Synchronization in a ring of four mutually coupled van der Pol oscillators: Theory and experiment

We investigate different states of synchronization in a ring of four mutually coupled van der Pol oscillators. The stability analysis and numerical simulation are performed to determine the suitable coupling parameters leading to high-quality synchronization. The consequences of parameter mismatch are also highlighted. Experimental realization is then used to show the existence of complete and partial synchronization.

Figure 1

Maximal values taken by the three modes $y_1$ (full line), $y_2$ (dashed), and $y_3$ (dotted).

Figure 2

The coupling coefficient boundary for complete synchronization vs the amplitude $F$ of the external excitation for $d_j=0.31$. Full line shows analytical result and dashed line shows numerical result.

Figure 3

Circuit of a ring of four mutually coupled van der Pol oscillators using operational amplifiers and multipliers.

Figure 4

(Color online) Phase plane of the first oscillator in the regular state (without external excitation) before the coupling; (a) experimentally obtained, (b) numerically obtained.

Figure 5

(Color online) Phase planes showing the absence of synchronization in the ring. Experimental results are shown on the left and numerically simulated plot on the right; (a) $x_3$ vs $x_1$, (b) $x_4$ vs $x_2$.

Figure 6

(Color online) Phase planes of complete synchronization in the system. Experimental results are shown on the left and numerically simulated plot on the right; (a) $x_3$ vs $x_1$, (b) $x_4$ vs $x_2$, (c) $x_2$ vs $x_1$.

Figure 7

(Color online) Phase plane of the first oscillator in a complete synchronization in regular state; (a) experimentally obtained, (b) numerically obtained.

Figure 8

(Color online) Phase plane of the first oscillator in chaotic state (with external excitation) before the coupling; (a) experimental results (b) numerical results.

Figure 9

(Color online) Phase planes illustrating the absence of synchronization in the ring. Experimental results are shown on the left and numerically simulated plot on the right; (a) $x_3$ vs $x_1$, (b) $x_4$ vs $x_2$, (c) $x_2$ vs $x_1$.

Figure 10

(Color online) Phase planes illustrating the partial synchronization in the ring. Experimental results are shown on the left and numerically simulated plot on the right; (a) $x_3$ vs $x_1$, (b) $x_4$ vs $x_2$, (c) $x_2$ vs $x_1$.

Figure 11

(Color online) Phase planes illustrating complete synchronization in the ring. Experimental results are shown on the left and numerically simulated plot on the right; (a) $x_3$ vs $x_1$, (b) $x_4$ vs $x_2$, (c) $x_2$ vs $x_1$.

Figure 12

(Color online) Phase plane of the first oscillator when the ring is in the complete synchronization state; (a) experimentally obtained, (b) numerically obtained.