International Journal of Mathematical Analysis Vol 9, 2015, no 38, 1869-1875 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/1012988/ijma201554135 Direction and Stability of Hopf Bifurcation in a Delayed Model with Heterogeneous Fundamentalists Luca Guerrini Department of Management Polytechnic University of Marche, Italy Copyright c 2015 Luca Guerrini This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Abstract Using the normal form theory and center manifold argument, we derive explicit formulas determining the stability, direction and other properties of bifurcation periodic solutions of Gori et al [6 model s Mathematics Subject Classification: 34K18, 91B62 Keywords: Time delay; Center manifold; Hopf bifurcation 1 Introduction In recent years we have witnessed an increasing interest in dynamical systems with time delays, especially in applied mathematics and economics (see eg [1-8,[9), where it has been showed that the introduction of time delays allows the capturing of more complex dynamics, thus enriching the description of the whole system In [6 Gori et al have inquired whether complex dynamic phenomena obtained in discrete time models also hold in continuous time models with discrete time delays They considered a continuous time version of the model developed by Naimzada and Ricchiuti [10 augmented with discrete time delays Specifically, their model is described by a delay differential equation P = γ [ (F 1 P d ) (F 2 P d ) 2 + (F 2 P d ) (F 1 P d ) 2 (F 1 P d ) 2 + (F 2 P d ) 2 F (P d ), (1)
1870 Luca Guerrini where γ > 0, F 2 > F 1 > 0, and P d indicates the state of the variable P at time t τ, with τ 0 the time delay The analysis conducted by Gori et al [6 confirmed the existence of nonlinear asset price dynamics in a continuous time version (with discrete time delays) of the model proposed by Naimzada and Ricchiuti [10, characterised by the existence of fundamentalists with heterogeneous expectations on the value of a risky asset In particular, it was proved the existence of a value τ 0 > 0 of the time delay τ such that the positive equilibria P = F 1 and P = F 2 of Eq (1) are locally asymptotically stable for τ [0, τ 0 ), and unstable for τ > τ 0 Furthermore, a Hopf bifurcation occurs at P when τ = τ 0 However, in their analysis, Gori et al [6 did not investigate the stability of the periodic solution In this paper, using the normal form theory and center manifold argument due to Hassard et alt [9, we derive explicit formulas determining the stability, direction and other properties of bifurcation periodic solutions 2 Direction and stability of the Hopf bifurcation In this section, we shall derive the explicit formulae for determining the direction, stability and period of these periodic solutions bifurcating from the equilibrium point P at the critical value τ 0, using techniques from normal form and center manifold theory (see Hassard et al [9) For convenience, we rescale the time t t/τ to normalize the delay, and let τ = τ 0 + µ, µ R, so that µ = 0 is the Hopf bifurcation value for Eq (1) in terms of the new bifurcation parameter µ Let u(t) = P (τt) P, and apply Taylor expansion to the right-hand side of Eq (1) at the trivial equilibrium Separating the linear from the non-linear terms, Eq (1) can be written as a functional differential equation in C = C([, 0, R) as u = L µ (u t ) + f(µ, u t ), (2) where u t = u(t + θ), for θ [, 0, and L µ : C R, f : R C R are given by L µ (ϕ) = (τ 0 + µ) γf P d ϕ(), ϕ C, and [ 1 f(µ, ϕ) = (τ 0 + µ) 2 F P d P d ϕ() 2 + 1 6 F P d P d P d ϕ 1 () 3 +, (3) where FP d P d = F Pd P d (P ) and FP d P d P d = F Pd P d P d (P ) By the Riesz representation theorem, there exists a bounded variation function η(θ, µ), θ [, 0, such that L µ ϕ = dη(θ, µ)ϕ(θ), for ϕ C
Hopf bifurcation in a delayed model 1871 For ϕ C, define A(µ)(ϕ) = and dϕ(θ), θ [, 0), dθ dη(r, µ)ϕ(r), θ = 0, { 0, θ [, 0), R(µ)(ϕ) = f(µ, ϕ), θ = 0 Then Eq (2) is equivalent to abstract differential equation u t = A(µ)u t + R(µ)u t, where u t = u(t + θ), for θ [, 0 For ψ C, define dψ(r), r (0, 1, A dr ψ(r) = dη(ζ, µ)ψ( ζ), r = 0 For ϕ C and ψ C, define the bilinear form < ψ(r), ϕ(θ) >= ψ(0)ϕ(0) θ= θ ξ=0 ψ(ξ θ)dη(θ)ϕ(ξ)dξ, where η(θ) = η(θ, 0) The A(0) and A are adjoint operators From the discussion in Gori et al [6, we know that if µ = 0, then (1) undergoes a Hopf bifurcation and the associated characteristic equation of Eq (1) has a pair of simple imaginary roots ±iω 0 τ 0 It is immediate that ±iω 0 τ 0 are eigenvalues of A(0), and so they are also eigenvalues of A By direct computation, we can obtain q(θ) = (1, ρ) T e iω 0τ 0 θ, with ρ complex, namely the eigenvector of A(0) corresponding to iω 0 τ 0, and similarly q (r) = D(σ, 1)e iω 0τ 0 r, the eigenvector of A corresponding to iω 0 τ 0, where the value of D is chosen to guarantee < q, q >= 1 As well, one has < q, q >= 0 Define On the center manifold C, z =< q, u t > and W (t, θ) = u t (θ) 2Re {zq(θ)} (4) W (t, θ) = W (z, z, θ) = W 20 (θ) z2 2 + W 11(θ)z z + W 02 (θ) z2 2 +, (5) where z and z are local coordinates for the center manifold in the direction of q and q Noting that W is also real if u t is real, we consider only real solutions For solutions u t C of Eq (1), we have z = iω 0 τ 0 z + q (0)f (0, W (z, z, 0)2Re {zq(0)}) def = iω 0 τ 0 z + q (0)f 0 (z, z)
1872 Luca Guerrini where f 0 (z, z) = f(0, u t ), with f defined as in (3) Set q (0)f 0 (z, z) by g(z, z) Then From (4), g(z, z) = q (0)f 0 (z, z) = g 20 z 2 2 + g 11z z + g 02 z 2 2 + g 21 z 2 z 2 + (6) u t (θ) = W (t, θ) + 2Re {zq(θ)} = W 20 (θ) z2 2 + W 11(θ)z z + W 02 (θ) z2 2 + + zq(θ) + z q(θ), so that substituting it into f(0, u t ) we obtain z 2 f 0 (z, z) = f(0, u t ) = f z 2 2 + f z 2 z zz z + f z 2 2 + f z 2 z z 2 z 2 + (7) Thus, a comparison of the coefficients of (7) with those in (6) yields g 20 = D( σ, 1)f z 2, g 02 = D( σ, 1)f z 2, g 11 = D( σ, 1)f z z, g 21 = D( σ, 1)f z2 z In order to determine g 21, we focus on the computation of W 20 (θ) and W 11 (θ) From (2) and (4), we can get W = u t żq z q { AW 2Re { q (0)f 0 q(θ)}, θ [, 0), = AW 2Re { q (0)f 0 q(0)} + f 0, θ = 0 Let W = AW + H(z, z, θ), (8) where H(z, z, θ) = H 20 (θ) z2 2 + H 11(θ)z z + H 02 (θ) z2 2 + (9) In view of (5), we get AW = AW 20 (θ) z2 2 + AW 11(θ)z z + AW 02 (θ) z2 2 + (10) Differentiating both sides of (5) with respect to t, we have W = W z ż + W z z (11) According to (10) and (11), comparing coefficients, we can obtain [A 2iω 0 τ 0 W 20 (θ) = H 20 (θ), AW 11 (θ) = H 11 (θ), (12)
Hopf bifurcation in a delayed model 1873 From (8), we get Thus, we have H(z, z, θ) = q (0)f 0 q(θ) q (0) f 0 q(θ) = gq(θ) ḡ q(θ) H 20 (θ) = g 20 q(θ) ḡ 02 q(θ), H 11 (θ) = g 11 q(θ) ḡ 11 q(θ) (13) Then, together with (12),(13), we can derive that Therefore, one has W 20 (θ) = 2iω 0 τ 0 W 20 (θ) + g 20 q(θ) + ḡ 02 q(θ) W 20 (θ) = g 20 q(0)e iω 0τ 0 θ ḡ02 q(0)e iω 0τ 0 θ + E 1 e 2iω 0τ 0 θ, (14) iω 0 τ 0 3iω 0 τ 0 with E 1 R Similarly, we have W 11 (θ) = g 11 q(0)e iω 0τ 0 θ ḡ11 q(0)e iω 0τ 0 θ + E 2, iω 0 τ 0 iω 0 τ 0 with E 2 R Next, we focus on the computation of E 1 and E 2 From the definition of A and (12), we have dη(θ)w 20 (θ) = 2iω 0 τ 0 W 20 (θ) H 20 (θ) (15) and dη(θ)w 11 (θ) = H 11 (θ) In addition, we note [ [ iω 0 τ 0 e iω 0τ 0 θ dη(θ) q(0) = 0, iω 0 τ 0 e iω 0τ 0 θ dη(θ) q(0) = 0 From (8) and (9), we have H 20 (0) = g 20 q(0) ḡ 02 q(θ) + f z 2, H 11 (0) = g 11 q(0) ḡ 11 q(0) + f z z (16) Then, substituting (14) and (16) into (15) gives [ 2iω 0 τ 0 e 2iω 0τ 0 θ dη(θ) E 1 = f z 2 Similarly, we get [ dη(θ) E 2 = f z z
1874 Luca Guerrini In conclusion, each g ij is computed Thus, we can calculate the following values [ c 1 (0) = i g 11 g 20 2 g 11 2 g 02 2 2ω 0 τ 0 3 + g 21 2, µ 2 = Re {c 1(0)} Re {τ 0 λ (τ 0 )}, β 2 = 2Re {c 1 (0)}, T 2 = Im {c 1(0)} + µ 2 Im {λ (τ 0 )} ω 0 τ 0, These formulae give a description of the Hopf bifurcation periodic solutions of (3) at τ = τ 0 on the center manifold Theorem 21 The periodic solution is forward (resp backward) if µ 2 > 0 (resp µ 2 < 0); the bifurcating periodic solutions are orbitally asymptotically stable if β 2 < 0 (unstable β 2 > 0); the periods of the bifurcating periodic solutions increase (resp decrease) if T 2 > 0 (resp T 2 < 0) References [1 LV Ballestra, L Guerrini and G Pacelli, Stability switches and bifurcation analysis of a time delay model for the diffusion of a new technology, International Journal of Bifurcation and Chaos, 24 (2014) http://dxdoiorg/101142/s0218127414501132 [2 C Bianca, M Ferrara and L Guerrini, Hopf bifurcations in a delayed-energy-based model of capital accumulation, Applied Mathematics & Information Sciences, 7 (2013), 139-143 http://dxdoiorg/1012785/amis/070116 [3 C L Dalgaard and H Strulik, Energy distribution and economic growth, Resource and Energy Economics, 33 (2011), 782-797 http://dxdoiorg/101016/jreseneeco201104004 [4 M Ferrara, L Guerrini and M Sodini, Nonlinear dynamics in a Solow model with delay and non-convex technology, Applied Mathematics and Computation, 228 (2014), 1-12 http://dxdoiorg/101016/jamc201311082 [5 L Gori, L Guerrini and M Sodini, Hopf bifurcation in a Cobweb model with discrete time delays, Discrete Dynamics in Nature and Society, 2014 (2014), 1-8 http://dxdoiorg/101155/2014/137090
Hopf bifurcation in a delayed model 1875 [6 L Gori, L Guerrini and M Sodini, Heterogeneous fundamentalists in a continuous time model with delays, Discrete Dynamics in Nature and Society, 2014 (2014), 1-6 http://dxdoiorg/101155/2014/959514 [7 L Gori, L Guerrini and M Sodini, A continuous time Cournot duopoly with delays, Chaos, Solitons & Fractals, forthcoming 2015 http://dxdoiorg/101016/jchaos201501020 [8 L Guerrini and M Sodini, Persistent fluctuations in a dual model with frictions: the role of delays, Applied Mathematics and Computation, 241 (2014), 371-379 http://dxdoiorg/101016/jamc201405028 [9 B Hassard, D Kazarino, and Y Wan, Theory and application of Hopf bifurcation, Cambridge University Press, 1981 [10 A K Naimzada and G Ricchiuti, Heterogeneous fundamentalists and imitative processes, Applied Mathematics and Computation, 199 (2008), 171-180 http://dxdoiorg/101016/jamc200709061 Received: April 29, 2015; Published: July 14, 2015