Stochastic switching in delay-coupled oscillators

A delay is known to induce multistability in periodic systems. Under influence of noise, coupled oscillators can switch between coexistent orbits with different frequencies and different oscillation patterns. For coupled phase oscillators we reduce the delay system to a nondelayed Langevin equation, which allows us to analytically compute the distribution of frequencies and their corresponding residence times. The number of stable periodic orbits scales with the roundtrip delay time and coupling strength, but the noisy system visits only a fraction of the orbits, which scales with the square root of the delay time and is independent of the coupling strength. In contrast, the residence time in the different orbits is mainly determined by the coupling strength and the number of oscillators, and only weakly dependent on the coupling delay. Finally we investigate the effect of a detuning between the oscillators. We demonstrate the generality of our results with delay-coupled FitzHugh-Nagumo oscillators.

Figure 1

Graphical determination of the different coexisting frequencies of a single oscillator with delayed feedback [Eq. (2)]. The intersections with the thick decreasing slopes of the sine function correspond to stable orbits and are marked with a circle. The coloring of the circles relates to the probability distribution $p\left[\omega \right(t\left)\right]$ of the corresponding stochastic oscillator (shown in Fig. 3): the probability that the oscillator has a frequency $\omega \left(t\right)\approx {\omega }_k$ is large for the most central frequencies ${\omega }_k\approx {\omega }_0$, marked with a black circle, while the probability to find the system's frequency $\omega \left(t\right)$ close to the outer frequencies ${\omega }_k\approx {\omega }_0\pm \kappa$, marked with an empty circle, is negligible. Parameters are ${\omega }_0=6,\kappa =2,\tau =10$, and $D=0.5$.

Figure 2

(a) Phase evolution $\varphi \left(t\right)-{\omega }_{0}t$ of a Kuramoto oscillator with delayed feedback and noise. We subtracted the natural frequency ${\omega }_{0}t$ for better visibility of the mode hoppings. (b) The frequency measure $\omega \left(t\right)=\left[\varphi \right(t)-\varphi (t-\tau \left)\right]/\tau$ is a good indicator for the mode hoppings. Parameters are ${\omega }_0=6,\kappa =2,\tau =10$, and $D=0.5$

Figure 3

The frequency distributions (gray) for (a) an oscillator with feedback, (b) two coupled identical identical oscillators, and (c) two detuned oscillators. The analytical approximations [Eqs. (6) and (12)] are plotted in black, the (blue) dashdotted lines show the respective Gaussian envelopes. The parameters are $\kappa =2,\tau =10,{\omega }_0=6,D=0.5$, and (c) $\Delta =0.8$.

Figure 4

(a) Logarithm of the residence time distribution $ln(p\left(T\right))$, for a Kuramoto oscillator with delayed feedback, for the orbits with frequencies ${\omega }_2$ and ${\omega }_3$. (b) Mean residence time of the orbits ${\omega }_{2,3,4,5}$ versus their frequency for a single oscillator (upper black dots) together with the theoretical approximation [Eq. (7)] (upper dashed pink curve). The lower pink dots and the lower blue dashed curve represent the mean average residence times of the orbits and their theoretical approximation [Eq. (14)], respectively, for two identical coupled systems. The parameters are $\kappa =3,\tau =10,{\omega }_0=6$, and $D=0.5$.

Figure 5

Potential for a Kuramoto oscillator with feedback [Eq. (5)]. Parameters are ${\omega }_0=6,\kappa =2,\tau =10$.

Figure 6

Graphical determination of the locking frequencies of two (a) identical and (b) nonidentical delay-coupled phase oscillators. Intersection with the thick full (dashed) line correspond stable in-phase (antiphase) orbits. The filling of the black (magenta) circles relates to the relative probability that the in-phase (antiphase) orbit is visited by the corresponding stochastic system, darker labeling corresponds to a higher probability to find a frequency $\omega \left(t\right)\approx {\omega }_k$. The corresponding probability distributions are shown in Figs. 3 and 3. In panel (b) a stable orbit is labeled as in-phase if $-\pi /2<\delta <\pi /2$. Just like for a single feedback oscillator, the frequencies ${\omega }_k\approx {\omega }_0$ are most often visited, the width of the frequency distribution is, however, smaller. Parameters are ${\omega }_0=6,\kappa =2,\tau =10$, and (b) $\Delta =0.8$.

Figure 7

(a) The phase evolutions ${\varphi }_1-{\omega }_{0}t$ (pink) and ${\varphi }_2-{\omega }_{0}t$ (black) of two identical noisy Kuramoto oscillators coupled with delay. We subtracted the natural frequency ${\omega }_0$ for better visibility of the mode hoppings. (b) The time evolution of the frequency $\omega \left(t\right)=[{\varphi }_1\left(t\right)+{\varphi }_2\left(t\right)-{\varphi }_1(t-\tau )-{\varphi }_2(t-\tau )]/\left(2\tau \right)$ for two coupled oscillators (black), together with the phase differences $x_1\left(t\right)/\tau =[{\varphi }_1\left(t\right)-{\varphi }_2(t-\tau )]/\tau$ (upper pink curve) and $x_2\left(t\right)/\tau ={\varphi }_2\left(t\right)-{\varphi }_1(t-\tau )$ (lower blue curve). The dashed lines indicate the mode hoppings. Parameters are ${\omega }_0=6,\kappa =3,\tau =10$, and $D=0.5$.

Figure 8

Two-dimensional potential for two coupled Kuramoto oscillators, without (a) and with (b) detuning. The arrows indicate the two pathways for a transition between two frequencies; thicker arrows correspond to more probable pathways. Parameters are ${\omega }_0=6,\kappa =2,\tau =10$, and (b) $\Delta =0.8$

Figure 9

(a) Time trace and (b) frequency evolution of a FitzHugh-Nagumo oscillator with feedback. Parameters are $\epsilon =0.01,a=0.9,k=0.2,\tau =20$, and $\tilde{D}=0.0145$.

Figure 10

Frequency distribution $p\left(\omega \right)$ for one (a) and two (b) coupled FitzHugh-Nagumo oscillators with delay, with $\epsilon =0.01,a=0.9,k=0.2,\tau =40$, and $\tilde{D}=0.0145$. The blue dashed curve shows the Gaussian envelope for the corresponding Kuramoto oscillator(s) with ${\omega }_0=2.55,D=0.2$, and $\tau =40$. In panel (c) the corresponding average residence times are shown for one (upper black dots) and two (lower pink dots) oscillators, the dashed curves represent the Kuramoto approximation for one [Eq. (7), upper pink curve] and two [Eq. (14), lower blue curve] elements for ${\omega }_0=2.55,D=0.2,\tau =40$, and $\kappa =0.815$.