where and is the Jacobian evaluated at the steady state . The Jacobian is given by the matrix:
where the derivatives of the functions have to be evaluated at steady state.
According to Ref. 9, the eigenvalues of Jacobian matrices of the following type:
are given by:
where is the sum of the determinants
Moreover such a Jacobian has a pair of purely imaginary eigenvalues if and only if the relations
For our model the constant and the determinant are given by
In the remainder of this section we make use of a linear utility function and assume . Then we obtain:
The bifurcation condition (28) is equivalent to
Choosing as bifurcation parameter the critical value can be calculated from (34)
Note that the steady state values of do not depend on the parameter . By choosing the parameter values , , and the function , we can calculate the steady state and if holds, then the critical value can be calculated from (34). In this case the Jacobian evaluated at the equilibrium has a pair of purely imaginary eigenvalues . Moreover, the crossing velocity
Therefore we conclude that isolated periodic solutions will exist either for or . To determine the stability of the cycles and the direction of the bifurcation further computations either analytically or numerically are necessary. As the analytical proof of the stability of cycles generated by a Hopf bifurcation is rather cumbersome even in very simple models (see, e.g., Ref. 11), we will present a numerical example leading to a stable cycle.
Specify the following functions Furthermore choose the parameter values , and . For these values the steady state is given by
According to (35) the critical value can be computed. To determine the stability of cycles and the direction of the bifurcation further numerical investigations with the code BIFDD (see Ref. 12) were carried out. It turned out that stable cycles occur for . Using the boundary value problem solver COLSYS (see Refs. 13 and 14) a stable limit cycle could be found, e.g., for . This cycle in the -plane is illustrated in figure 1.