Research Article Center Manifold Reduction and Perturbation Method in a Delayed Model with a Mound-Shaped Cobb-Douglas Production Function

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1 Abstract and Applied Analysis Volume 2013 Article ID pages Research Article Center Manifold Reduction and Perturbation Method in a Delayed Model with a Mound-Shaped Cobb-Douglas Production Function Massimiliano Ferrar Luca Guerrini 2 and Giovanni Molica Bisci 3 1 Department of Law and Economics University Mediterranea of Reggio Calabria Via dei Bianchi 2 (Palazzo Zani) 89127ReggioCalabriaItaly 2 Department of Management Polytechnic University of Marche Piazza Martelli Ancona Italy 3 Department of PAU University Mediterranea of Reggio Calabria Via Melissari Reggio Calabria Italy Correspondence should be addressed to Luca Guerrini; luca.guerrini@univpm.it Received 29 October 2013; Accepted 20 November 2013 Academic Editor: Carlo Bianca 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 closertotherealworldthereisalwaysadelaybetweenthe 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] a continuous-time neoclassical growth model with time delay was developed similarly in spirit and functional form to Day s [17] discretetime model. Specifically they have proposed the following delay differential equation: k = αk+βk d (1 k d ) (1) 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.aswellweuse k d to indicate the state of the variable k at time t τ τ represents the time delay inherent in the production process. According to Matsumoto and Szidarovszky [5] (1) has a unique positive steady state k = β α β (2)

2 2 Abstract and Applied Analysis if β>α. In case β>3α this equilibrium is locally asymptotically stable for τ<τ and unstable for τ>τ τ = cos 1 (α/ (2α β)) with ω ω = (β α) (β 3α). (3) 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 [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 noticedthatthestudyofthetypeofhopfbifurcationisreally 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)thataregeneratedat the positive equilibrium when τ=τ.weletiω be the corresponding purely imaginary root of the characteristic equationofthelinearizedequationof(1) atthepositiveequilibrium. 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 let τ=τ +μ μ Rsothat μ=0isthehopfbifurcationvaluefor(1). First we use the transformation x=k k sothat(1)becomes x= αx+(2α β)x d βx 2 d. (4) Let C = C([ τ 0]R) be the Banach space of continuous mappings from [ τ 0] into R equipped with supremum norm. Let x t = x (t + θ) forθ [ τ 0]. Then (4) canbe written as x (t) =L μ (x t )+F(μ x t ) (5) the linear operator L μ and the function F are given by L μ (φ) = αφ (0) +(2α β)φ( τ) F (μ φ) = βφ( τ) 2 with φ C. By the Riesz representation theorem there exists a bounded variation function η(θ μ) θ [ τ 0]suchthat (6) 0 L μ φ= dη (θ μ) φ (θ) (7) τ η (θ μ) = αδ (θ) + (2α β) δ (θ+τ) (8) with δ representing the Dirac delta function. Next for φ C define dφ (θ) { dθ θ [ τ 0) A (μ) (φ) = 0 { dη (r μ) φ (r) θ = 0 { τ R (μ) (φ) = { 0 θ [ τ 0) F (μ φ) θ = 0. As a result (5) can be expressed as (9) x t =A(μ)x t +R(μ)x t. (10) For ψ C =C([0τ ] R) the adjoint operator A of A is defined as dψ (r) r (0τ A { dr ] (μ) ψ (r) = 0 { dη (ζ μ) ψ ( ζ) r = 0. { τ (11) 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: 0 θ ψ φ = ψ (0) φ (0) ψ (ξ θ) dη (θ) φ (ξ) dξ θ= τ ξ=0 (12) for φ C and ψ C dη(θ) = dη(θ 0) and ψ represents the complex conjugate operation of ψ.thevectors q and q can be normalized by the conditions q q =1and q q = 0. A direct computation shows that q (θ) =e iω θ θ [ τ 0] (13) q (r) =Be iω r r [0τ ] (14) B= Let z= q x t and 1 1+(2α β)τ e iω τ. (15) W (t θ) =x t (θ) 2Re {zq (θ)}. (16) On the center manifold C 0 W(t θ) = W(z z θ)with W (z zθ) =W 20 (θ) z2 2 +W 11 (θ) zz +W 02 (θ) z2 2 + (17) z and z are local coordinates for C 0 in the direction of q and q respectively.foranyx t C 0 solution of (10) we have z= q x t = q A(μ)x t +R(μ)x t (18) =iω z+q (0) F 0 (z z) =iω z+g(z z)

3 Abstract and Applied Analysis 3 F 0 (z z) = F(0 x t ) and g(z z) = BF 0 (z z).noting from (16)that it follows that g (z z) x t (θ) =W(z zθ) +zq(θ) + z q (θ) (19) = βbe 2iω τ 2βBzz βbe 2iω τ βb {[2W 11 ( τ )e iω τ +W 20 ( τ )e iω τ ] z +[2W 11 ( τ )e iω τ +W 02 ( τ )e iω τ ]z }. (20) Expanding g(z z) in powers of z and zthatis g (z z) =g g 11zz+g g z (21) and comparing the above coefficients with those in (20) we get g 20 = 2βBe 2iω τ g 02 = 2βBe 2iω τ g 11 = 2βB g 21 = 2Bβ [2W 11 ( τ )e iω τ +W 20 ( τ )e iω τ ]. (22) In order to compute g 21 we need to know W 20 (0) W 20 ( τ ) and W 11 (0) W 11 ( τ ) first. From (16) one has W= x t zq z q = { AW 2 Re {BF 0 q (θ)} θ [ τ 0) { AW 2 Re {BF { 0 }+F 0 θ = 0 =AW+H(z z θ) (23) A comparison of the coefficients of (24)and(25)gives H 20 (θ) = g 20 q (θ) g 02 q (θ) (27) H 11 (θ) = g 11 q (θ) g 11 q (θ). Thus (26)becomes W 20 (θ) =2iω W 20 (θ) +g 20 q (θ) + g 02 q (θ) (28) which is solved by W 20 (θ) = g 20 e iω θ g 02 e iω θ +E iω 3iω 1 e 2iω θ. (29) Similarly from W 11 (θ) =g 11 q (θ) + g 11 q (θ) (30) we derive W 11 (θ) = g 11 iω e iω θ g 11 iω 0 e iω θ +E 2 (31) (E 1 E 2 ) is a constant vector. In order to compute W 20 and W 11 theconstantse 1 and E 2 are needed. From (23) we have Thus H (z z 0) = 2Re {BF 0 q (0)}+F 0. (32) H 20 (0) = g 20 g 20 B 2βe 2iω τ H 11 (0) = g 11 g 11 B 2β. (33) On the center manifold we have W=W z z+w z z.replacing W z W z and ż z we obtain a second expression for W.Acomparison of the coefficients of this equation with those in (23) for θ=0 leads us to the following: (A 2iω )W 20 (0) = H 20 (0) AW 11 (0) = H 11 (0). (34) H (z z θ) =H 20 (θ) z2 2 +H 11 (θ) zz+h 02 (θ) z (24) Recalling (23) it follows that H (z z θ) = 2Re {BF 0 q (θ)} = gq(θ) g q (θ) = (g g 11zz+g )q(θ) (25) Since AW 20 (0) = αw 20 (0) +(2α β)w 20 ( τ ) AW 11 (0) = αw 11 (0) +(2α β)w 11 ( τ ) from the previous analysis we arrive at αw 20 (0) +(2α β)w 20 ( τ ) 2iω W 20 (0) =g 20 q (0) + g 20 q (0) +2βe 2iω τ (35) (g g 11 zz+g 02 On the other hand W 20 (θ) =2iω W 20 (θ) H 20 (θ) 2 AW 11 (θ) = H 11 (θ). + )q (θ). (26) αw 11 (0) +(2α β)w 11 ( τ )=g 11 q (0) + g 11 q (0) +2β. (36) Hence E 1 and E 2 can be computed from (29)and(31)asθ= 0andweobtain E 1 = F 1 α+(2α β)e 2iω τ 2iω (37)

4 4 Abstract and Applied Analysis F 1 = ( α 2iω )( g 20 iω + g 02 3iω ) +(2α β)( g 20 e iω τ + g 02 e iω τ ) iω 3iω +g 20 + g 02 +2βe 2iω τ E 2 = F 2 α+(2α β) (38) F 2 =α( g 11 g 11 ) (2α β)( g 11 e iω τ g 11 e iω τ ) iω iω iω iω +g 11 + g 11 +2β. (39) Basedontheaboveanalysisall g ij have been obtained. Consequently we can compute the following quantities: C 1 (0) = i 2ω (g 11 g 20 2 g 11 2 g 02 2 )+ g μ 2 = Re [C 1 (0)] Re {λ (τ )} β 2 =2Re [C 1 (0)] T 2 = Im [C 1 (0)]+μ 2 Im [λ (τ )] ω (40) 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 C 1 (0) μ 2 β 2 andt 2 be defined in (40). (i) Thebifurcatingperiodicsolutionissupercriticalbifurcating as Re[C 1 (0]) > 0 and it is subcritical bifurcating as Re[C 1 (0]) < 0. (ii) The bifurcating periodic solutions are stable if Re[C 1 (0]) < 0 and unstable if Re[C 1 (0]) > 0. (iii) As τ increases the period of bifurcating periodic solutions increases if T 2 >0whileitdecreasesifT 2 <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 onedimensional 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 s=ωt (41) so that solutions of (4)whichare2π/ω periodic in t become 2π periodic in s. This change of variables results in the following form of (4): dx (s) ω =a ds 0 x (s) + x (s ωτ) +a 2 x(s ωτ) 2 (42) the terms a 0 anda 2 are given by a 0 = α<0 =2α β<0 a 2 = β<0. (43) The idea is now to expand the solution of (42) inapower series in a suitable smallness parameter εthatis x (s) =x 0 (s) ε+x 1 (s) ε 2 +x 2 (s) ε 3 + (44) and to solve for the unknown functions x j (s) recursively. In this context the definition of the x j (s) (j = ) is clear. As already mentioned ε represents a small quantity so that we can expand the frequency ω and the delay τ in powers of ε according to ω=ω(ε) =ω 0 +ω 1 ε+ω 2 ε 2 + (45) τ=τ(ε) =τ 0 +τ 1 ε+τ 2 ε 2 + wehaveset τ 0 =τ ω 0 =ω. (46) In addition we also have to consider a corresponding expansion of the time delayed term x(s ωτ) which is achieved by x (s ωτ) =x 0 (s ωτ) ε+x 1 (s ωτ) ε 2 +x 2 (s ωτ) ε 3 + x j (s ωτ) stands for x j (s ωτ) =x j (s ω 0 τ 0 ) x j (s ω 0τ 0 ) [(ω 1 τ 0 +ω 0 τ 1 )ε+(ω 2 τ 0 +ω 1 τ 1 +ω 0 τ 2 )ε 2 + ] (47) x j (s ω 0τ 0 )[(ω 1 τ 0 +ω 0 τ 1 )ε+ ] 2 (48) 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 ε wegetthe following three equations: dx O (ε) :ω 0 (s) 0 =a ds 0 x 0 (s ω 0 τ 0 )+ x 0 (s ω 0 τ 0 ) (49) O(ε 2 dx ):ω 1 (s) 0 a ds 0 x 1 (s) x 1 (s ω 0 τ 0 ) dx = ω 0 (s) 1 a ds 1 x 0 (s ω 0τ 0 )(ω 1 τ 0 +ω 0 τ 1 ) +x 2 0 (s) +a 2x 2 0 (s ω 0τ 0 ) (50)

5 Abstract and Applied Analysis 5 O(ε 3 ):ω 0 dx 2 (s) ds = ω 2 dx 0 (s) ds a 0 x 2 (s) x 2 (s ω 0 τ 0 ) x 0 (s ω 0τ 0 )(ω 2 τ 0 +ω 1 τ 1 +ω 0 τ 2 ) +2a 2 x 0 (s ω 0 τ 0 )x 1 (s ω 0 τ 0 ) ω 2 dx 0 (s) ds x 0 (s ω 0τ 0 )(ω 2 τ 0 +ω 1 τ 1 +ω 0 τ 2 ) 2a 2 x 0 (s ω 0 τ 0 )x 0 (s ω 0τ 0 )(ω 1 τ 0 +ω 0 τ 1 ) x 0 (s ω 0τ 0 )(ω 1 τ 0 +ω 0 τ 1 ) 2. Wetakethesolutionof (49) as follows: (51) x 0 (s) =A 0 sin s+b 0 cos s (52) A 0 and B 0 are constants. Next we substitute (52)into (49) and derive that A 0 and B 0 are arbitrary. Without loss of generalityweimposetheinitialconditionsx 0 (0) = 0 and x 0 (0) = 1 and get from (52)that Nextwelookforasolutionto(50)as x 0 (s) = sin s. (53) x 1 (s) =A 1 sin s+b 1 cos s+c 1 sin (2s) +D 1 cos (2s) +E 1 (54) the coefficients A 1 B 1 C 1 D 1 ande 1 are constants. Substituting (53) and(54) in(50) andequatingthecoefficients of the resonant terms sin s coss sin(2s) andcos(2s) we find that ω 1 =τ 1 =0 C 1 = M 1M 3 +M 2 M 4 M1 2 +M2 2 D 1 = M 2M 3 M 1 M 4 M1 2 E 1 = 1+a 2 +M2 2 2(a 0 + ) (55) be the solution of (51) with A 2 B 2 C 2 D 2 E 2 F 2 andg 2 being constants. Using (53) (57) and (58)into(51) after trigonometric simplifications have been performed we obtain (ω 0 A 2 +ω 2 ) cos s ω 0 B 2 sin s+2ω 0 C 2 cos (2s) 2ω 0 D 2 sin (2s) +3ω 0 E 2 cos (3s) 3ω 0 F 2 sin (3s) =[ω 0 (ω 2 τ 0 +ω 0 τ 2 ) ω 0 B 2 +N 1 ] sin s +[a 0 (ω 2 τ 0 +ω 0 τ 2 )+ω 0 A 2 +N 2 ] cos s +[a 0 C 2 + (C 2 N 4 +D 2 N 3 ) a 2 (A 1 N 3 B 1 N 4 )] sin (2s) +[a 0 D 2 + ( C 2 N 3 +D 2 N 4 ) a 2 (A 1 N 4 +B 1 N 3 )] cos (2s) +[a 0 E 2 + (E 2 N 5 +F 2 N 6 )] sin (3s) +[a 0 F 2 + (F 2 N 5 E 2 N 6 )] cos (3s) +a 0 G 2 + G 2 +a 2 A 1 N 1 = 2E 1a 0 a 2 +C 1 a 2 ω 0 D 1 a 0 a 2 N 2 = 2E 1a 2 ω 0 D 1 a 2 ω 0 C 1 a 0 a 2 N a 3 = 2a 0ω 0 1 N 4 = 2a2 0 a2 1 a 2 1 N 5 = 4a3 0 3a 0a 2 1 a1 3 N 6 = 3a2 1 ω 0 4ω 3 0 a1 3. a 2 1 (59) (60) Comparing the coefficients of the terms sin s coss sin(2s) cos(2s) sin(3s) and cos(2s) we get the following expressions: with A 1 and B 1 being arbitrary and M 1 = 2ω 0 ( a 0 ) M 2 = (a 0 + )( 2a 0 ) M 3 = a 2 (a 2 1 2a2 0 ) a2 1 2a1 2 M 4 = ω 0a 0 a 2. (56) ω 2 = N 2ω 0 N 1 a 0 ω 0 τ 2 = N 1 (a 0 τ 0 1) N 2 ω 0 τ 0 ω0 2. (61) Summing up all the above analysis the bifurcated periodic solution of (4) has an approximation of the form For simplicity we let A 1 =B 1 =0.Hence(54)becomes x 1 (s) =C 1 sin (2s) +D 1 cos (2s) +E 1 (57) C 1 D 1 ande 1 are given in (55). Finally let x 2 (s) =A 2 sin s+b 2 cos s+c 2 sin (2s) +D 2 cos (2s) +E 2 sin (3s) +F 2 cos (3s) +G 2 (58) x (s) = τ τ 0 τ 2 x 0 (s) + τ τ 0 τ 2 x 1 (s) + (62) τ τ 0 +τ 2 ε 2 ω ω 0 +ω 2 ε 2 with x 0 (s) and x 1 (s) given in (53) and(57) respectively. Here the parameters τ 2 and ω 2 determine the direction of the Hopf bifurcation and the period of the bifurcating periodic solution respectively. We have the following result.

6 6 Abstract and Applied Analysis Theorem 2. The Hopf bifurcation of (1) at the equilibrium point k when τ=τ is supercritical (resp. subcritical) if τ 2 > 0 (resp. τ 2 <0) and the bifurcating periodic solutions exist for τ>τ (resp. τ<τ ). In addition its period decrease (resp. increases) as τ increases if ω 2 >0(resp. ω 2 <0). Remark 3. Let β=4αand α=1.then M 2 =0 C 1 =M 3 = 1 16 D 1 = M 4 = 2 3 E 1 = 1 2 τ 0 = 2π (63) 3 3. As direct calculation shows that (61) yieldsω 2 >0and τ 2 < 0. In this case the Hopf bifurcation is subcritical and the bifurcating periodic solutions exist for τ<τ.moreoverits 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 [5] 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 unstableafterwards.atthiscriticalvaluehopfbifurcationoccurs. 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. References [6] A. Matsumoto F. Szidarovszky and H. Yoshida Dynamics in linear cournot duopolies with two time delays Computational Economicsvol.38no.3pp [7] H. d Albis E. Augeraud-Veron and A. Venditti Business cycle fluctuations and learning-by-doing externalities in a one-sector model JournalofMathematicalEconomics vol. 48 no. 5 pp [8] M.BambiG.FabbriandF.Gozzi Optimalpolicyandconsumption smoothing effects in the time-to-build AK model Economic Theoryvol.50no.3pp [9] R.BoucekkineG.FabbriandP.Pintus Ontheoptimalcontrol of a linear neutral differential equation arising in economics Optimal Control Applications & Methodsvol.33no.5pp [10] A. Matsumoto and F. Szidarovszky Nonlinear delay monopoly with bounded rationality Chaos Solitons and Fractalsvol.45 no. 4 pp [11] L. V. Ballestra L. Guerrini and G. Pacelli Stability switches and Hopf bifurcation in a Kaleckian model of business cycle Abstract and Applied Analysis vol Article ID pages [12] C. Bianca and L. Guerrini On the Dalgaard-Strulik model with logistic population growth rate and delayed-carrying capacity Acta Applicandae Mathematicaevol.128pp [13] C. Bianca M. Ferrara and L. Guerrini Stability and bifurcation analysis in a Solow growth model with two time delays Abstract and Applied Analysis.Inpress. [14] L. Guerrini and M. Sodini Nonlinear dynamics in the Solow modelwithboundedpopulationgrowthandtime-to-build technology Abstract and Applied Analysisvol.2013ArticleID pages [15] L. Guerrini and M. Sodini Dynamic properties of the Solow model with increasing or decreasing population and time-tobuild technology Abstract and Applied Analysis.Inpress. [16] A. Matsumoto and F. Szidarovszky Asymptotic behavior of a delay differential neoclassical growth model Sustainabilityvol. 5 no. 2 pp [17] R. Day Irregular growth cycles American Economic Review vol. 72 pp [18] B. D. Hassard N. D. Kazarinoff and Y. H. Wan Theory and Applications of Hopf Bifurcation vol. 41 Cambridge University Press Cambridge Mass USA [19] A. H. Nayfeh Introduction to Perturbation Techniques John Wiley & Sons New York NY USA [20] N. MacDonald Time Lags in Biological Modelsvol.27ofLecture Notes in Biomathematics Springer Berlin Germany [1] P. K. Asea and P. J. Zak Time-to-build and cycles Economic Dynamics & Controlvol.23no.8pp [2] P. J. Zak Kaleckian lags in general equilibrium Review of Political Economyvol.11pp [3] M. Szydłowski Time to build in dynamics of economic models. II. Models of economic growth Chaos Solitons and Fractalsvol.18no.2pp [4] M. Szydłowski and A. Krawiec A note on Kaleckian lags in the Solow model Review of Political Economy vol.16no.4pp [5] A. Matsumoto and F. Szidarovszky Delay differential neoclassical growth model Economic Behavior and Organizationvol.78no.3pp

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