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 [1], Zak [2], Szydłowski [3], Szydłowski and Krawiec [4], Matsumoto and Szidarovszky [5], Matsumoto et al. [6], d’Albis et al. [7], Bambi et al. [8], Boucekkine et al. [9], Matsumoto and Szidarovszky [10], Ballestra et al. [11], Bianca and Guerrini [12], Bianca et al. [13], Guerrini and Sodini [14, 15], and Matsumoto and Szidarovszky [16]. 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 [5], where a continuous-time neoclassical growth model with time delay was developed similarly in spirit and functional form to Day’s [17] discrete-time model. Specifically, they have proposed the following delay differential equation:
(1)k˙=-αk+βkd(1-kd),
wherekis the per capita per labor andα, βare positive parameters. In order to simplify the notation, we omit the indication of time dependence for variables and derivatives referred to as timet. As well, we usekdto indicate the state of the variablekat timet-τ, whereτrepresents the time delay inherent in the production process. According to Matsumoto and Szidarovszky [5], (1) has a unique positive steady state
(2)k*=β-αβ,
if β>α. In caseβ>3α, this equilibrium is locally asymptotically stable forτ<τ*and unstable forτ>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 asτis further increased. Using the theory of normal form and center manifold (see [18]), 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 whenτ=τ*. We letiω*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. [18]. For notational convenience, letτ=τ*+μ, whereμ∈ℝ, so thatμ=0is the Hopf bifurcation value for (1). First we use the transformationx=k-k*, so that (1) becomes
(4)x˙=-αx+(2α-β)xd-βxd2.
Let C=C([-τ*,0],ℝ) be the Banach space of continuous mappings from[-τ*,0]into ℝ equipped with supremum norm. Let xt=x(t+θ), forθ∈[-τ*,0].Then, (4) can be written as
(5)x˙(t)=Lμ(xt)+ℱ(μ,xt),
where the linear operatorLμand the function ℱ are given by
(6)Lμ(φ)=-αφ(0)+(2α-β)φ(-τ),ℱ(μ,φ)=-βφ(-τ)2,
withφ∈C. By the Riesz representation theorem, there exists a bounded variation functionη(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),
withδrepresenting the Dirac delta function. Next, forφ∈C, 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),ℱ(μ,φ),θ=0.
As a result, (5) can be expressed as
(10)x˙t=A(μ)xt+R(μ)xt.
Forψ∈C~=C([0,τ*],ℝ), the adjoint operatorA*ofAis 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 forA(0)(resp., forA*(0)) corresponding toτ*; 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*0∫ξ=0θψ¯(ξ-θ)dη(θ)φ(ξ)dξ,
forφ∈Candψ∈C~, wheredη(θ)=dη(θ,0)andψ¯represents the complex conjugate operation of ψ. The vectorsqandq*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*.
Letz=〈q*,xt〉and
(16)W(t,θ)=xt(θ)-2Re{zq(θ)}.
On the center manifoldC0,W(t,θ)=W(z,z¯,θ), with(17)W(z,z¯,θ)=W20(θ)z22+W11(θ)zz¯+W02(θ)z¯22+⋯,
wherezandz¯are local coordinates forC0in the direction ofq*andq¯*, respectively. For anyxt∈C0solution of (10), we have
(18)z˙=〈q*,x˙t〉=〈q*,A(μ)xt+R(μ)xt〉=iω*z+q¯*(0)ℱ0(z,z¯)=iω*z+g(z,z¯),
whereℱ0(z,z¯)=ℱ(0,xt)andg(z,z¯)=B¯ℱ0(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}.
Expandingg(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¯ℱ0q(θ)},θ∈[-τ*,0),AW-2Re{B¯ℱ0}+ℱ0,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¯ℱ0q(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 computeW20 and W11, the constantsE1andE2are needed. From (23), we have
(32)H(z,z¯,0)=-2Re{B¯ℱ0q(0)}+ℱ0.
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 forW˙. A comparison of the coefficients of this equation with those in (23), forθ=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,E1andE2can be computed from (29) and (31) asθ=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, allgijhave 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, andT2be defined in (40).

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

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

Asτincreases, 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. [18]. 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 are2π/ωperiodic intbecome2πperiodic ins. 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 termsa0, a1, anda2are 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 parameterɛ, that is,
(44)x(s)=x0(s)ɛ+x1(s)ɛ2+x2(s)ɛ3+⋯,
and to solve for the unknown functionsxj(s)recursively. In this context, the definition of the xj(s)(j=0,1,2,…)is clear. As already mentioned,ɛrepresents a small quantity so that we can expand the frequencyωand the delayτin powers ofɛaccording 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+⋯,
wherexj(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 tos.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,
whereA0andB0are constants. Next we substitute (52) into (49) and derive that A0 andB0are arbitrary. Without loss of generality, we impose the initial conditionsx0(0)=0andx0′(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, andE1are constants. Substituting (53) and (54) in (50) and equating the coefficients of the resonant terms sins, coss, sin(2s), andcos(2s), we find that
(55)ω1=τ1=0,C1=M1M3+M2M4M12+M22,D1=M2M3-M1M4M12+M22,E1=-1+a22(a0+a1),
withA1andB1being 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, andE1are 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), andcos(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-τ0τ2x0(s)+τ-τ0τ2x1(s)+⋯,
whereτ≈τ0+τ2ɛ2,ω≈ω0+ω2ɛ2, withx0(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 pointk*whenτ=τ*is supercritical (resp., subcritical), ifτ2>0(resp.,τ2<0)and the bifurcating periodic solutions exist forτ>τ*(resp.,τ<τ*). In addition, its period decrease (resp., increases) as τ increases, ifω2>0 (resp., ω2<0).

Remark 3.

Letβ=4αandα=1.Then
(63)M2=0,C1=M3=-116,D1=-M4=-23,E1=-12,τ0=2π33.
As direct calculation shows that (61) yieldsω2>0andτ2<0.In this case, the Hopf bifurcation is subcritical and the bifurcating periodic solutions exist forτ<τ*. Moreover, its period decreases asτincreases.

4. Conclusions

In this paper, we consider the special neoclassical growth model with fixed time delay introduced and examined by Matsumoto and Szidarovszky’s [5], 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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