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## 4.1 A Discrete Chaotic Map

Firstly we show an example of the result for a decrete system described as follows: (8)

where, and are parameters. Assume that is the fundamental solution matrix of the variational equation for Eq.(8), then the characteristic equation becomes (9)

One can calculate easily all factors in the Jacobian matrix Eqs. (6) and (7). Figure 1 shows a bifurcation diagram of the system (8). and indicate period-doubling and tangent bifurcation of the -periodic solution, respectively. They are solved by the conventional method. Dashed curves show isoclines with several arguments, and the thick line shows NS bifurcation obtained by the new method discussed in Sect. 3.

NS bifurcation parameter value is calculated in , this is a codimension-2 bifurcation with the tangent bifurcation . As increases gradually to , the whole NS bifurcation curve is obtained. In the case of , we have , this is a codimension-2 bifurcation point with period-doubling bifurcation. Therefore, the region in which a stable fixed point exist is surrounded by , , and curves.

The origin is a super stable fixed point ( , ), and all isoclines are started from this point. Along the isocline, is gradually changed; keeps the fixed point stable. The intersection of an isocline and curve possibly also connect to the cusp point of frequency entrainment region, e.g., cross points of curve and isoclines , and are connected with a cusp points of , , and periodic region, respectively. Then the diagram forms Arnold's tongue structure. But not all cusp points connect in this way, and we will mention it in Sec. 4.3. If the argument is rational for , the linearized space around the stable fixed point just before NS bifurcation preserves These cross points on the parameter plane can be solved with any-precision accuracy because of convergence ability of Newton's method. Indeed, it is noted that NS bifurcation parameter values can be solved analytically for this system; , . Thus we can confirm the validity and ability of the proposed method by comparing with this analytical solution.

Figures 2 and 3 show the orbits started from the neighborhood of the fixed point on the NS bifurcation parameter values with the angular and , respectively. The orbits spend much iterations to approach since becomes a center at this parameter values. From these figures, we can see that the linear space around the fixed point is deeply related to the periodic motion after the NS bifurcation.     Next: 4.2 A couple of Up: 4 Examples Previous: 4 Examples
tetsushi