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# Stability Analysis

Next we linearize the canonical system to analyze the stability behavior of the steady state. This yields

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

hold.

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.

Figure 1: Cycle of fear for Vampires