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Massimiliano Ferrara, Luca Guerrini, Giovanni Molica Bisci, "Center Manifold Reduction and Perturbation Method in a Delayed Model with a Mound-Shaped Cobb-Douglas Production Function", Abstract and Applied Analysis, vol. 2013, Article ID 738460, 6 pages, 2013. https://doi.org/10.1155/2013/738460
Center Manifold Reduction and Perturbation Method in a Delayed Model with a Mound-Shaped Cobb-Douglas Production Function
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.
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 , Zak , Szydłowski , Szydłowski and Krawiec , Matsumoto and Szidarovszky , Matsumoto et al. , d’Albis et al. , Bambi et al. , Boucekkine et al. , Matsumoto and Szidarovszky , Ballestra et al. , Bianca and Guerrini , Bianca et al. , Guerrini and Sodini [14, 15], and Matsumoto and Szidarovszky . 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 , where a continuous-time neoclassical growth model with time delay was developed similarly in spirit and functional form to Day’s  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 , (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 ), 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. . 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. . 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.
In this paper, we consider the special neoclassical growth model with fixed time delay introduced and examined by Matsumoto and Szidarovszky’s , 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.
The authors would like to thank the referees for their valuable comments.
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