has explicit parameters and and is a map for these parameters. Then the fixed point of Eq.(1) is given as

(2) |

(3) |

Let
be the Jacobian matrix of around
and be the corresponding characteristic equation:

(4) |

To calculate the location of the fixed point with specified
argument, the following simultaneous equation should be
solved with
by using Newton's method:

Note that all factors of the matrix in Eq.(6) are obtained by solving the variational equation of . If is fixed as an appropriate value within , the isocline is drawn in the parameter plane - by plotting the solution as the incremental parameter changes.

Now suppose that the parameter region in which the fixed point exists is already obtained. Firstly specifying or , we obtain an isocline which splits the parameter region into two parts; the part has is a sink, the other has a stable node. Therefore this isocline can be regarded as a bifurcation curve. For any rational ratio with of , the corresponding isocline shows instantaneous phase around the fixed point and it is deeply related to the NS bifurcation and existence of frequency entrainment regions. Along this isocline, the stability is indexed by , i.e., within , we have a stable sink, and shows a super stable sink.

平成15年6月16日