4.3 Forced BVP oscillator

Next example is the forced BVP oscillator:

$$\begin{array}{l} \dot{x} = y + 1.6 x - x^3\\ \dot{y} = -x - 0.77 y + B_0 + B \cos t \end{array}$$ (11)

Figure 6 shows bifurcation diagram of periodic solutions. NS bifurcation curve is connected to period doubling bifurcation curve with $\theta=\pi$ isocline. However, not all frequency entrainment regions touch the NS curve like above two examples. Only $G^6$ touches $NS$ curve and $\pi/6$ curve, see Fig.8 It is not found a relationship between the period of the entrainment region and the argument of isoclines $\pi/2$. In this case, these entrainment regions cannot regarded as Arnold's tongues. To clarify the relationship among NS bifurcation, entrainment regions and isocline for this example is very important.

図 6: Bifurcation diagram of Eq. (11).

図 7: Enlargement of Fig. (6).

図 8: Enlargement of Fig. (6).