4.1 A Discrete Chaotic Map

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

$$\begin{array}{l} x_1(k+1) = x_2(k) + a x_1(k)\\ x_2(k+1) = x_1^2(k) + b \end{array}\quad k = 0, 1, 2, \cdots$$ (8)

where, $a$ and $b$ are parameters. Assume that $\mbox{\boldmath$\ X $}$ is the fundamental solution matrix of the variational equation for Eq.(8), then the characteristic equation becomes

$$\begin{array}{l} \Re\chi = r^2(\cos^2\theta - \sin^2\theta) ... ...ta - \mathop{\rm tr}\nolimits \mbox{\boldmath$ X $} \end{array}$$ (9)

One can calculate easily all factors in the Jacobian matrix Eqs. (6) and (7).

図 1: Bifurcation diagram of Eq. (8)

Figure 1 shows a bifurcation diagram of the system (8). $I^m$ and $G^m$ indicate period-doubling and tangent bifurcation of the $m$-periodic solution, respectively. They are solved by the conventional method[1]. Dashed curves show isoclines with several arguments, and the thick line $N\!S$ shows NS bifurcation obtained by the new method discussed in Sect. 3.

NS bifurcation parameter value is calculated $(a, b) = (2, 0.25)$ in $\theta=0$, this is a codimension-2 bifurcation with the tangent bifurcation $G^1$. As $\theta$ increases gradually to $\pi$, the whole NS bifurcation curve is obtained. In the case of $\theta=\pi$, we have $(a, b)= (-1.75, -2)$, this is a codimension-2 bifurcation point with period-doubling bifurcation. Therefore, the region in which a stable fixed point exist is surrounded by $I^1$, $G^1$, and $N\!S$ curves.

The origin $O$ is a super stable fixed point ($\mu_1=0$, $\mu_2 =0$), and all isoclines are started from this point. Along the isocline, $r$ is gradually changed; $\vert r\vert<1$ keeps the fixed point stable. The intersection of an isocline and $NS$ curve possibly also connect to the cusp point of frequency entrainment region, e.g., cross points of $NS$ curve and isoclines $\pi/2$, $2\pi/5$ and $\pi/3$ are connected with a cusp points of $4$, $5$, and $6$ 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 $\theta$ is rational for $2\pi$, 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; $b = 0.5 a - 0.75$, $-2 \le a \le 2$. 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 $\pi/3$ and $2\pi/5$, respectively. The orbits spend much iterations to approach $\mbox{\boldmath$\ u $}_0$ since $\mbox{\boldmath$\ u $}_0$ 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.

図 2: Phase portrait on the NS bifurcation curve with $\theta = \pi /3$. $a = 1.0$, $b= -0.25$, $\protect\mbox{\boldmath $\ u $}_0 = (-0.5, 0.0)$. The orbit is obtained after 12,000 iterations from $(x_1(0), x_2(0)) = (-0.45, -0.1)$. Arrows show the order of mapping.

図 3: Phase portrait on the NS bifurcation curve with $\theta = 2\pi /5$. $a = 0.618033$, $b= -0.440983$, $\protect\mbox{\boldmath $\ u $}_0 = (-0.5, -0.19098)$. The orbit is obtained after 12,000 iterations from $(x_1(0), x_2(0)) = (-0.490083, -0.215154)$.