High-order synchronization, transitions, and competition among Arnold tongues in a rotator under harmonic forcing

We consider a rotator whose equation of motion for the angle $\theta$ consists of the zeroth and first Fourier modes. Numerical analysis based on the trailing of saddle-node bifurcations is used to locate the $n:1$ Arnold tongues where synchronization occurs. Several of them are wide enough for high-order synchronization to be seen in passive observations. By sweeping the system parameters within a certain range, we find that the stronger the dependence of $\dot{\theta }$ on $\theta$, the wider the regions of synchronization. Use of a synchronization index reveals a vast number of very narrow $n:m$ Arnold tongues. A competition phenomenon among the tongues is observed, in that they “push” and “squeeze” one another: as some tongues widen, others narrow. Two mechanisms for transitions between different $n:m$ synchronization states are considered: slow variation of the driving frequency, and the influence of low-frequency noise on the rotator.

Figure 1

(Color online) Arnold tongues for the externally driven rotator (6), with parameters $\left(a_0,a_1\right)$ equal to (7.08116,2) in (a); (9.85714,7.14286) in (b); and (16.46637,15) in (c). The frequency $\nu$ and amplitude $B$ of the external driving are plotted on the abscissa and ordinate axes, respectively. The tongues are $n:1$, with $n$ increasing from 1 to 9 as we move from right to left.

Figure 2

(Color online) Widths of the Arnold tongues as functions of parameters $a_1$ and ${\nu }_0$: (a) 1:1; (b) 1:2; (c) 1:3; and (d) 1:4. They are expressed in units of the driving frequency $\nu$, at driving amplitude $B=1.5$. Note the differences in ordinate scale.

Figure 3

(Color online) Detail of the synchronization regions inside a small interval. The parameters $\left(a_0,a_1\right)$ are (a) (9.85714, 7.14286); (b) (7.08116,2). The tongues are, from left to right: (a) 5:1, 33:8, 41:10, 4:1, 39:10, 35:9, 31:8, 27:7, 23:6, 19:5, and 34:9; (b) 5:1, 29:7, 33:8, 37:9, 41:10, 4:1, 39:10, 35:9, 31:8, and 27:7.

Figure 4

(Color online) (a) Tongues 4:1 (right) and 33:8 (second from the right) for parameters $\left(a_0,a_1\right)=\left(9.85714,7.14286\right)$. For clarity, the width of the 33:8 tongue in (a) is shown as larger than it really is: this plot is just intended to show the location. (b) Width of the 33:8 tongue, plotted as a function of the driving amplitude $B$. The location of the 33:8 tongue in (a), and its width in (b), were obtained by trailing the bifurcations in their “flexible” meaning, using a threshold of 0.001.

Figure 5

(Color online) An $n:1$ synchrogram for the forced rotator while varying the driving frequency. Several $n:1$ synchronization epochs can be observed. The straight line plots the instantaneous frequency of the external driving (right-hand ordinate axis).

Figure 6

(Color online) Sections of an $n:2$ synchrogram for the forced rotator while varying the driving frequency, illustrating a 5:2 synchronization epoch in (a) and a 3:2 epoch in (b). The straight line plots the instantaneous frequency of the external driving (right-hand ordinate axis).

Figure 7

(Color online) (a) Transitions between 5:1 and 4:1 synchronization states, caused by the added low-frequency component. (b) A 9:2 epoch for time $\sim 600$ in the transition between 5:1 and 4:1.