Embedding the dynamics of a single delay system into a feed-forward ring

We investigate the relation between the dynamics of a single oscillator with delayed self-feedback and a feed-forward ring of such oscillators, where each unit is coupled to its next neighbor in the same way as in the self-feedback case. We show that periodic solutions of the delayed oscillator give rise to families of rotating waves with different wave numbers in the corresponding ring. In particular, if for the single oscillator the periodic solution is resonant to the delay, it can be embedded into a ring with instantaneous couplings. We discover several cases where the stability of a periodic solution for the single unit can be related to the stability of the corresponding rotating wave in the ring. As a specific example, we demonstrate how the complex bifurcation scenario of simultaneously emerging multijittering solutions can be transferred from a single oscillator with delayed pulse feedback to multijittering rotating waves in a sufficiently large ring of oscillators with instantaneous pulse coupling. Finally, we present an experimental realization of this dynamical phenomenon in a system of coupled electronic circuits of FitzHugh-Nagumo type.

Figure 1

(a) Branch of periodic solutions for a single Van-der-Pol oscillator with delayed feedback (6). Solid and dashed parts indicate stability or instability. (b) Branches of rotating waves for a feed-forward ring of $N=3$ Van-der-Pol oscillators with delayed coupling (6). Colored parts indicate stability for $M=0$ (blue), $M=1$ (red), and $M=2$ (green).

Figure 2

(a) Branch of periodic solutions for a single Van-der-Pol oscillator with delayed feedback (6). Solid and dashed parts indicate stability and instability, respectively. The red thick part of the line corresponds to periodic solutions that are stable when they reappear for sufficiently large delays. Gray lines indicate the resonance conditions $\tau =MT/N$ for $N=6$ and $M=1,\cdots ,5$ and their intersections with the branch indicates resonant periodic solutions leading to rotating waves. Stable rotating waves exist for (b) $M=4$ and (c) $M=5$.

Figure 3

(a) The PRC and (b) the circuitry of the electronic FitzHugh-Nagumo oscillator.

Figure 4

The bifurcation diagram of a single oscillator with pulse delayed feedback (18), period versus the delay. Black solid line corresponds to stable regular regimes, black dashed line to unstable regular regime, green lines to jittering regimes. Red solid lines depict the interspike intervals that constitute the jittering regimes. Stars denote the multijitter bifurcation points. The bifurcation diagram is combined with a family of lines $\tau =MT/N$ with $N=5$. The intersection points correspond to rotating waves in a ring of pulse-coupled oscillators with no delays (20). Hollow points correspond to unstable, solid to stable waves.

Figure 5

Regular rotating waves in Eq. (20): (a) $M=0$, (b) $M=2$, and (c) $M=3$. Points denote the moments of spikes of the oscillators.

Figure 6

Two different jittering rotating waves observed in Eq. (20). The top panels depict the moments of spike generation, the bottom interspike intervals.

Figure 7

Two different jittering rotating waves experimentally observed in a ring of five electronic FitzHugh-Nagumo oscillators. The top panels depict the interspike intervals, bottom ones the output voltages of the oscillators.