Basins of attraction in a ring of overdamped bistable systems with delayed coupling

Abstract

Theoretical and experimental works reveal that coupling similar overdamped bistable systems can lead, under certain conditions that depend on the topology of connections and the number of units, to self-induced large-amplitude oscillations that emerge through a global bifurcation of heteroclinic connections between saddle-node equilibria. This critical observation has led to new mechanisms for weak (compared to the energy barrier height) signal detection and amplification. While the mathematical models and related devices governed by bistable potential functions may assume instantaneous coupling, in practice we must account for the fact that even high-speed, high-precision, circuit components can introduce a delay in the coupling signal. Thus, in this manuscript we investigate the behavior of a ring of overdamped bistable systems with delayed nearest-neighbor connections. Related work, without delay, shows that large-amplitude oscillations and nontrivial synchronous equilibria can coexist near the onset of the oscillations. Our study shows that a delay-induced Hopf bifurcation occurs from the synchronous equilibria but, generically, the small amplitude oscillations that appear are unstable. Thus, delay has the effect of decreasing the size of the basin of attraction of nontrivial synchronous equilibria, which in turn, makes the basin of attraction of the stable large-amplitude oscillations larger. Collectively, this is a positive effect because the sensor device depends mainly on large amplitude oscillations, so a small delay can make it easier to induce the device to oscillate on its own. As a “test bed”, we use the model equations of a CCFM device with N fluxgates. The results are, however, generic and applicable to all rings of overdamped bistable units unidirectionally coupled.