The Cai Model with Time Delay: Existence of Periodic Solutions and Asymptotic Analysis

Transcription

1 Appl Math Inf Sci 7, No 1, (2013) 21 Applied Mathematics & Information Sciences An International Journal Natural Sciences Publishing Cor The Cai Model with Time Delay: Existence of Periodic Solutions Asymptotic Analysis Carlo Bianca 1, Massimiliano Ferrara 2 Luca Guerrini 3 1 Dipartimento di Scienze Matematiche, Politecnico, Corso Duca Degli Abruzzi 24, Torino, Italy 2 Dipartimento di Giurisprudenza ed Economia, Università Mediterranea di Reggio Calabria, Reggio Calabria, Italy 3 Dipartimento di Matematica per le Scienze Economiche e Sociali, Università di Bologna, Viale Filopanti 5, Bologna, Italy Received: 7 Jun 2012; Revised 21 Sep 2012; Accepted 23 Sep 2012 Published online: 1 January 2013 Abstract: The economic growth model with endogenous labor shift under a dual economy proposed by Cai Applied Mathematics Letters 21, (2008) is generalized in this paper by introducing a time delay in the physical capital By choosing the delay as a bifurcation parameter, it is proved that the delayed model has unique nonzero equilibrium a Hopf bifurcation is proven to exist as the delay crosses a critical value Moreover the direction of the Hopf bifurcation the stability of the bifurcating periodic solutions are investigated in this paper by applying the center manifold theorem the normal form theory Keywords: Delay, Hopf bifurcation, dual economy, oscillations, center manifold theorem 1 Introduction Recently an economic growth model with endogenous labor shift under a dual economy has been proposed by Cai 11 Specifically the role of the industry (sector I) of the agriculture (sector II) are taken into account the economy is based on the assumption that the labor shift from the agricultural sector to the industrial sector is a necessary process for developing countries to realize industrialization Accordingly, the mathematical model of Cai 11 reads: K = sf (K, L 0 + M) δk, (1) M = G (w (K, L 0 + M) w 0 ) H(M), : K denotes the physical capital, F is a neoclassical production function, δ is the depreciation rate of capital stock, s is the saving rate, M represents the total labor shifted from the sector I to the sector II, L 0 is the initial labor in the agricultural sector, w (K, L 0 + M) is the wage rate of sector I, w 0 is the survival wage rate of sector II From the mathematical viewpoint G H are C 1 functions of their arguments Moreover G(0) = H(0) = 0, G ( ) > 0, H ( ) > 0, lim H(M) = M The main result of paper 11 is the proof of the existence uniqueness of an asymptotically stable equilibrium point of the mathematical model (1) Moreover Cai shows that the initial capital labor in the industrial sector have a substantial effect on the capital growth of the industrial sector the labor shift This paper is concerned with a generalization of the mathematical model (1) Specifically the production occurs with a delay while new capital is installed, namely at time t the productive capital stock happens to be K(t τ) = K d It is worth stressing that the idea of production taking time was firstly studied by Hayek 16, with an analysis based on Böhm-Bawerk s work 10, later rigorously formulated by Kalecki 18 Other works-related were done by Frisch Holme 13, Kydl Prescott 20, Rustichini 23 Corresponding author: carlobianca@politoit Natural Sciences Publishing Cor

2 22 C Bianca, M Ferrara, L Guerrini: Bifurcations stability analysis of the delayed Cai model Nowadays, the introduction of a lag has become a common approach in biology economics, see, among others, the review papers 1,2, papers 3,9,12,14,15,19,21, 24,25 the references therein Assuming, for simplicity, the following Cobb-Douglas production function: F (K d, L 0 + M) = K α d (L 0 + M) 1 α, α (0, 1), we have that the economy is now described by the following system of two non-linear delayed differential equations: K = skd α(l 0 + M) 1 α δk, M = G ((1 α)kd α(l 0 + M) α w 0 ) H(M) (2) It is known that the critical points of the system (2) correspond to those with vanishing delay In particular it is easy to show (see Cai 11) that there exists a unique non-trivial equilibrium (K, M ) such that: sk α (L 0 + M ) 1 α = δ, G ( (1 α)k α (L 0 + M ) α w 0 ) = H(M ) In this paper the asymptotic dynamics of the system (2) is studied in terms of local stability Taking the delay as a bifurcation parameter, a Hopf bifurcation is proven to exist as the delay crosses a critical value Additionally an explicit algorithm is established for determining the direction of Hopf bifurcation the stability or instability of the bifurcating branch of periodic solutions The present paper is organized as follows Section 2 deals with the local stability result of the positive equilibrium the occurrence of the Hopf bifurcation is also investigated Section 3 is devoted to investigations on the direction of Hopf bifurcation the stability of bifurcating periodic solutions by using normal form center manifold method introduced by Hassard et al 17 Finally, conclusions are drawn in Section 4 2 Asymptotic stability existence of Hopf bifurcation The first step for studying the local stability of the equilibrium (K, M ) is to linearize the system (2) near (K, M ) Therefore the linear system reads: K = a(k K ) + b(m M ) + c(k d K ), (3) M = d(m M ) + n(k d K ), a, b, c, d, n have the following expressions: a = δ < 0, b = (1 α)δk (L 0 + M ) > 0, d = (1 α)αδk (L 0 + M ) 2 G M s H M < 0, n = (1 α)αδ(l 0 + M ) G K d s > 0, c = αδ > 0, G v = G v ( (1 α)k α (L 0 + M ) α w 0 ), v Kd, M}, H M = H M (M ) The characteristic equation of the system (3) is λ 2 (a + d)λ + ad + c(d λ) bn e λτ = 0 (4) It is well known that the steady state of the system (3) is asymptotically stable if all roots of Eq (4) have negative real parts, unstable if Eq (4) has a root with positive real part We shall now investigate the distribution of the roots of Eq (4) First, we consider the case τ = 0 In this case, the characteristic equation (4) reduces to λ 2 (a + c + d)λ + (a + c)d bn = 0 (5) By applying the Routh-Hurwitz criterion, we obtain the following lemma Lemma 21 All roots of Eq (5) have negative real parts if only if a + c + d < 0 (a + c)d bn > 0 Hereafter we assume that the conditions stated in the above Lemma hold By Corollary 24 in Ruan Wei paper 22, it follows that if instability occurs for a particular value of the delay τ > 0, a characteristic root of (4) must intersect the imaginary axis Suppose that (4) has a purely imaginary root iω, with ω > 0 Then, by separating real imaginary parts in (5), we have ω 2 ad = (cd bn) cos ωτ cω sin ωτ, (6) (a + d)ω = (cd bn) sin ωτ + cω cos ωτ Therefore ω 4 + (a 2 c 2 + d 2 )ω 2 + (ad) 2 (cd bn) 2 = 0 (7) Setting p = a 2 c 2 + d 2, q = (ad) 2 (cd bn) 2 r = ω 2, Eq (7) rewrites as Lemma 22 r 2 + pr + q = 0 (8) i) If q < 0, or q = 0 p < 0, Eq (8) has exactly one positive root ii) If q > 0, p < 0 p 2 > 4q, Eq (8) has two positive roots iii) If p < 0 p 2 = 4q, Eq (8) has only one positive root iv) If q 0 p 0 or p < 0 p 2 < 4q, Eq (8) has no real roots or non-positive roots Without loss of generality, we can assume that Eq (8) has two positive roots r 1 r 2, with the possibility of Natural Sciences Publishing Cor

3 Appl Math Inf Sci 7, No 1, (2013) / wwwnaturalspublishingcom/journalsasp 23 r 1 = r 2 Then Eq (7) has only two positive roots ω 1 = r1 ω 2 = r 2 From (6), we can define τ l = 1 } cω 3 arcsin l + acd + (a + d)(n c)b ω l ω l c 2 ωl 2, + (cd bn)2 l 1, 2} Setting τ 0 = τ l0 min l 1,2} τ l} ω 0 = ω l0, then (τ 0, ω 0 ) solves Eqs (6) Lemma 23 Let τ = τ 0 Then Eq (5) has a unique pair of simple purely imaginary roots ±iω 0 Furthermore, d (Reλ) > 0 if g(ω0) 2 > 0, τ=τ0 g(ω 2 0) = (a + d) 2 (cd bn) 2 + 2(ω 2 0 ad) +c 2 ω 4 0 (ad) 2 Proof Let λ(τ) = µ(τ) + iω(τ) denote the root of (5) such that µ(τ 0 ) = 0 ω(τ 0 ) = ω 0 Substituting λ(τ) into Eq (5) differentiating both sides of it with respect to τ yields: 2λ c(d λ) bn τe λτ (a + d) ce λτ } dλ = c(d λ) bn λe λτ (9) From Eq (9) using Eq (5), we have ( ) dλ a + d 2λ = τ=τ 0 λ λ 2 (a + d)λ + ad c λ c(d λ) bn τ λ Therefore d (Reλ) sign τ=τ0 } = sign Re = sign (a + d) 2 (cd bn) 2 + 2(ω 2 0 ad) +c 2 ω 4 0 (ad) 2}, ( ) } dλ τ=τ 0 then the proof Applying Lemmas 21 23, we can draw the following conclusions on the stability of the equilibrium point (K, M ) of the system (2) the existence of Hopf bifurcation at (K, M ) Theorem 24 If condition iv) in Lemma 22 is satisfied, then the equilibrium (K, M ) is locally asymptotically stable for all values of τ 0 If either conditions i) or ii) or iii) of Lemma 22 hold g(ω 2 0) 0, then there exists a critical value τ 0 > 0 such that the equilibrium (K, M ) is locally asymptotically stable when τ 0, τ 0 ), it is unstable when τ > τ 0 Moreover, a Hopf bifurcation occurs at (K, M ) when τ = τ 0 Remark 25 If g(ω 2 0) 0, then d (Reλ) / τ=τ0 > 0 The proof is by contradiction Assume d (Reλ) / < 0 for τ = τ 0 τ close to τ 0 Then Eq (5) has a characteristic root λ(τ) = µ(τ) + iω(τ) with µ(τ) > 0, which contradicts the fact that all roots of Eq (5) have negative real parts when τ < τ 0 Thus, when the delay τ near τ 0 is increased, the root of Eq (5) crosses the imaginary axis from left to right Remark 26 If g(ω 2 0) = 0, then the equilibrium (K, M ) is locally asymptotically stable while τ 0, τ 0 ) However, when τ = τ 0, we cannot conclude to stability or instability of the equilibrium In this case, from Lemma 23, we have d (Reλ) / τ=τ0 = 0 3 Direction stability of Hopf bifurcation In this section, we investigate the direction of Hopf bifurcation the stability of the bifurcating periodic solutions based on the normal form approach theory center manifold theory introduced by Hassard et al 17 For notational convenience, let τ = τ 0 +µ, µ R Then µ = 0 is the Hopf bifurcation value for the system (2) By the transformation x = K K, y = M M, t = t/τ, our system is equivalent to the following functional differential equation system in C = C(, 0, R 2 ): u = L µ (u t ) + f(µ, u t ), (10) u = (u 1, u 2 ) T = (x, y) T R 2, u t (θ) = u(t+θ) C, L µ : C R 2, f : R C R 2 are defined, respectively, as follows: L µ (ϕ) = (τ 0 + µ) a b 0 d ϕ(0) + (τ 0 + µ) f (1) f(µ, ϕ) = (τ 0 + µ) f (2) c 0 ϕ() n 0 Here, ϕ = (ϕ 1, ϕ 2 ) C, the expansion of f near the equilibrium point is given by f (1) = 1 (α 1)cK ϕ 1 () (1 α)c(l 0 + M ) ϕ 1 ()ϕ 2 (0) αb(l 0 + M ) ϕ 2 (0) (1 α)(2 α)ck 2 ϕ 1 () 3 3! +3(α 2 1)(L 0 + M ) ck ϕ 1 () 2 ϕ 2 (0) +3α(α 1)c(L 0 + M ) 2 ϕ 1 ()ϕ 2 (0) 2 +α(1 + α)b(l 0 + M ) 2 ϕ 2 (0) 3 Natural Sciences Publishing Cor

4 24 C Bianca, M Ferrara, L Guerrini: Bifurcations stability analysis of the delayed Cai model f (2) = 1 P 2 xd x d ϕ 1 () 2 the eigenvector q(θ) of A belonging to iω 0 τ 0 eigenvector q (s) of A belonging to the eigenvalue iω 0 τ 0 +2Px d yϕ 1 ()ϕ 2 (0) + Pyyϕ 2 (0) 2 Then, it is not difficult to show that + 1 P 3! xd x d x d ϕ 1 () 3 + 3Px d x d yϕ 1 () 2 ϕ 2 (0) q(θ) = (1, ρ) T e iω0τ0θ, q (s) = B(1, σ)e iω0τ0s, +3Px d yyϕ 1 ()ϕ 2 (0) 2 + Pyyyϕ 2 (0) 3 We use the notation ρ = ne iω 0τ 0 P (x d, y) G (w (x d + K, L 0 + y + M ) w 0 ) H(y+M ), iω 0 d, σ = b iω 0 d, with Px d x d P xd x d (K, L 0 + M ) so on By the Riesz representation theorem, there exists a matrix function whose components are functions η(θ) of the bounded variation in θ, 0 such that: L µ ϕ = 0 In fact, we can choose η(θ, µ) = (τ 0 + µ) dη(θ, µ)ϕ(θ), ϕ C a b 0 d D(θ)+(τ 0 + µ) c 0 D(θ+1), n 0 D(θ) = 0 if θ 0, D(θ) = 1 if θ = 0 For ϕ C(, 0, R 2 ), we define A(µ)ϕ = R(µ)ϕ = 0 dϕ, θ, 0), dθ dη(θ, µ)ϕ(θ), θ = 0, 0, θ, 0), f(µ, ϕ), θ = 0 Then system (10) can be written as follows: u = A(µ)u t + R(µ)u t, (11) u t = u(t + θ), for θ, 0 For ψ C(0, 1,, (R 2 ) ), we define the adjoint operator A of A as A ψ(s) = 0 dψ(s), s (0, 1, ds dη(r, µ)ψ( r), s = 0, the following bilinear inner product < ψ(s), ϕ(θ) > = ψ(0)ϕ(0) 0 θ= θ ξ=0 ψ(ξ θ)dη(θ)ϕ(ξ)dξ, η(θ) = η(θ, 0) the bar means complex conjugate transpose of a vector From the discussions in the previous section, we know that ±iω 0 τ 0 are eigenvalues of A Thus, they are also eigenvalues of A Next, we calculate B = ρσ + τ 0 (nσ + c)e iω 0τ 0 Moreover, < q (s), q(θ) >= 1 < q (s), q(θ) >= 0 Next, we study the stability of bifurcated periodic solutions By using the approach of Hassard et al 17, we first compute the coordinates to describe the center manifold at µ = 0 We define z =< q, u t > W (t, θ) = u t (θ) 2Re zq(θ) (12) On the center manifold, we have W (t, θ) = W (z, z, θ) = W 20 (θ) z2 2 + W 11(θ)z z + W 02 (θ) z2 2 + (13) In fact, z z are local coordinates for the center manifold in the direction of q q Note that W is real if u t is real We consider only real solutions For the solution u t, since µ = 0, from (11) we have z = iω 0 τ 0 z+ < q (θ), f (0, W (z, z, θ)2re zq(θ)) > = iω 0 τ 0 z + q (0)f (0, W (z, z, 0)2Re zq(0)) (14) We rewrite the above equation as z = iω 0 τ 0 z + g(z, z), (15) g(z, z) = q (0)f 0 (z, z) (16) = g 20 (θ) z2 2 + g 11(θ)z z + g 02 (θ) z2 2 +g 21 (θ) z2 z 2 + (17) From Eq (11) Eq (14), we have W = u t żq z q = AW 2Re q (0)fq(θ)}, θ, 0), AW 2Re q (0)fq(0)} + f, θ = 0, Rewrite Eq (18) as follows (18) W = AW + Ω(z, z, θ), (19) Natural Sciences Publishing Cor

5 Appl Math Inf Sci 7, No 1, (2013) / wwwnaturalspublishingcom/journalsasp 25 Ω(z, z, θ) = Ω 20 (θ) z2 2 + Ω 11(θ)z z +Ω 02 (θ) z2 2 + (20) Substituting the corresponding series into Eq (19) comparing the coefficients, we obtain (A 2iω 0 τ 0 ) W 20 (θ) = Ω 20 (θ), AW 11 (θ) = Ω 11 (θ) Noticing that q(θ) = (1, ρ) T e iω 0τ 0 θ, q (0) = B(1, σ), from (12) we get u 1t (0) = z + z + W (1) (t, 0), u 2t (0) = ρz + ρz + W (2) (t, 0), u 1t () = e iω 0τ 0 z + e iω 0τ0 z + W (1) (t, 0), u 2t () = e iω0τ0 ρz + e iω0τ0 ρz + W (2) (t, 0) According to Eq (14) Eq (15), we have g(z, z) = q (0)f 0 (z, z) By Eq (13), exping comparing the coefficients with those in Eq (16), it follows: g 20 = τ 0 B (α 1)cK e 2iω0τ0 +2α(1 α)δ 2 K (L 0 + M ) ρe iω 0τ 0 αb(l 0 + M ) ρ 2 + σ(p x d e iω 0τ 0 + P y ρ) (2)} (P x d e iω 0τ 0 + P y ρ) (2) P x d x d e 2iω 0τ 0 +2P x d ye iω0τ0 ρ + P yyρ 2 It remains to compute W 20 (θ) W 11 (θ) that appear in g 21 From Eq (18) Eq (19), we have Ω(z, z, θ) = q (0)f 0 q(θ) q (0) f 0 q(θ) = gq(θ) gq(θ) Comparing the coefficients with Eq (20) gives Ω20 (θ) = g 20 q(θ) ḡ 02 q(θ), Ω 11 (θ) = g 11 q(θ) ḡ 11 q(θ) Hence, from Eqs (22) it follows W 20 (θ) = 2iω 0 τ 0 W 20 (θ) + g 20 q(θ) + ḡ 02 q(θ), W 11 (θ) = g 11 q(θ) + ḡ 11 q(θ) Solving with respect W 20 (θ) W 11 (θ), we obtain W 20 (θ) = ig 20 ω 0 τ 0 q(0)e iω0τ0θ + iḡ 02 3ω 0 τ 0 q(0)e iω 0τ 0 θ + E 1 e 2iω 0τ 0 θ, (22) g 02 = τ 0 B (α 1)cK e 2iω 0τ 0 +2α(1 α)δ 2 K (L 0 + M ) ρe iω0τ0 αb(l 0 + M ) ρ 2 + σ(p x d e iω0τ0 + P y ρ) (2)} g 11 = τ 0 B (α 2 1)(L 0 + M ) ck + σp x d x d +(e iω 0τ0 ρ + e iω 0τ 0 ρ) (1 α)(l 0 + M ) c + σp x d y + ρ 2 αb(l 0 + M ) + σp yy } g 21 = W (1) 20 () e iω 0τ 0 (α 1)cK + σp x d x } d ρ(1 + σ) + + W (1) 11 2 () 2e iω0τ0 (α 1)cK +W (2) + σp x d x d +2ρ(1 + σ)} 11 (0) 2e iω 0τ 0 (1 + σ) + 2ρ αb(l 0 + M ) +σpyy } (2) + W 20 (0) e iω 0 τ 0 (1 + σ) + ρ αb(l 0 + M ) + σp yy} + σp x d yy}, (21) + σ(p x d e iω 0τ 0 + Py ρ) (2), W 11 (θ) = ig 11 ω 0 τ 0 q(0)e iω0τ0θ + iḡ 11 ω 0 τ 0 q(0)e iω0τ0θ + E 2, E 1, E 2 R 2 are constant vectors Following Hassard et al 17, by using the found expressions for g 20 g 11, we can build the following two equations: 0 2iω 0 τ 0 I e 2iω 0τ 0 θ dη(θ, 0) E 1 = M(2)E 1 = 2α(1 α)δ 2 K (L 0 + M ) ρe iω 0τ 0 +(α 1)cK e 2iω0τ0 αb(l 0 + M ) ρ 2 + σ(p x d e iω0τ0 + P y ρ) (2) 0 dη(θ, 0) E 2 = M(0)E 2 +τ 0 B (e 2iω 0τ0 ρ + 2ρ) (α 2 1)(L 0 + M ) ck + σp x = 2α(1 α)δ 2 K (L 0 + M ) ρe iω 0τ 0 d x d y + (2 ρ 2 e iω0τ0 + e iω0τ0 ρ 2 ) α(α 1)c(L 0 + M ) 2 +(α 1)cK e 2iω 0τ 0 αb(l 0 + M ) ρ 2 Natural Sciences Publishing Cor

6 26 C Bianca, M Ferrara, L Guerrini: Bifurcations stability analysis of the delayed Cai model M(ζ) = iω 0ζ a ce iω 0τ 0 ζ b, ζ 0, 2} ne iω 0τ 0 ζ iω 0 ζ d Solving this set of equations, we can derive E 1 E 2, respectively According to the above analysis, W 20 (0) W 11 (0) are determined Thus, all g ij are known we can compute the following quantities: c 1 (0) = i 2ω 0 τ 0 µ 2 = Re c 1(0) Re λ (τ 0 ), β 2 = 2Re c 1 (0), g 11 g 20 2 g 11 2 g 02 2 τ 2 = Im c 1(0) + µ 2 Im λ (τ 0 ) ω 0 τ g 21 2, Using the method of Hassard et al 17, we obtain the following result Theorem 31 1 The sign of µ 2 determines the direction of Hopf bifurcation: if µ 2 > 0 (µ 2 < 0), then the Hopf bifurcation is supercritical (subcritical) the bifurcating periodic solutions exist for τ > τ 0 (τ < τ 0 ) 2 β 2 determines the stability of bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if β 2 < 0 (β 2 > 0) 3 τ 2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if τ 2 > 0 (τ 2 < 0) 4 Conclusion In this paper we have investigated the local stability Hopf bifurcation of the positive equilibrium of an endogenous labor shift model under a dual economy with time delay Specifically, we have generalized the model analyzed in Cai 11 by assuming the growth of capital at time t to be a function of the amount of capital held at time t τ Regarding the delay τ as a parameter applying the local Hopf bifurcation theory, we have studied the existence of periodic oscillations for the delayed model In detail, we have showed that if some conditions are satisfied, then different scenarios arise The positive equilibrium may be asymptotically stable for all τ 0, or as the delay τ increases, it can lose its stability, a Hopf bifurcation occurs at the positive equilibrium, ie a family of periodic orbits bifurcates from it Furthermore, by using the normal form theory center manifold theorem, we have derived an explicit algorithm sufficient conditions for the stability of the bifurcating periodic solutions It is worth stressing that further generalizations of the model proposed in the present paper can be performed by introducing a thermostat which can guarantee the conservation of some quantities in the system, see papers 5,8, an internal structure that takes care of the ability of the interacting entities in the system, see papers 4,6 Finally the analysis should be addressed to the comparison with empirical data following the suggestions proposed in the paper 7 Acknowledgement The first author acknowledges the financial support by the FIRB project-rbid08pp3j-metodi matematici e relativi strumenti per la modellizzazione e la simulazione della formazione di tumori, competizione con il sistema immunitario, e conseguenti suggerimenti terapeutici References 1 PK Asea PJ Zak, Journal of Economic Dynamics Control 23, (1999) 2 CTH Baker, GA Bocharov CAH Paul, Journal of Theoretical Medicine 2, (1997) 3 M Bambi, Journal of Economic Dynamics Control 32, (2008) 4 C Bianca, Appl Math Inf Sci 6, (2012); H Eleuch, Pankaj K Jha Yuri V Rostovtsev, Math Sci Lett 1, 1 (2012) 5 C Bianca, Mathematical Methods in the Applied Sciences, doi:101002/mma C Bianca, Nonlinear Analysis: Real World Applications 13, (2012) 7 C Bianca, Physics of Life Reviews, 8 C Bianca, Physics of Life Reviews, 9 C Bianca, M Ferrara L Guerrini, Appl Math Inf Sci 7, 1-5 (2013) 10 E Böhm-Bawerk, The Positive Theory of Capital, translated by William A Smart, Macmillan & Co Ltd, D Cai, Applied Mathematics Letters 21, (2008) 12 M Ferrara L Guerrini, Far East Journal of Dynamical Systems 19, (2012) 13 R Frisch H Holme, Econometrica 3, (1935), 14 L Guerrini, International Journal of Mathematical Analysis 6, (2012) 15 L Guerrini, Mathematics in Engineering, Science Aerospace 3, (2012) 16 F von Hayek, Prices Production, George Routledge & Sons, Ltd, Broadway House, Carter Lane, BD Hassard, ND Kazarinoff YH Wan, Theory Application of Hopf Bifurcation, M Kalecki, Econometrica 3, (1935) 19 A Krawiec M Szydłowski, Review of Political Economy 16, (2004) 20 FE Kydl EC Prescott, Econometrica 50, (1982) Natural Sciences Publishing Cor

7 Appl Math Inf Sci 7, No 1, (2013) / wwwnaturalspublishingcom/journalsasp A Matsumoto F Szidarovszky, Journal of Economic Behavior Organization 78, (2011) 22 S Ruan J Wei, Dynamics of Continuous, Discrete Impulsive Systems A: Mathematical Analysis 10, (2003) 23 A Rustichini, Journal of Dynamics Differential Equation 1, (1989) 24 M Szydłowski, Chaos, Solitons & Fractals 14, (2002) 25 PJ Zak, Review of Political Economy 11, (1999) Carlo Bianca received the PhD degree in Mathematics for Engineering Science at Politecnico of Turin His research interests are in the areas of applied mathematics mathematical physics including the mathematical methods models for complex systems, mathematical billiards, chaos, anomalous transport in microporous media numerical methods for kinetic equations He has published research articles in reputed international journals of mathematical engineering sciences He is referee editor of mathematical journals Massimiliano Ferrara is Associate Professor of Mathematical Economics at University Mediterranea of Reggio Calabria He was the Founder Director of MEDAlics (2010) Research Center of Mediterranean Relations - Vice Rector at University Dante Alighieri of Reggio Calabria Visiting Professor at Harvard University, Cambridge (USA), Morgan State University in Baltimore (USA), Western Michigan University (USA), New Jersey Institute of Technology in Newark (NJ) (USA) He received the PhD degree in Mathematical Economics at University of Messina (Italy) He is referee Editor of several international journals in the frame of pure applied mathematics, applied economics His main research interests are: dynamical systems, patterns of growth sustainable development, mathematical economics, game theory, optimization theory, applied economics, differential geometry applications, geometric dynamics applications Luca Guerrini is Associate Professor of Mathematical Economics at the University of Bologna, Italy He holds an MA PhD in Pure Mathematics from the University of California, Los Angeles, USA His research interests are in the areas of pure applied mathematics as well as mathematical economics He has published extensively in internationally refereed journals Natural Sciences Publishing Cor

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