Stability of Limit Cycle in a Delayed IS-LM Business Cycle model

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1 Applied Mathematical Sciences, Vol. 2, 28, no. 5, Stability of Limit Cycle in a Delayed IS-LM Business Cycle model A. Abta, A. Kaddar H. Talibi Alaoui Université Chouaib Doukkali Faculté des Sciences Département de Mathématiques et Informatique B.P:2, El Jadida, Morocco a.kaddar@yahoo.fr, Abstract In this paper, we extend the work of J. Cai [] in the nonlinear case using the investment function of the Kaldor-type. Actually we investigate the direction of the Hopf bifurcation the stability or instability of the bifurcating branch of periodic solutions using the methods presented by O. Diekmann et al. in [3]. In the end we show numerical application. Mathematics Subject Classification: 37G5, 9B62 Keywords: IS-LM business cycle, delayed differential equations, Hopf bifurcation, periodic solutions Introduction mathematical models Kaldor was probably the first economist to realize the importance of the nonlinear mechanism of the economy to introduce his first model (in 94, [7]) by an ordinary differential equations as follows { dy = α[i(y (t),k(t)) S(Y (t),k(t))], = I(Y (t),k(t)), dk where Y is the gross product, K is the capital stock, α is the adjustment coefficient in the goods market, I(Y,K) is the investment function S(Y,K) is the saving function. In this model the nonlinearity of investment saving function leads to limit cycle solution (see also [2, 5, 3] for more information). In (977, [2]) Torre revised updated this model by replacing the capital

2 246 A. Abta, A. Kaddar H. Talibi Alaoui stock K(t) with the interest rate R(t) to formulate the following stard IS-LM business cycle model { dy = α[i(y (t),r(t)) S(Y (t),r(t))], = β[l(y (t),r(t)) M], dr where M is the constant money supply, β is the adjustment coefficient in money market L is the dem for money. In (989, [4]), Gabisch Lorenz considered an augmented IS-LM business cycle model as follows dy = α[i(y (t),k(t),r(t)) S(Y (t),r(t))], dk = I(Y (t),k(t),r(t)) δk(t), = β[l(y (t),r(t)) M], dr where δ is the depreciation rate of capital stock. Based on the Kalecki s idea of time delay (see [8,, ] for more information), Cai (in 25, []) presented the following delayed IS-LM model: dy = α[i(y (t),k(t),r(t)) S(Y (t),r(t))], dk = I(Y (t τ),k(t),r(t)) δk(t), = β[l(y (t),r(t)) M], dr with τ is the time delay needed for new capital to be installed, he investigated the local stability the local Hopf bifurcation for () in the linear case (the functions I, S, L are linear). In the following analysis we will consider the nonlinear case by using the investment function of the Kaldor-type. The dynamics are studied in terms of local stability of the description of the Hopf bifurcation, that is proven to exists as the delay (taken as a parameter of bifurcation) cross some critical value. Additionally we establish an explicit algorithm for determining the direction of the Hopf bifurcation the stability or instability of the bifurcating branch of periodic solutions using the methods presented by O. Diekmann et al. in [3]. In the end, we give some numerical simulations which show the existence the nature of the periodic solutions. 2 Steady state local stability analysis As in [9], we assume that the investment function I, is given by I(Y,K,R)=I(Y ) δ K β R. The saving function S, the dem for money L are given by S(Y,R)=l Y + β 2 R, ()

3 delayed IS-LM business cycle model 246 L(Y,R) =l 2 Y β 3 R, with δ,l,l 2,β,β 2,β 3 are positive constants. Then system () becomes: dy = α[i(y (t)) δ K (β + β 2 )R(t)) l Y ], dk = I(Y (t τ)) (δ + δ )K(t)) β R(t), = β[l 2 Y (t) β 3 R(t) M]. dr (2) 2. Steady state In the following proposition, we give a sufficient conditions for the existence uniqueness of positive equilibrium E of the system (2). Proposition 2. Suppose that (i): There exists a constant L> such that I(Y ) L for all Y R. (ii): I (Y ) ρ< for all Y R, where ρ is given by ρ = ((δ + δ )β 2 + δβ )l 2 + (δ + δ )l. (3) δβ 3 δ Then there exists a unique equilibrium E =(Y,K,R ) of system (2), where Y is the positive solution of K,R are given by I(Y ) ρy + [(δ + δ )β 2 + δβ ] M δβ 3 = (4) K = β 2l 2 + l β 3 δβ 3 Y β 2 δβ 3 M, (5) R = l 2Y M. (6) β 3 proof. (Y,K,R) is a steady-state of (2) if dy = dk = dr =, that is I(Y ) δ K (β + β 2 )R l Y =, I(Y ) (δ + δ )K β R =, l 2 Y β 3 R M =, (7)

4 2462 A. Abta, A. Kaddar H. Talibi Alaoui Let us assume that Y>, K>, R> satisfy (5). Then K = β 2l 2 + l β 3 δβ 3 Y β 2 δβ 3 M, (8) R = l 2Y M, (9) β 3 I(Y ) ρy + [(δ + δ )β 2 + δβ ] M = () δβ 3 where ρ is defined in (3). In view of hypotheses i) ii) of proposition 2. it s clear that equation () has a unique solution Y >. This concludes the proof. 2.2 Local stability analysis of E We will study the stability of the positive equilibrium E. Firstly we recall that E is asymptotically stable if all roots of the characteristic equation associated to the linearized system of (2) have negative real parts, the stability is lost only if characteristic roots cross the imaginary axis, that is if purely imaginary roots appear. Let y = Y Y,k= K K r = R R. By linearizing system (2) around E =(Y,K,R ), we obtain dy = α(i (Y ) l )y(t) αδ k(t) α(β + β 2 )r(t), dk = I (Y )y(t τ) (δ + δ )k(t) β r(t), dr = βl 2y(t) ββ 3 r(t), () The characteristic equation associated to system () takes the general forme with where P (λ) + Q(λ)exp( λτ) = (2) P (λ) =λ 3 + Aλ 2 + Bλ + C Q(λ) =Dλ + E, A = δ + δ + ββ 3 α(i (Y ) l ), B =(δ + δ )[ββ 3 α(i (Y ) l )] + αβ[(β + β 2 )l 2 β 3 (I (Y ) l )], C = αβ[β δl 2 + β 2 (δ + δ )l 2 β 3 (δ + δ )(I (Y ) l )], D = αδ I (Y ),

5 delayed IS-LM business cycle model 2463 E = αββ 3 δ I (Y ). Recall that the equilibrium of (2) is asymptotically stable if all roots of (2) have negative real parts, the stability is lost only if characteristic roots cross the imaginary axis, that is if pure imaginary roots appear. In order to investigate the local stability of the steady state, we begin by considering the case without delay τ =. This case is of importance, because it can be necessary that the nontrivial positive equilibrium of (2) is stable when τ = to be able to obtain the local stability for all nonnegative values of the delay, or to find a critical values which could destabilize the equilibrium. When τ = the characteristic equation (9) reads as λ 3 + Aλ 2 +(B + D)λ +(C + E) =, (3) hence, according to the Routh-Hurwitz criterion, we have the following lemma. Lemma 2. For τ =, the equilibrium E is locally asymptotically stable if only if (H): A>; (H2): C + E>; (H3): A(B + D) (C + E) > ; where A, B, C, D, E, are defined in (2). We assume in the sequel, that hypotheses (H), (H2), (H3) are true, we return to the study of equation (2) with τ>. Let λ = iω, ω R, rewrite (2) in terms of its real imaginary parts as Aω 2 C = E cos(ωτ)+dω sin(ωτ) (4) ω 3 Bω = Dω cos(ωτ) E sin(ωτ) (5) It follows by taking the sum of squares that ω 6 + aω 4 + bω 2 + c =, (6) with a = A 2 2B, b = B 2 2AC D 2,c= C 2 E 2, where A; B; C; D; E are given in (2). Then equation (6) becomes h(z) :=z 3 + az 2 + bz + c =, (7) with z = ω 2. Suppose that equation (7) has simple positive roots. Without loss of generality, assume that it has three positive roots, denoted by z,z 2 z 3, respectively. Then equation (3) has three positive roots, say ω = z ; ω 2 = z 2 ; ω 3 = z 3

6 2464 A. Abta, A. Kaddar H. Talibi Alaoui Let τ l = [arccos( (Aω2 l C)(F Dωl 2)+(ω3 l Bω l )Eω l )],l =, 2, 3. ω l (Dω l F ) 2 + E 2 ωl 2 Then ±iω l is a pair of purely imaginary roots of equation (3) with τ = τ l, l=,2,3. Define τ = τ l = min (τ l),ω = ω l. (8) l=,2,3 We have the following theorem. Theorem 2.2 Assume that (H), (H2) (H3) hold (see lemma 2.). If one of the following hypotheses is true: (H4) c< h (ω ) ;(H5) a<, b, c>, a 2 > 3b, Δ < ; (H6) b<, c>, Δ < ; where Δ is defined by Δ= 4 27 b3 27 a2 b a3 c 2 3 abc + c2. (9) ω is defined by (8). Then τ > when τ [,τ ) the steady state E is locally asymptotically stable, when τ = τ, system (2) will undergo a Hopf bifurcation. Moreover, we have Re( dλ dτ )(τ ) >. Proof. Similar to the proof in []. 3 Direction of Hopf Bifurcation In the last section we have a Hopf bifurcation at a critical value τ of the delay. Thus (see [6]), there exists ɛ > such that for each ɛ<ɛ, system (2) has a family of periodic solutions p(ɛ) with period T = T (ɛ), for the parameter values τ = τ(ɛ) such that p() =, T () = 2π ω τ = τ(), where τ = ω arccos (Aω2 C)+(ω3 Bω )Dω, with A; B; C; D; E are given in (2) ω E 2 ω 2 is the least simple positive root of (6). In this section we use a formula on the direction of the Hopf bifurcation given by Diekmann in [3] to formulate an explicit algorithm about the direction the stability of the bifurcating branch of periodic solutions of (4). Normalizing the delay τ by scaling t t effecting the change U(t) = τ Y (τt), V(t) =K(τt), W (t) =R(τt), the system (4) is transformed into du = ατ[i(u(t)) δ V (t) (β + β 2 )W (t) l U(t)], dv = τ(i(u(t )) (δ + δ )V (t)) β W (t), = βτ(l 2 U(t) β 3 W (t) M). dw (2)

7 delayed IS-LM business cycle model 2465 By the translation Z(t) =(U, V, W ) (Y,K,R ), system (2) is written as a functional differential equation in C := C([, ], R 3 ), Z(t) =L(τ)Z t + h(z t,τ), (2) where L(τ) :C R 3 the linear operator h: C R R 3 the nonlinear part of (2) are given respectively by: L(τ)ϕ = τ h(ϕ, τ) =τ Let α[i (Y ) l )ϕ () δ ϕ 2 () (β + β 2 )ϕ 3 ()] I (Y )ϕ ( ) (δ + δ )ϕ 2 () β ϕ 3 () β[l 2 ϕ () β 3 ϕ 3 ()] α[i(ϕ () + Y ) I (Y )ϕ () δ K (β + β 2 )R l Y )] I(ϕ ( ) + Y ) I (Y )ϕ () (δ + δ )K β R βl 2 Y ββ 3 R β M) L := L(τ ):C R 3. Using the Riesz representation theorem (see [6]), we obtain Lϕ = dη(θ)ϕ(θ) (22) where, dη(θ) =τ α(i (Y ) l )δ(θ) αδ δ(θ) α(β + β 2 )δ(θ) I (Y )δ(θ +) (δ + δ )δ(θ) β δ(θ) βl 2 δ(θ) ββ 3 δ(θ) (23) δ(.) denotes the Dirac function. Let A(τ) denotes the generator of semigroup generated by the linear part of (2) A = A(τ ). Then, { dϕ (θ) for θ [, ) Aϕ(θ) = dθ (24) Lϕ for θ = for ϕ =(ϕ,ϕ 2 ) C. From Theorem 2., a Hopf bifurcation occurs at the critical value τ = τ. By the Taylor expansion of the time delay function τ(ε) near the critical value τ, we have τ(ε) =τ + τ 2 ε 2 + o(ε 2 ). (25) The sign of τ 2 determines either the bifurcation is supercritical (if τ 2 > ) periodic orbits exist for τ > τ, or it is subcritical (if τ 2 < ) periodic

8 2466 A. Abta, A. Kaddar H. Talibi Alaoui orbits exist for τ < τ. formula, The term τ 2 may be calculated (see [3]) using the τ 2 = Re(c) Re(qD 2 M (iω,τ )p, (26) where M is the characteristic matrix of the linear part of (2), M (λ, τ) = λ τα(i (Y ) l ) ταδ τα(β + β 2 ) τi (Y ) exp( λ) λ + τ(δ + δ ) τβ τβl 2 λ + τββ 3, (27) D 2 M (iω,τ ) denotes the derivative of M with respect to τ at τ = τ, the constant c is defined as follows c = 2 qd3 h(,τ )(P 2 (θ), P (θ)) +qd 2 h(,τ )(e. M (,τ )D 2 h(,τ )(P (θ), P(θ)),P(θ)) + 2 qd2 h(,τ )(e 2iω. M (2iω,τ )D 2 h(,τ )(P (θ),p(θ)), P(θ)) where D i h, i =2, 3, denotes the i th derivative of h with respect to ϕ, P (θ) denotes the eigenvector of A, P (θ) denotes it conjugate eigenvector p, q are defined later. To study the direction of Hopf bifurcation, one needs to calculate the second third derivatives of nonlinear part of (2)with respect to ϕ, D 2 h(ϕ, τ)ψχ = τ αi (ϕ () + Y )ψ ()χ () I (ϕ ( ) + Y )ψ ( )χ ( ) (28) D 3 h(ϕ, τ)ψχυ = τ αi (ϕ () + Y )ψ ()χ ()υ () I (ϕ ( ) + Y )ψ ( )χ ( )υ ( ) (29) Then Dh(,τ 2 )ψχ = τ αi (Y )ψ ()χ () D 3 h (,τ )ψχυ = [ τ αi (Y )ψ ()χ ()υ () ( ) + τ I (Y )ψ ( )χ ( ) + τ I (Y )ψ ( )χ ( )υ ( ) ( (3) )]. (3)

9 delayed IS-LM business cycle model 2467 ψ =(ψ,ψ 2,ψ 3 ),χ=(χ,χ 2,χ 3 ),υ =(υ,υ 2,υ 3 ) C([, ], R 3 ). As iω is a solution of (2) at τ = τ, then iω is an eigenvalue of A there exist a corresponding eigenvector of the form P (θ) =pe iω θ where p = (p,p 2,p 3 ) C 2, satisfy the equations: with Then one may assume calculate p 2 = Mp = M = M (iω,τ ). (32) p =, [ τ 2αβl 2(β + β 2 )( iω + τ ββ 3 ) + iω τ αδ ω 2 + τ 2 β 2 β3 2 τ α(i (Y ) l )], p 3 = τ βl 2 ( iω + τ ββ 3 )). ω 2 + τ 2 β 2 β3 2 So, from (3) (3), we have Dh(,τ 2 )(P (θ), P(θ)) = τ I (Y ) D 2 h(,τ )(P (θ),p(θ)) = τ I (Y ) D 3 h(,τ )(P 2 (θ), P (θ)) = τ I (Y ) α α exp( 2iω ) α exp( iω ) (33) (34). (35) Now, consider A, a conjugate operator of A, A : C([, ], R 2 ) R 2, defined by, { A dψ (s), for s (, ] ψ(s) = ds ψ( s)dη(s), for s = (36) ψ =(ψ,ψ 2 ) C([, ], R 2 ). Let Q(s) =qe iω s be the eigenvector for A associated to the eigenvalue iω, q =(q,q 2,q 3 ) T. One needs to choose q such that the inner product (see [6]), <Q,P >=,

10 2468 A. Abta, A. Kaddar H. Talibi Alaoui where <Q,P >= Q()P () If we take q 2 =, q 3 =, then θ q = from (35), we have Q(ξ θ)dη(θ)p (ξ)dξ. 2 qd3 h(,τ )(P 2 (θ), P (θ)) = ατ I (Y ) 2 (37) From (27), (3),(33), we deduce, qd 2 h(,τ )(e. M (,τ )D 2 h(,τ )(P (θ), P (θ)),p(θ)) = τ 4 α 2 ββ 3 δi (Y ) 2, det M (,τ ) (38) where det M (,τ )=τ 3 αβ[ β 3(I (Y ) l )(δ+δ )+β 3 δ I (Y ) β δ l 2 +l 2 (β +β 2 )(δ+δ )] where 2 qd2 h(,τ )(e 2iω. M (2iω,τ )D 2 h(,τ )(P (θ),p(θ)), P(θ)) = α 2 τ 2 I (Y ) 2 2(C 2 + C 2 2) [(C C 3 + C 2 C 4 )+i(c C 4 C 2 C 3 )], (39) C = τ α(i (Y ) l )(4ω 2 τ 2 (δ + δ )ββ 3 ) 4ω 2 τ (ββ 3 + δ + δ ) +τ 2 αδ I (Y )[τ ββ 3 cos(2ω )+2ω sin(2ω )]+τ 3 αβ(β +β 2 )l 2 (δ+δ ) τ 3 βl 2αδ β, C 2 =2ω (τ 2 ββ 3 (δ + δ ) 4ω 2 ) 2ω τ 2 α(ββ 3 + δ + δ )(I (Y ) l ) +τ 2 αδ I (Y )[2ω cos(2ω ) τ ββ 3 sin(2ω )] + 2τ 2 αβ(β + β 2 )l 2 ω, C 3 = 4ω 2 + τ 2 ββ 3 (δ + δ ) τ 2 ββ 3 δ cos(2ω ) 2τ δ ω sin(2ω ), C 4 =2ω τ (ββ 3 + δ + δ ) 2τ δ ω cos(2ω )+τ 2 δ ββ 3 sin(2ω ).

11 delayed IS-LM business cycle model 2469 Then Re(c) = ατ I (Y ) + τ 2α2 I (Y ) 2 2 2(C 2 + C2) (C C C 2 C 4 )+ α2 τ 4ββ 3δI (Y ) 2. (4) det M (,τ ) Now, from (27) we have Re(qD 2 M (iω,τ )p) =Re( dλ dτ )(τ ), from theorem 2., we have Re( dλ dτ )(τ ) >. Consequently we deduce the following result : Theorem 3. Assume (H), (H2), (H3). If one of the previous conditions (H4), or (H5) or (H6) (see theorem 2.2)holds, then, (a) the Hopf bifurcation occurs as τ crosses τ to the right (supercritical Hopf bifurcation) if Re(c) > to the left (subcritical Hopf bifurcation) if Re(c) < ; (b) the bifurcating periodic solutions is stable if Re(c) > unstable if Re(c) < ; where Re(c) is given by (4). Note that, Theorem 3. provides an explicit algorithm for detecting the direction stability for the Hopf bifurcated branch of periodic solution given by theorem Application From section 3, we can determine the direction of a Hopf bifurcation the stability of the bifurcating periodic solutions through the formula (4). In this section, we give a numerical simulation supporting the theoretical analysis. Consider the following Kaldor-type investment function: I(Y )= exp(y ) + exp(y ). When α =3;β =2;δ =.; δ =.5; M =.5 l =.2; l 2 =.; β = β 2 = β 3 =.2, the system (2) has the positive equilibrium E =(.534,.44,.67). It follows from section 3, that τ = Re(c) = Thus from theorem 2. we know that when τ<τ,e is asymptotically stable, this property is depicted in the numerical simulation by Fig.. When τ passes through the critical value τ,e loses its stability a family of unstable periodic solutions bifurcating from E occurs (see Fig.2).

12 247 A. Abta, A. Kaddar H. Talibi Alaoui t Figure : The steady state E of (2) is stable when τ = t Figure 2: A family of periodic solution bifurcating from E occurs when τ =.7975.

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