Discretization of frequencies in delay coupled oscillators

We study the dynamics of two mutually coupled chaotic oscillators with a time delayed coupling. Due to the delay, the allowed frequencies of the oscillators are shown to be discretized. The phenomenon is observed in the case when the delay is much larger than the characteristic period of the solitary uncoupled oscillator.

Figure 1

(a)–(c) Mean frequencies ${\Omega }_1$ and ${\Omega }_2$ for instantaneously coupled systems $k=0.005,\tau =0$. (d)–(f) The same for delay coupled systems with $\tau =3000$ and $k=0.005$. Different rows correspond to different values of $c$, as indicated in the figure. $c=8.5$ and 7 stand for the chaotic and $c=4$ for the regular case.

Figure 2

Dependence of ${\Omega }_2-{\Omega }_1$ on the detuning. $k=0.005,c=4$. $\tau =$ (a) 0; (b) 3000.

Figure 3

Largest Lyapunov exponents as functions of $\Delta \omega$. The left panel corresponds to the instantaneous coupling and the right one to the delayed case with $\tau =3000$. The dashed vertical lines mark the parameter value at which the phase synchronization transition happens.

Figure 4

Evolution of phase differences of delay coupled oscillators. Parameter values for (a) $c=4$; orbit $C,{\omega }_1=0.998,{\omega }_2=1.002$ (phase synchronized case); $B,{\omega }_1=0.9962,{\omega }_2=1.0038$ (minimal allowed frequency difference); $C,{\omega }_1=0.9961,{\omega }_2=1.0039$. The corresponding points are also indicated in Fig. 2b. Parameter values for (b) $A_1,{\omega }_1=0.9955,{\omega }_2=1.0045$; $B_1,{\omega }_1=0.9954,{\omega }_2=1.0045$; $C_1,{\omega }_1=0.9963,{\omega }_2=1.0037$; $D_1,{\omega }_1=0.9975,{\omega }_2=1.0025$.

Figure 5

Evolution of phase differences modulo $2\pi$ for delay coupled oscillators.

Figure 6

Frequencies of system (2) for different values of $\tau$ versus $\Delta$. (a) the case with $\tau =400$; (b) $\tau =1000$. In both figures $k=0.01$.