)φ(θ), θ [ τ,]. For Φ C 1 ([ τ,],ir 4 ) we define: dφ(θ) dθ, θ [ τ,) A(µ)Φ(θ) = dη(,µ)φ(), θ =, τ, θ [ τ,) R(µ)Φ(θ) = F(µ,Φ), θ =. We can rewrie (6) in he following vecor form: u = A(µ)u +R(µ)u (12) where u = (u 1,u 2,u 3,u 4 ) T, u = u(+θ) for θ [ τ,]. For Ψ C 1 ([,τ ],IR 4 ), we define he adjunc operaor A of A by: dψ(s) A ds, s (,τ ] Ψ(s) = dη T (,)ψ( ), s =. τ 7

8 We define he following bilinear form: < Ψ(θ),Φ(θ) >= Ψ T ()Φ() τ ξ= Ψ T (ξ θ)dη(θ)φ(ξ)dξ, where η(θ) = η(θ,). We assume ha ±iω are eigenvalues of A(). Thus, hey are also eigenvalues of A. We can easily obain: where v = (v 1,v 2,v 3,v 4 ) T, Φ(θ) = ve λ 1θ, θ [ τ,] (13) v 1 = 1,v 2 = β(a 4 +b 4 e λ 1τ mλ 1 ),v 4 = a 4 +b 4 e λ 1τ, λ 1 (λ 1 βγ γ 1 ) λ 1 v 3 = a λ 1 +δ [ρ 1e λ 1τ β(a 4 +b 4 e λ1τ mλ 1 ) +ρ 1 ] λ 1 (βγ γ 1 λ 1 ) is he eigenvecor of A() corresponding o λ 1 = iω and where w = (w 1,w 2,w 3,w 4 ), Ψ(s) = we λ 2s, s [, ) w 1 = λ 2 βγ γ 1 1 αaρ 1 η,w 2 = 1 η,w 3 =,w 4 = β 1 λ 2 η η = λ 1 βγ γ 1 λ 1 τ e λ1τ e λ1τ +1 (1+αb 1 )+v aαρ 1 λ 2 2 β λ 1 τ e λ 1τ e λ 1τ +1 (v 4 +b 4 ) 1 λ 1 λ 2 1 is he eigenvecor of A() corresponding o λ 2 = iω. Wecanverifyha: < Ψ(s),Φ(s) >= 1,< Ψ(s), Φ(s) >=< Ψ(s),Φ(s) >=, < Ψ(s), Φ(s) >= 1. Using he approach of Hassard [4], we nex compue he coordinaes o describe he cener manifold Ω a µ =. Le u = u (+θ),θ [ τ,) be he soluion of equaion (12) when µ = and z() =< Ψ,u >, w(,θ) = u (θ) 2Re{z()Φ(θ)}. On he cener manifold Ω, we have: w(,θ) = w(z(), z(),θ) 8

9 where w(z, z,θ) = w 2 (θ) z2 2 +w 11(θ)z z +w 2 (θ) z2 2 +w 3(θ) z in which z and z are local coordinaes for he cener manifold Ω in he direcion of Ψ and Ψ and w 2 (θ) = w 2 (θ). Noe ha w and u are real. For soluion u Ω of equaion (12), since µ =, we have: where where wih and ż() = λ 1 z()+g(z, z) (14) g(z, z) = Ψ()F(w(z(), z(),)+2re(z()φ())) = z() 2 = g 2 +g 11 z() z()+g 2 z() 2 z() 2 z() +g g 2 = F 1 2 w 1 +F 2 2 w 2,g 11 = F 1 11 w 1 +F 2 11 w 2,g 2 = F 1 2 w 1 +F 2 2 w 2, (15) F 1 2 = αa(ρ 2 +2ρ 11 v 2 +ρ 2 v 2 2 ),F1 11 = αa(ρ 2 +ρ 11 ( v 2 +v 2 )+ρ 2 v 2 v 2 ), F 2 2 = γ γ 2 v 2 2,F 2 11 = γ γ 2 v 2 v 2,F 1 2 = F 1 2,F 2 2 = F 2 2, where g 21 = F 1 21 w 1 +F 2 21 w 2 (16) F 1 21 = αaρ 2(2w 1 11 ()+w1 2 ())+αaρ 11(2w 2 11 ()+w2 2 ()+2w1 11 () v 2+ w 1 2()v 2 )+αaρ 2 (2w 2 11()v 2 +w 2 2() v 2 )+αa(ρ 3 +2ρ 21 v 2 +2ρ 12 v ρ 3 v 2 2 v 2 +ρ 12 v 2 2 +ρ 21 v 2 ) F 2 21 = γ γ 2 (2w 2 11()v 2 +w 2 2() v 2 )+γ γ 3 v 2 2 v 2. The vecors w 2 (θ), w 11 (θ) wih θ [ τ,] are given by: w 2 (θ) = g 2 ve λ1θ ḡ2 λ 1 3λ 1 v λ 2θ e +E 1 e 2λ 1θ w 11 (θ) = g 11 ve λ1θ ḡ11 (17) ve λ2θ +E 2 λ 1 λ 1 9

10 where E 1 = (A+e λ 1τ B 2λ 1 I) 1 F 2, E 2 = (A+B) 1 F 11, where F 2 = (F 1 2,F2 2,F3 2,)T, F 11 = (F 1 11,F2 11,F3 11,)T. Based on he above analysis and calculaion, we can see ha each g ij in (15), (16) are deermined by he parameers and delay from sysem (4). Thus, we can explicily compue he following quaniies: and C 1 () = i (g 2 g 11 2 g ω 3 g 2 2 )+ g 21 2 µ 2 = Re(C 1()) Reλ (τ ),T 2 = Im(C 1())+µ 2 Imλ (τ ),β 2 = 2Re(C 1 ()). ω λ = λ(q 2 λ 2 +q 1 λ+q )e λτ 3λ 2 +2p 2 λ+p 1 +e λτ (2q 2 λ+q 1 τ(q 2 λ 2 +q 1 λ+q )) In summary, his leads o he following resul: (18) Theorem 3. In formulas (18), µ 2 deermines he direcion of he Hopf bifurcaion: if µ 2 > (< ), hen he Hopf bifurcaion is supercriical (subcriical) and he bifurcaing periodic soluions exi for τ > τ (< τ ); β 2 deermines he sabiliy of he bifurcaing periodic soluions: he soluions are orbially sable (unsable) if β 2 < (> ); and T 2 deermines he period of he bifurcaing periodic soluions: he period increases (decreases) if T 2 > (< ). 4. Numerical example. In his secion we find he waveform plos hrough he formula: X(+θ)=z()Φ(θ)+ z() Φ(θ)+ 1 2 w 2(θ)z 2 ()+w 11 (θ)z() z()+ 1 2 w 2(θ) z() 2 +X, wherez()ishesoluionof(14),φ(θ)isgivenby(13),w 2 (θ),w 11 (θ),w 2 (θ) = w 2 (θ) are given by (17) and X = (Y,r,K,M ) T is he equilibrium sae. For he numerical simulaions we use Maple 9.5. We consider sysem (4) wih a =.38, α =.96, β = 1, α 1 =.5, α 2 =.83, γ = 1, d =.1, s =.3, 1

11 r 2 =.3, δ =.2, m =.5, g = 5. The equilibrium poin is: Y =5, r = , K =675, M = In wha follows we consider wo differen shares ǫ of delay ax revenues: ε =.3 and ε =.8. For ε =.3 we obain: µ 2 = , β 2 = , T 2 = , ω = , τ = Then he Hopf bifurcaion is supercriical, he soluions are orbially unsable and he period of he soluion is increasing. The wave plos are given in he following figures: Fig.1.Waveplo (, Y()) Fig.2.Waveplo (, r()) Fig.3.Waveplo (, K()) E E E E E E E2 6.75E E E E E E Fig.4.Waveplo (, M()) Fig.5.Waveplo (, I()) Fig.6.Waveplo (, L()) For ε =.8 we obain: µ 2 = , β 2 = , T 2 = , ω = , τ = Then he Hopf bifurcaion is subcriical, he soluions are orbially sable and he period of he soluion is decreasing. The wave plos are given in he following figures: 11

12 Fig.7.Waveplo (, Y()) Fig.8.Waveplo (, r()) Fig.9.Waveplo (, K()) 5.214E E E2 5E E E2 6.75E E E E E E Fig.1.Waveplo (, M()) Fig.11.Waveplo (, I()) Fig.12.Waveplo (, L()) Conclusions. From he analysis of he model wih coninuous ime, i resuls ha he model acceps a limi cycle. The naure of he limi cycle is given by he coefficiens (18) which include he parameers of he model. We esablish he naure of he limi cycle. Because he expressions of he parameer and he coefficiens (18) are difficul o analyze direcly, he use of Maple 9.5 was essenial. The paper s resuls confirm ha a series of economical processes, where a variable wih ime delay inervenes, have a limi cycle, hus, allowing a predicion concerning he evoluion of he model. 12

13 References [1] Cai J, Hopf bifurcaion in he IS-LM business cycle model wih ime delay, Elecronic Journal of Differenial Equaions, 25(15):1-6. [2] De Cesare L, Sporelli M, A dynamic IS-LM model wih delayed axaion revenues, Chaos, Solions and Fracals, 25(25): [3] Hale J. K, Lunel S.M. V, Inroducion o Funcional Differenial Equaion, Springer-Verlag, New York, Applied Mahemaical Sciences, 1993(99). [4] Hassard B.D, Kazarinoff N.D, Wan Y.H, Theory and applicaions of Hopf bifurcaion, Cambridge Universiy Press, Cambridge, [5] Liao X, Li C, Zhou S, Hopf bifurcaion and chaos in macroeconomic models wih policy lag, Chaos, Solions and Fracals, 25(25): [6] Sasakura K, On he dynamic behavior of Schinasi s business cycle model, J. Macroecon, 1994(16): [7] Schinasi G.J, A nonlinear dynamic model of shor run flucuaions, Rev. Econ. Sud.,1981(48): [8] Schinasi G.J, Flucuaions in a dynamic inermediae-run IS-LM model: applicaions of he Poicar-Bendixon heorem, J. Econ Theory., 1982(28): [9] Szydlowski M, Krawiec A, The sabiliy problem in he Kaldor-Kalecki business cycle model, Chaos, Solions and Fracals, 25(25): [1] Takeuchi Y, Yamamura T, Sabiliy analysis of he Kaldor model wih ime delays: moneary policy and governmen budge consrain, Nonlinear Analysis, 24(5): [11] Torre V, Exisence of limi cycles and conrol in complee Keynesian sysems by heory of bifurcaions, Economerica, 1977(45):