   Next: 3 Calculation of NS Up: Calculation of the Isocline Previous: 1 Introduction

# 2 Description of the Problem

Let be a map described as follows: (1) has explicit parameters and and is a map for these parameters. Then the fixed point of Eq.(1) is given as (2)

Assume now that this fixed point has at least a pair of complex conjugate multipliers described by: (3)

where, is the radius, is the imaginary unit and is the argument. If is specified, the location and the parameter value of the fixed point forms an isocline as the another parameter value changes.

Let be the Jacobian matrix of around and be the corresponding characteristic equation: (4)

where and show the real and imaginary part of the characteristic equation.

To calculate the location of the fixed point with specified argument, the following simultaneous equation should be solved with by using Newton's method: (5)

Then the Jacobian matrix of the Eq.(5) is written as follows. (6)

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.   Next: 3 Calculation of NS Up: Calculation of the Isocline Previous: 1 Introduction
tetsushi