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The systems described nonlinear differential and difference equations
frequently exhibit quasi-periodic solution (torus) via Neimark-Sacker
(abbr. NS)
bifurcation. Besides this quasi-periodic solution is often locked
into various periodic motions by changing parameters. This phenomenon
is called frequency entrainment and characterized as Arnold's tongue
structure of in the two-parameter bifurcation diagram[1]
[2].
Sometimes chaotic oscillation is given from the quasi-periodic
solution via the torus breakdown. It is important to obtain
parameter values of the Neimark-Sacker bifurcation for
understanding the route to chaos.
Several numerical methods for NS bifurcation are established
(summarized in Ref.[3]),
and the relationship between entrainment region of periodic
solutions and NS bifurcation are investigated in detail[1].
The keyword of this relationship is `argument' of the complex multipliers
of the fixed point concerned with the NS bifurcation.
In this paper, we propose a numerical method to obtain
the accurate location and the parameter value
of the fixed point with a specified
argument of complex multipliers. As the parameter changes,
such solution draws as an isocline in the parameter
space. We can calculate isoclines corresponding to
the specified arguments, then the stability
and instantaneous phase around the fixed point is clarified.
As an application, we also propose a new calculation method
of NS bifurcation parameter values with specified argument.
In addition, cross points of a cusp in a periodic
entrainment region and the NS bifurcation curve is
identified by this application.

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**Up:** Calculation of the Isocline
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*tetsushi *

平成15年6月16日