Abstract and Applied Analysis

Volume 2013 (2013), Article ID 738460, 6 pages

http://dx.doi.org/10.1155/2013/738460

Research Article

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

Massimiliano Ferrara,¹ Luca Guerrini,² and Giovanni Molica Bisci³

¹Department of Law and Economics, University Mediterranea of Reggio Calabria, Via dei Bianchi 2 (Palazzo Zani), 89127 Reggio Calabria, Italy

²Department of Management, Polytechnic University of Marche, Piazza Martelli 8, 60121 Ancona, Italy

³Department of PAU, University Mediterranea of Reggio Calabria, Via Melissari 24, 89124 Reggio Calabria, Italy

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.

Abstract.

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:

whereis the per capita per labor and, are positive parameters. In order to simplify the notation, we omit the indication of time dependence for variables and derivatives referred to as time. As well, we useto indicate the state of the variableat time, whererepresents the time delay inherent in the production process. According to Matsumoto and Szidarovszky [5], (1) has a unique positive steady state

if . In case, this equilibrium is locally asymptotically stable forand unstable for, where

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 asis 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 when. We letbe 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, let, where, so thatis the Hopf bifurcation value for (1). First we use the transformation, so that (1) becomes

Let be the Banach space of continuous mappings frominto equipped with supremum norm. Let , forThen, (4) can be written as

where the linear operatorand the function are given by

with. By the Riesz representation theorem, there exists a bounded variation function,, such that

where

withrepresenting the Dirac delta function. Next, for, define

As a result, (5) can be expressed as

For, the adjoint operatorofis defined as

Let (resp.,) denote the eigenvector for(resp., for) corresponding to; namely, (resp.,). To construct the coordinates to describe the center manifold near the origin, we define an inner product as follows:

forand, whereandrepresents the complex conjugate operation of . The vectorsandcan be normalized by the conditions and . A direct computation shows that

where

Letand

On the center manifold,, with

whereandare local coordinates forin the direction ofand, respectively. For anysolution of (10), we have

whereand. Noting from (16) that

it follows that

Expandingin powers of and , that is,

and comparing the above coefficients with those in (20), we get

In order to compute , we need to know , and , first. From (16), one has

where

Recalling (23), it follows that

On the other hand,

A comparison of the coefficients of (24) and (25) gives

Thus, (26) becomes

which is solved by

Similarly, from

we derive

whereis a constant vector. In order to compute and , the constantsandare needed. From (23), we have

Thus,

On the center manifold, we have . Replacing , and , , we obtain a second expression for. A comparison of the coefficients of this equation with those in (23), for, leads us to the following:

Since

from the previous analysis we arrive at

Hence,andcan be computed from (29) and (31) as, and we obtain

where

where

Based on the above analysis, allhave been obtained. Consequently, we can compute the following quantities:

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 , , , andbe defined in (40).

(i)The bifurcating periodic solution is supercritical bifurcating as, and it is subcritical bifurcating as .(ii)The bifurcating periodic solutions are stable if and unstable if .(iii)Asincreases, the period of bifurcating periodic solutions increases if , while it decreases, if .

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

so that solutions of (4) which areperiodic inbecomeperiodic in. This change of variables results in the following form of (4):

where the terms, , andare given by

The idea is now to expand the solution of (42) in a power series in a suitable smallness parameter, that is,

and to solve for the unknown functionsrecursively. In this context, the definition of the is clear. As already mentioned,represents a small quantity so that we can expand the frequencyand the delayin powers ofaccording to

where we have set

In addition, we also have to consider a corresponding expansion of the time delayed term , which is achieved by

wherestands for

with primes representing differentiation with respect toApplying the expansions for and to (42) and collecting terms for the distinct orders of , we get the following three equations:

We take the solution of (49) as follows:

whereandare constants. Next we substitute (52) into (49) and derive that andare arbitrary. Without loss of generality, we impose the initial conditionsand and get from (52) that

Next, we look for a solution to (50) as

where the coefficients , , , , andare constants. Substituting (53) and (54) in (50) and equating the coefficients of the resonant terms , , , and, we find that

withandbeing arbitrary and

For simplicity, we let . Hence, (54) becomes

where , , andare given in (55). Finally, let

be the solution of (51), with , , , , , , and being constants. Using (53), (57), and (58) into (51), after trigonometric simplifications have been performed, we obtain

where

Comparing the coefficients of the terms, , , , , , and, we get the following expressions:

Summing up all the above analysis, the bifurcated periodic solution of (4) has an approximation of the form

where,, withand given in (53) and (57), respectively. Here, the parameters and 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 pointwhenis supercritical (resp., subcritical), if(resp.,)and the bifurcating periodic solutions exist for(resp.,). In addition, its period decrease (resp., increases) as increases, if (resp., ).

Remark 3. LetandThen

As direct calculation shows that (61) yieldsandIn this case, the Hopf bifurcation is subcritical and the bifurcating periodic solutions exist for. Moreover, its period decreases asincreases.

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.

References

P. K. Asea and P. J. Zak, “Time-to-build and cycles,” Journal of Economic Dynamics & Control, vol. 23, no. 8, pp. 1155–1175, 1999.

P. J. Zak, “Kaleckian lags in general equilibrium,” Review of Political Economy, vol. 11, pp. 321–330, 1999.

M. Szydłowski, “Time to build in dynamics of economic models. II. Models of economic growth,” Chaos, Solitons and Fractals, vol. 18, no. 2, pp. 355–364, 2003.

M. Szydłowski and A. Krawiec, “A note on Kaleckian lags in the Solow model,” Review of Political Economy, vol. 16, no. 4, pp. 501–506, 2004.

A. Matsumoto and F. Szidarovszky, “Delay differential neoclassical growth model,” Journal of Economic Behavior and Organization, vol. 78, no. 3, pp. 272–289, 2011.

A. Matsumoto, F. Szidarovszky, and H. Yoshida, “Dynamics in linear cournot duopolies with two time delays,” Computational Economics, vol. 38, no. 3, pp. 311–327, 2011.

H. d'Albis, E. Augeraud-Veron, and A. Venditti, “Business cycle fluctuations and learning-by-doing externalities in a one-sector model,” Journal of Mathematical Economics, vol. 48, no. 5, pp. 295–308, 2012.

M. Bambi, G. Fabbri, and F. Gozzi, “Optimal policy and consumption smoothing effects in the time-to-build AK model,” Economic Theory, vol. 50, no. 3, pp. 635–669, 2012.

R. Boucekkine, G. Fabbri, and P. Pintus, “On the optimal control of a linear neutral differential equation arising in economics,” Optimal Control Applications & Methods, vol. 33, no. 5, pp. 511–530, 2012.

A. Matsumoto and F. Szidarovszky, “Nonlinear delay monopoly with bounded rationality,” Chaos, Solitons and Fractals, vol. 45, no. 4, pp. 507–519, 2012.

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. 2013, Article ID 689372, 8 pages, 2013.

C. Bianca and L. Guerrini, “On the Dalgaard-Strulik model with logistic population growth rate and delayed-carrying capacity,” Acta Applicandae Mathematicae, vol. 128, pp. 39–48, 2013.

C. Bianca, M. Ferrara, and L. Guerrini, “Stability and bifurcation analysis in a Solow growth model with two time delays,” Abstract and Applied Analysis. In press

L. Guerrini and M. Sodini, “Nonlinear dynamics in the Solow model with bounded population growth and time-to-build technology,” Abstract and Applied Analysis, vol. 2013, Article ID 836537, 6 pages, 2013.

L. Guerrini and M. Sodini, “Dynamic properties of the Solow model with increasing or decreasing population and time-to-build technology,” Abstract and Applied Analysis. In press

A. Matsumoto and F. Szidarovszky, “Asymptotic behavior of a delay differential neoclassical growth model,” Sustainability, vol. 5, no. 2, pp. 440–455, 2013.

R. Day, “Irregular growth cycles,” American Economic Review, vol. 72, pp. 406–414, 1982.

B. D. Hassard, N. D. Kazarinoff, and Y. H. Wan, Theory and Applications of Hopf Bifurcation, vol. 41, Cambridge University Press, Cambridge, Mass, USA, 1981.

A. H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, New York, NY, USA, 1981.

N. MacDonald, Time Lags in Biological Models, vol. 27 of Lecture Notes in Biomathematics, Springer, Berlin, Germany, 1978.