1 Introduction

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.