Journal of Nonlinear Dynamics

Volume 2015, Article ID 278234, 11 pages

http://dx.doi.org/10.1155/2015/278234

## Complete Coefficient Criteria for Five-Dimensional Hopf Bifurcations, with an Application to Economic Dynamics

Faculty of Engineering, University of Patras, 26504 Rion, Greece

Received 7 May 2015; Accepted 23 June 2015

Academic Editor: Dibakar Ghosh

Copyright © 2015 Christos Douskos and Panagiotis Markellos. 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

Paper presents a complete mathematical characterization of coefficient criteria for five-dimensional Hopf bifurcations and an example of the application of these criteria to a model of economic dynamics. The application illustrates that the proposed criteria are practical and useful in determining the existence or nonexistence of Hopf bifurcations of five-dimensional dynamical systems in entire ranges of the system’s parameters.

#### 1. Introduction

Hopf bifurcations occur in dynamical systems leading to cyclical fluctuations emerging from equilibrium states, and the Hopf bifurcation theorem is a useful tool to establish the emergence of such fluctuations. The theorem describes conditions to be satisfied by the eigenvalues of the Jacobian matrix of the linearized system around an equilibrium, for cycles to bifurcate from the equilibrium. Although the theorem is of local validity, it provides useful information about the system’s potential for oscillatory behaviour for a range of a bifurcation parameter. Interpreting the theorem in terms of conditions to be satisfied by the coefficients of the characteristic polynomial at the equilibrium, that is establishing “coefficient criteria” for Hopf bifurcations, facilitates the detection of cycles generated locally at equilibrium and the application of this detection process globally in entire ranges of the system parameters.

In the field of economic dynamics Hopf bifurcations are of interest for the mathematical modelling of endogenous business cycles. Several authors have used this theorem to study the appearance of business cycles in continuous time economic models. For example, Asada [1] and Asada and Yoshida [2] treated three- and four-dimensional Hopf bifurcations by means of coefficient criteria. In recent years the tendency is to consider higher-dimensional dynamics in macroeconomic modelling (see, e.g. [3]), but complete coefficient criteria for Hopf bifurcations have not been proposed so far for -dimensional systems with .

The highest dimensionality for which complete coefficient criteria are presently available is , and the relevant criteria were provided by Asada and Yoshida [2]. Liu [4] has provided a general criterion for -dimensional Hopf bifurcations, , which however is not complete as it applies only to the so called “simple” Hopf bifurcation. This is a special kind of Hopf bifurcation occurring when all eigenvalues of the Jacobian matrix have negative real parts, except the single pair of imaginary eigenvalues characterizing the bifurcation instance, and the simple Hopf bifurcation is therefore related to a change of equilibrium stability.

In this paper we present a complete mathematical characterization of coefficient criteria for five-dimensional Hopf bifurcations, not only for simple Hopf bifurcations. Our proposed criterion for simple Hopf bifurcations is marginally more concise and informative, providing the imaginary eigenvalues and the approximate period of the cycles directly from the coefficients of the characteristic polynomial but is essentially equivalent to Liu’s criterion for . The main contribution of this paper is the establishment of the coefficient criterion for the existence or nonexistence of nonsimple five-dimensional Hopf bifurcations. We also present an application of our criteria to a model of economic dynamics. The application illustrates that these can be practically and usefully employed to determine the existence or nonexistence of Hopf bifurcations of a five-dimensional dynamical system in entire ranges of the system parameters.

The paper is organized into the following sections. Section 2 contains the Hopf bifurcation theorem for -dimensional systems, and Section 3 presents a complete characterization of coefficient criteria for five-dimensional Hopf bifurcations by means of four mathematical propositions covering simple as well as nonsimple Hopf bifurcations. Section 4 contains an application of the proposed criteria to a five-dimensional two-region business cycle model suggested by Asada et al. [5]. In Section 5 we give an example of Hopf bifurcation, established in Section 4, by presenting diagrams illustrating the orbital behaviour of the system associated with the bifurcation. Section 6 contains some concluding remarks. Finally, Appendices A to D contain the proofs of the mathematical propositions.

#### 2. The Hopf Bifurcation Theorem

We employ here a version of the Hopf bifurcation theorem from Guckenheimer and Holmes [6] with minor changes in notation. Consider the -dimensional dynamical system:with a bifurcation parameter.

Theorem 1. *Suppose that the dynamical system (1) has the following properties.*(I)*The system has a smooth curve of equilibria:*(II)*The characteristic equation* *where is the Jacobian matrix of the system at , has a pair of imaginary roots,* *and no other roots with zero real parts.*(III)*The real part of satisfies**Then, there exists a continuous function with , and for sufficiently small values of there exists a continuous family of nonconstant periodic solutions of the dynamical system. As , the family of periodic solutions collapses at the equilibrium point , with their period tending to*

*
In the present case the Jacobian matrix of the system is*

*
with characteristic equation:*

*
where the coefficients depend on the parameter . The aim of this paper is to establish conditions that must be satisfied by these five coefficients for a Hopf bifurcation to occur at the critical value of the bifurcation parameter .*

*3. Coefficient Criteria for *

*The following propositions provide the complete characterization of the Hopf bifurcation in the case of five-dimensional dynamical systems. Proofs of these propositions are provided in Appendices A to D.*

*Lemma 2. The polynomial has exactly one pair of imaginary roots , , if and only if one of the following sets of conditions, coefficient criteria, is satisfied: , with ,, , and , with ,, , and , , with .*

*Theorem 3. The polynomial has one pair of imaginary roots , and all other roots with nonzero real parts if and only if one of the following coefficient criteria is satisfied::(), , :(), .*

*Theorem 4. The polynomial has one pair of imaginary roots , and all other roots with negative real parts if and only if the following coefficient criterion is satisfied::(), .*

*Theorem 5. Under conditions () or (), condition (III) of Theorem 1 is equivalent to ,and under conditions () condition (III) of Theorem 1 is equivalent to ,where the derivatives are with respect to the bifurcation parameter .*

*Theorem 3 is a complete characterization of condition (II) of Theorem 1, for nonsimple Hopf bifurcation, in the present case of a five-dimensional system. Theorem 4 regards the special case of simple Hopf bifurcation. The proposed here criterion for the simple five-dimensional Hopf bifurcation is marginally more concise and informative but essentially equivalent to Liu’s criterion for -dimensional simple Hopf bifurcations in the present case (see Remark 6 in Appendix C).*

*Finally, Theorem 5 provides a complete characterization of condition (III) of the Hopf bifurcation theorem in the case of a five-dimensional system. The above Theorems 3 and 5 are entirely new results to the authors’ knowledge and complement Theorem 4 providing a complete mathematical characterization of the coefficient criteria for Hopf bifurcations in five-dimensional dynamical systems.*

*The application in the following section illustrates that the proposed criteria are practical and useful in determining the existence or nonexistence of Hopf bifurcations of five-dimensional dynamical systems in entire ranges of the system’s parameters and are therefore useful for the analytical investigation of cyclical behaviour in five-dimensional dynamical systems.*

*4. Application to Economic Dynamics*

*4. Application to Economic Dynamics*

*In this section, we present an application of the coefficient criteria stated in the previous section to a typical model of five-dimensional macroeconomic dynamics. For the application we consider a continuous time version of the Kaldorian two-region discrete time business cycle model proposed by Asada et al. [5].*

*The continuous time version employed here for the application of the proposed coefficient criteria is described by the following five nonlinear differential equations: where dots denote derivatives with respect to time and we have abbreviatedThe function is a particular case of the Kaldorian -shaped direct dependence of the investment function on income given by The subscript is the index number of a region and denotes the time. The price levels are considered fixed and the meanings of the symbols are as follows. is the real regional income, is the real physical capital stock, and is the nominal money stock. The exchange rate is fixed and the total nominal money supply of the two regions is taken to be fixed at .*

*The parameters of the model are the adjustment speed of the goods market of each region, the degree of capital mobility, and the degree of interregional trade, . We note that under the specifications and functional forms adopted in the formulation of the system of equations (9), the two regional economies are assumed quite similar; any dissimilarity will be due to the possibly unequal speeds of adjustment , . For a full description of the model and its economic foundations see Asada et al. [5].*

*The system has a smooth curve of equilibria described by the equilibrium values:The equilibrium values and depend on the trade parameter only, while the equilibrium values of the other variables depend on both and . All equilibrium values are positive for and .*

*In this application we first focus our search for Hopf bifurcations by reducing the four-dimensional space of parameters of the model to a series of two-dimensional parameter subspaces of the basic parameters and . This is done by assuming that the other two parameters and take equal values and considering a succession of values for Alternatively, we set a plausible value, for example, , for and consider a succession of values for . We then apply our coefficient criteria to each parameter plane by creating appropriate plots representing these criteria graphically.*

*In applying our criterion for nonsimple Hopf bifurcations, these plots include the curve (bold), the curve (thin) representing the boundary of the region (light-shaded) in which , and if required also the curve . According to our criterion Hopf bifurcations occur at the points of the curve provided that these points are inside the region where and correspond to such values of and that the condition , equivalently , is also satisfied.*

*In applying our criterion for nonsimple Hopf bifurcations, the plots include the curves and as before, and if required also the curves , , . According to our criterion Hopf bifurcations occur at the points of intersection of the curves and which correspond to such values of and that the condition is also satisfied with , .*

*If a nonsimple Hopf bifurcation appears possible at some point under criterion or under criterion , the bifurcation can be verified by application of the corresponding transversality condition or . For this last verification either one of the model parameters may be tested as the bifurcation parameter of Theorem 1.*

*A simple Hopf bifurcation is a special kind of nonsimple Hopf bifurcation. Once a nonsimple Hopf bifurcation at the point has been verified as above under criterion , computation of the characteristic polynomial at the equilibrium values of the system variables corresponding to this pair of parameter values and checking the conditions , , reveals whether the bifurcation is “simple” or not. This last step, to distinguish between nonsimple and simple Hopf bifurcation, can be performed globally for all points of the bifurcation curve, that is, of those segments of the curve which are inside the region where and correspond to such values of and that the condition , equivalently , is also satisfied.*

*Following the above guidelines, we firstly apply our coefficient criteria to several parameter planes corresponding to different values of . Figure 1 shows the results for . In the left column diagrams of Figure 1 we see that the curve segments of (bold) do not cross with the curve segments of (thin); therefore Hopf bifurcations do not exist under the coefficient criterion in the considered region of the parameter plane. However, the leftmost branch of the curve lies within the region for which , and the curve does not appear, therefore Hopf bifurcations may exist under the coefficient criterion in the considered region of the parameter plane at the points of the leftmost branch of the curve .*