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.

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 [

The highest dimensionality for which complete coefficient criteria are presently available is

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 paper is organized into the following sections. Section

We employ here a version of the Hopf bifurcation theorem from Guckenheimer and Holmes [

Suppose that the dynamical system (

The system has a smooth curve of equilibria:

The characteristic equation

where

and no other roots with zero real parts.

The real part

In the present case

with characteristic equation:

where the coefficients

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

The polynomial

The polynomial

The polynomial

Under conditions (

Theorem

Finally, Theorem

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.

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. [

The continuous time version employed here for the application of the proposed coefficient criteria is described by the following five nonlinear differential equations:

The parameters of the model are

The system has a smooth curve of equilibria described by the equilibrium values:

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

In applying our criterion

In applying our criterion

If a nonsimple Hopf bifurcation appears possible at some point

A simple Hopf bifurcation is a special kind of nonsimple Hopf bifurcation. Once a nonsimple Hopf bifurcation at the point

Following the above guidelines, we firstly apply our coefficient criteria to several

Plots of the curves

For verification we adopt the parameter

Plots of quantities related to the coefficient criteria

In the same computation we determine the values of the quantities

Let us now consider a fixed value

Plots of the curves

In this section, we give an example of Hopf bifurcation, established in Section

Employing numerical integration for a value of

Numerical simulation of an orbit starting near the repelling cycle of the subcritical Hopf bifurcation.

The “projection” of the five-dimensional orbit in the

Schematic representation of the repelling cycles of the Hopf bifurcation regarding the relevant values

Interpreting the Hopf bifurcation 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 the equilibrium and makes feasible the application of this detection process globally, in the space of the system’s parameters, to acquire preliminary information about the system’s potential for oscillatory behaviour.

In this paper, we presented a complete mathematical characterization of coefficient criteria for five-dimensional Hopf bifurcations. The criterion proposed here for simple Hopf bifurcations is marginally more concise and informative (see relations (

We also suggested a possible manner of conducting the application of our coefficient criteria by presenting an application to a model of nonlinear macroeconomic dynamics. The application showed that our 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. Finally we presented an example of a bifurcation detected by using our criteria, which turned out to be a subcritical simple Hopf bifurcation, and illustrated by numerical simulation the cyclical behaviour of the system variables due to the repelling cycle associated with the subcritical bifurcation.

It is well known that the polynomial

If

If one of the quantities

Note that if

Under the conditions

In the case of conditions

Finally, under conditions

The remaining roots of

Note that under conditions

Also, it can be verified that our conditions

Hopf bifurcations occur under conditions

From the well-known Vieta formulas, giving the coefficients of the polynomial

We now consider the remaining possible case of Hopf bifurcation, namely

The authors declare that there is no conflict of interests regarding the publication of this paper.