AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 738460 10.1155/2013/738460 738460 Research Article Center Manifold Reduction and Perturbation Method in a Delayed Model with a Mound-Shaped Cobb-Douglas Production Function Ferrara Massimiliano 1 http://orcid.org/0000-0001-8489-5531 Guerrini Luca 2 Bisci Giovanni Molica 3 Bianca Carlo 1 Department of Law and Economics University Mediterranea of Reggio Calabria Via dei Bianchi 2 (Palazzo Zani) 89127 Reggio Calabria Italy unirc.it 2 Department of Management Polytechnic University of Marche Piazza Martelli 8 60121 Ancona Italy univpm.it 3 Department of PAU University Mediterranea of Reggio Calabria Via Melissari 24 89124 Reggio Calabria Italy unirc.it 2013 12 12 2013 2013 29 10 2013 20 11 2013 2013 Copyright © 2013 Massimiliano Ferrara et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Matsumoto and Szidarovszky (2011) examined a delayed continuous-time growth model with a special mound-shaped production function and showed a Hopf bifurcation that occurs when time delay passes through a critical value. In this paper, by applying the center manifold theorem and the normal form theory, we obtain formulas for determining the direction of the Hopf bifurcation and the stability of bifurcating periodic solutions. Moreover, Lindstedt’s perturbation method is used to calculate the bifurcated periodic solution, the direction of the bifurcation, and the stability of the periodic motion resulting from the bifurcation.

1. Introduction

In recent years, great attention has been paid to economic growth models with time delay. The reason is that, getting closer to the real world, there is always a delay between the time when information is obtained and the time when the decision is implemented. Different mathematical and computational frameworks have been proposed whose difficulty is strictly related to the phenomena of the system that has to be modeled. The inclusion of delay in these systems has illustrated more complicated and richer dynamics in terms of stability, bifurcation, periodic solutions, and so on. For examples, see Asea and Zak , Zak , Szydłowski , Szydłowski and Krawiec , Matsumoto and Szidarovszky , Matsumoto et al. , d’Albis et al. , Bambi et al. , Boucekkine et al. , Matsumoto and Szidarovszky , Ballestra et al. , Bianca and Guerrini , Bianca et al. , Guerrini and Sodini [14, 15], and Matsumoto and Szidarovszky . However, in some of these papers the formulas for determining the properties of Hopf bifurcation were not derived.

This paper is concerned with the study of Hopf bifurcation of the model system with a fixed time delay presented in Matsumoto and Szidarovszky , where a continuous-time neoclassical growth model with time delay was developed similarly in spirit and functional form to Day’s  discrete-time model. Specifically, they have proposed the following delay differential equation: (1)k˙=-αk+βkd(1-kd), where  k  is the per capita per labor and  α, β  are positive parameters. In order to simplify the notation, we omit the indication of time dependence for variables and derivatives referred to as time  t. As well, we use  kd  to indicate the state of the variable  k  at time  t-τ, where  τ  represents the time delay inherent in the production process. According to Matsumoto and Szidarovszky , (1) has a unique positive steady state (2)k*=β-αβ, if β>α. In case  β>3α, this equilibrium is locally asymptotically stable for  τ<τ*  and unstable for  τ>τ*, where (3)τ*=cos-1(α/(2α-β))ω*,withω*=(β-α)(β-3α). The change in stability will be accompanied by the birth of a limit cycle in a Hopf bifurcation. This limit cycle will start with zero amplitude and will grow as  τ  is further increased. Using the theory of normal form and center manifold (see ), we extend their analysis, providing formulas for determining the stability of the bifurcating periodic solutions and the direction of the Hopf bifurcation. Finally, even if the literature on economic models with delays is quite huge, we have noticed that the study of the type of Hopf bifurcation is really rare. Therefore, we have deepened this last point by using the perturbation method known as Lindstedt’s expansion (see, e.g., [19, 20]) and furnished a detailed analysis on approximation to the bifurcating periodic solutions.

2. Direction and Stability of Bifurcating Periodic Solutions

In this section, we study the direction, stability, and period of the bifurcating periodic solutions in (1) that are generated at the positive equilibrium when  τ=τ*. We let  iω*  be the corresponding purely imaginary root of the characteristic equation of the linearized equation of (1) at the positive equilibrium. The method we used is based on the normal form theory and the center manifold theorem introduced in Hassard et al. . For notational convenience, let  τ=τ*+μ, where  μ, so that  μ=0  is the Hopf bifurcation value for (1). First we use the transformation  x=k-k*, so that (1) becomes (4)x˙=-αx+(2α-β)xd-βxd2. Let C=C([-τ*,0],) be the Banach space of continuous mappings from  [-τ*,0]  into equipped with supremum norm. Let xt=x(t+θ), for  θ[-τ*,0].  Then, (4) can be written as (5)x˙(t)=Lμ(xt)+(μ,xt), where the linear operator  Lμ  and the function are given by (6)Lμ(φ)=-αφ(0)+(2α-β)φ(-τ),(μ,φ)=-βφ(-τ)2, with  φC. By the Riesz representation theorem, there exists a bounded variation function  η(θ,μ),  θ[-τ*,0], such that (7)Lμφ=-τ*0dη(θ,μ)φ(θ), where (8)η(θ,μ)=-αδ(θ)+(2α-β)δ(θ+τ), with  δ  representing the Dirac delta function. Next, for  φC, define (9)A(μ)(φ)={dφ(θ)dθ,θ[-τ*,0),-τ*0dη(r,μ)φ(r),θ=0,R(μ)(φ)={0,θ[-τ*,0),(μ,φ),θ=0. As a result, (5) can be expressed as (10)x˙t=A(μ)xt+R(μ)xt. For  ψC~=C([0,τ*],), the adjoint operator  A*  of  A  is defined as (11)A*(μ)ψ(r)={-dψ(r)dr,r(0,τ*],-τ*0dη(ζ,μ)ψ(-ζ),r=0. Let q(θ) (resp.,  q*(θ)) denote the eigenvector for  A(0)  (resp., for  A*(0)) corresponding to  τ*; namely, A(0)q(θ)=iω*q(θ)  (resp.,  A*(0)q*(r)=-iω*q*(r)). To construct the coordinates to describe the center manifold near the origin, we define an inner product as follows: (12)ψ,φ=ψ¯(0)φ(0)-θ=-τ*0ξ=0θψ¯(ξ-θ)dη(θ)φ(ξ)dξ, for  φC  and  ψC~, where  dη(θ)=dη(θ,0)  and  ψ¯  represents the complex conjugate operation of ψ. The vectors  q  and  q*  can be normalized by the conditions q*,q=1 and q*,q¯=0. A direct computation shows that (13)q(θ)=eiω*θ,θ[-τ*,0],(14)q*(r)=Beiω*r,r[0,τ*], where (15)B=11+(2α-β)τ*eiω*τ*. Let  z=q*,xt  and (16)W(t,θ)=xt(θ)-2Re{zq(θ)}. On the center manifold  C0,  W(t,θ)=W(z,z¯,θ), with(17)W(z,z¯,θ)=W20(θ)z22+W11(θ)zz¯+W02(θ)z¯22+, where  z  and  z¯  are local coordinates for  C0  in the direction of  q*  and  q¯*, respectively. For any  xtC0  solution of (10), we have (18)z˙=q*,x˙t=q*,A(μ)xt+R(μ)xt=iω*z+q¯*(0)0(z,z¯)=iω*z+g(z,z¯), where  0(z,z¯)=(0,xt)  and  g(z,z¯)=B¯0(z,z¯). Noting from (16) that (19)xt(θ)=W(z,z¯,θ)+zq(θ)+z¯q¯(θ), it follows that (20)g(z,z¯)=-βB¯e-2iω*τ*z2-2βB¯zz¯-βB¯e2iω*τ*z¯2-βB¯{[2W11(-τ*)e-iω*τ*+W20(-τ*)eiω*τ*]z2z¯+[2W11(-τ*)eiω*τ*+W02(-τ*)e-iω*τ*]zz¯2}. Expanding  g(z,z¯)  in powers of z and z¯, that is, (21)g(z,z¯)=g20z22+g11zz¯+g02z¯22+g21z2z¯2+, and comparing the above coefficients with those in (20), we get (22)g20=-2βB¯e-2iω*τ*,g11=-2βB¯,g02=-2βB¯e2iω*τ*,g21=-2B¯β[2W11(-τ*)e-iω*τ*+W20(-τ*)eiω*τ*]. In order to compute g21, we need to know W20(0), W20(-τ*) and W11(0), W11(-τ*) first. From (16), one has (23)W˙=x˙t-z˙q-z¯˙q¯={AW-2Re{B¯0q(θ)},θ[-τ*,0),AW-2Re{B¯0}+0,θ=0=AW+H(z,z¯,θ), where (24)H(z,z¯,θ)=H20(θ)z22+H11(θ)zz¯+H02(θ)z¯22+. Recalling (23), it follows that (25)H(z,z¯,θ)=-2Re{B¯0q(θ)}=-gq(θ)-g¯q¯(θ)  =-(g20z22+g11zz¯+g02z¯22+)q(θ)-(g¯20z¯22+g¯11zz¯+g02z22+)q¯(θ). On the other hand, (26)W˙20(θ)=2iω*W20(θ)-H20(θ),AW11(θ)=-H11(θ). A comparison of the coefficients of (24) and (25) gives (27)H20(θ)=-g20q(θ)-g¯02q¯(θ),H11(θ)=-g11q(θ)-g¯11q¯(θ). Thus, (26) becomes (28)W˙20(θ)=2iω*W20(θ)+g20q(θ)+g¯02q¯(θ), which is solved by (29)W20(θ)=-g20iω*eiω*θ-g¯023iω*e-iω*θ+E1e2iω*θ. Similarly, from (30)W˙11(θ)=g11q(θ)+g¯11q¯(θ), we derive (31)W11(θ)=g11iω*eiω*θ-g¯11iω0e-iω*θ+E2, where  (E1,E2)  is a constant vector. In order to compute  W20 and W11, the constants  E1  and  E2  are needed. From (23), we have (32)H(z,z¯,0)=-2Re{B¯0q(0)}+0. Thus, (33)H20(0)=-g20-g¯20B¯-2βe-2iω*τ*,  H11(0)=-g11-g¯11B¯-2β. On the center manifold, we have W˙=Wzz˙+Wz¯z¯˙. Replacing Wz, Wz¯ and z˙, z¯˙, we obtain a second expression for  W˙. A comparison of the coefficients of this equation with those in (23), for  θ=0, leads us to the following: (34)(A-2iω*)W20(0)=-H20(0),AW11(0)=-H11(0). Since (35)AW20(0)=-αW20(0)+(2α-β)W20(-τ*),  AW11(0)=-αW11(0)+(2α-β)W11(-τ*), from the previous analysis we arrive at (36)-αW20(0)+(2α-β)W20(-τ*)-2iω*W20(0)=g20q(0)+g¯20q¯(0)+2βe-2iω*τ*,-αW11(0)+(2α-β)W11(-τ*)=g11q(0)+g¯11q¯(0)+2β. Hence,  E1  and  E2  can be computed from (29) and (31) as  θ=0, and we obtain (37)E1=F1-α+(2α-β)e-2iω*τ*-2iω*, where (38)F1=(-α-2iω*)(g20iω*+g¯023iω*)+(2α-β)(g20iω*e-iω*τ*+g¯023iω*eiω*τ*)+g20+g¯02+2βe-2iω*τ*,E2=F2-α+(2α-β), where (39)F2=α(g11iω*-g¯11iω*)-(2α-β)(g11iω*eiω*τ*-g¯11iω*e-iω*τ*)+g11+g¯11+2β. Based on the above analysis, all  gij  have been obtained. Consequently, we can compute the following quantities: (40)C1(0)=i2ω*(g11g20-2|g11|2-|g02|23)+g212,μ2=-Re[C1(0)]Re{λ(τ*)},β2=2Re[C1(0)],T2=-Im[C1(0)]+μ2Im[λ(τ*)]ω*, which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value. We will summarize it in the following result.

Theorem 1.

Let C1(0), μ2, β2, and  T2  be defined in (40).

The bifurcating periodic solution is supercritical bifurcating as  Re[C1(0])>0, and it is subcritical bifurcating as Re[C1(0])<0.

The bifurcating periodic solutions are stable if  Re[C1(0])<0 and unstable if Re[C1(0])>0.

As  τ  increases, the period of bifurcating periodic solutions increases if T2>0, while it decreases, if T2<0.

3. Lindstedt’s Method

In the previous section, the direction and stability of the Hopf bifurcation were investigated by using the normal form theory and the center manifold theorem as in Hassard et al. . Specifically, the delay differential equation of our model was converted into an operator equation on a Banach space of infinite dimension and then simplified into a one-dimensional ordinary differential equations on the center manifold. Now we will use a different approach to investigate periodic solutions of (4), namely, of (1), which consists in applying Lindstedt’s perturbation method (see, e.g., [19, 20]).  To this end, we start stretching time with the transformation (41)s=ωt, so that solutions of (4) which are  2π/ω  periodic in  t  become  2π  periodic in  s. This change of variables results in the following form of (4): (42)ωdx(s)ds=a0x(s)+a1x(s-ωτ)+a2x(s-ωτ)2, where the terms  a0, a1, and  a2  are given by (43)a0=-α<0,a1=2α-β<0,a2=-β<0. The idea is now to expand the solution of (42) in a power series in a suitable smallness parameter  ɛ, that is, (44)x(s)=x0(s)ɛ+x1(s)ɛ2+x2(s)ɛ3+, and to solve for the unknown functions  xj(s)  recursively. In this context, the definition of the xj(s)  (j=0,1,2,)  is clear. As already mentioned,  ɛ  represents a small quantity so that we can expand the frequency  ω  and the delay  τ  in powers of  ɛ  according to (45)ω=ω(ɛ)=ω0+ω1ɛ+ω2ɛ2+,τ=τ(ɛ)=τ0+τ1ɛ+τ2ɛ2+, where we have set (46)τ0=τ*,ω0=ω*. In addition, we also have to consider a corresponding expansion of the time delayed term x(s-ωτ), which is achieved by (47)x(s-ωτ)=x0(s-ωτ)ɛ+x1(s-ωτ)ɛ2+x2(s-ωτ)ɛ3+, where  xj(s-ωτ)  stands for (48)xj(s-ωτ)=xj(s-ω0τ0)-xj(s-ω0τ0)×[(ω1τ0+ω0τ1)ɛ+(ω2τ0+ω1τ1+ω0τ2)ɛ2+]+12xj′′(s-ω0τ0)[(ω1τ0+ω0τ1)ɛ+]2-, with primes representing differentiation with respect to  s.  Applying the expansions for x(s) and x(s-ωτ)  to (42) and collecting terms for the distinct orders of ɛ, we get the following three equations: (49)O(ɛ):ω0dx0(s)ds=a0x0(s-ω0τ0)+a1x0(s-ω0τ0),(50)O(ɛ2):ω0dx1(s)ds-a0x1(s)-a1x1(s-ω0τ0)=-ω1dx0(s)ds-a1x0(s-ω0τ0)(ω1τ0+ω0τ1)+x02(s)+a2x02(s-ω0τ0),(51)O(ɛ3):ω0dx2(s)ds-a0x2(s)-a1x2(s-ω0τ0)  =-ω2dx0(s)ds-a1x0(s-ω0τ0)(ω2τ0+ω1τ1+ω0τ2)+2a2x0(s-ω0τ0)x1(s-ω0τ0)-ω2dx0(s)ds-a1x0(s-ω0τ0)(ω2τ0+ω1τ1+ω0τ2)-2a2x0(s-ω0τ0)x0(s-ω0τ0)(ω1τ0+ω0τ1)+12a1x0′′(s-ω0τ0)(ω1τ0+ω0τ1)2. We take the solution of (49) as follows: (52)x0(s)=A0sins+B0coss, where  A0  and  B0  are constants. Next we substitute (52) into (49) and derive that A0 and  B0  are arbitrary. Without loss of generality, we impose the initial conditions  x0(0)=0  and  x0(0)=1 and get from (52) that (53)x0(s)=sins. Next, we look for a solution to (50) as (54)x1(s)=A1sins+B1coss+C1sin(2s)+D1cos(2s)+E1, where the coefficients A1, B1, C1, D1, and  E1  are constants. Substituting (53) and (54) in (50) and equating the coefficients of the resonant terms sins, coss, sin(2s), and  cos(2s), we find that (55)ω1=τ1=0,C1=M1M3+M2M4M12+M22,D1=M2M3-M1M4M12+M22,E1=-1+a22(a0+a1), with  A1  and  B1  being arbitrary and (56)M1=2ω0(a1-a0)a1,M2=(a0+a1)(a1-2a0)a1,M3=a2(a12-2a02)-a122a12,M4=-ω0a0a2a1. For simplicity, we let A1=B1=0. Hence, (54) becomes (57)x1(s)=C1sin(2s)+D1cos(2s)+E1, where C1, D1, and  E1  are given in (55). Finally, let (58)x2(s)=A2sins+B2coss+C2sin(2s)+D2cos(2s)+E2sin(3s)+F2cos(3s)+G2 be the solution of (51), with A2, B2, C2, D2, E2, F2, and G2 being constants. Using (53), (57), and (58) into (51), after trigonometric simplifications have been performed, we obtain (59)(ω0A2+ω2)coss-ω0B2sins+2ω0C2cos(2s)-2ω0D2sin(2s)+3ω0E2cos(3s)-3ω0F2sin(3s)=[ω0(ω2τ0+ω0τ2)-ω0B2+N1]sins+[a0(ω2τ0+ω0τ2)+ω0A2+N2]coss+[a0C2+a1(C2N4+D2N3)-a2(A1N3-B1N4)]sin(2s)+[a0D2+a1(-C2N3+D2N4)-a2(A1N4+B1N3)]cos(2s)+[a0E2+a1(E2N5+F2N6)]sin(3s)+[a0F2+a1(F2N5-E2N6)]cos(3s)+a0G2+a1G2+a2A1, where (60)N1=-2E1a0a2+C1a2ω0-D1a0a2a1,N2=2E1a2ω0-D1a2ω0-C1a0a2a1,N3=2a0ω0a12,N4=2a02-a12a12,N5=-4a03-3a0a12a13,N6=-3a12ω0-4ω03a13. Comparing the coefficients of the terms, sins, coss, sin(2s), cos(2s), sin(3s), and  cos(2s), we get the following expressions: (61)ω2=N2ω0-N1a0ω0,τ2=N1(a0τ0-1)-N2ω0τ0ω02. Summing up all the above analysis, the bifurcated periodic solution of (4) has an approximation of the form (62)x(s)=τ-τ0τ2  x0(s)+τ-τ0τ2  x1(s)+, where  ττ0+τ2ɛ2,  ωω0+ω2ɛ2, with  x0(s)  and x1(s) given in (53) and (57), respectively. Here, the parameters τ2 and ω2 determine the direction of the Hopf bifurcation and the period of the bifurcating periodic solution, respectively. We have the following result.

Theorem 2.

The Hopf bifurcation of (1) at the equilibrium point  k*  when  τ=τ*  is supercritical (resp., subcritical), if  τ2>0  (resp.,  τ2<0)  and the bifurcating periodic solutions exist for  τ>τ*  (resp.,  τ<τ*). In addition, its period decrease (resp., increases) as τ increases, if  ω2>0 (resp., ω2<0).

Remark 3.

Let  β=4α  and  α=1.  Then (63)M2=0,C1=M3=-116,  D1=-M4=-23,E1=-12,τ0=2π33. As direct calculation shows that (61) yields  ω2>0  and  τ2<0.  In this case, the Hopf bifurcation is subcritical and the bifurcating periodic solutions exist for  τ<τ*. Moreover, its period decreases as  τ  increases.

4. Conclusions

In this paper, we consider the special neoclassical growth model with fixed time delay introduced and examined by Matsumoto and Szidarovszky’s , where a mound-shaped production function for capital growth was assumed in the dynamic equation. In their model, the stability can be lost at a certain value of the delay and the equilibrium remains unstable afterwards. At this critical value, Hopf bifurcation occurs. By applying the normal form theory and the center manifold theorem, we derive explicit formulae which determine the stability and direction of the bifurcating periodic solutions. Moreover, we employ Lindstedt’s perturbation theory to approximate the bifurcated periodic solution and provide approximate expressions for the amplitude and frequency of the resulting limit cycle as a function of the model parameters.

Conflict of Interests

The authors declare that there is no conflict of interests.

Acknowledgment

The authors would like to thank the referees for their valuable comments.

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