Collective oscillations of globally coupled bistable, nonresonant components

Bistable microelectrodes with an S-shaped current-voltage characteristic have recently been shown to oscillate under current control, when connected in parallel. In other systems with equivalently coupled bistable components, such oscillatory instabilities have not been reported. In this paper, we derive a general criterion for when an ensemble of coupled bistable components may become oscillatorily unstable. Using a general model, we perform a stability analysis of the ensemble equilibria, in which the components always group in three or fewer clusters. Based thereon, we give a necessary condition for the occurrence of collective oscillations. Moreover, we demonstrate that stable oscillations may persist for an arbitrarily large number of components, even though, as we show, any equilibrium with two or more components on the middle, autocatalytic branch is unstable.

Figure 1

Circuit representation and $y_k\text{−}u$ curve of an individual bistable component (a) under $u$ control and (b) under $y_k$ control. (c) Circuit representation of many bistable components under $y_{\mathrm{tot}}$ control.

Figure 2

(a) and (b) Sketch of the two cases in which sustained oscillations can occur with a fixed $y_{\mathrm{tot}}$. A red dot represents a cluster of components.

Figure 3

(a) and (b) Equivalent circuits of a state space model with $x\in \mathbb{R}$, close to equilibrium. (c) Exemplary illustration of the three equilibria of Eqs. (30) and (31) for $n=3$, $a=0.05$, $b=0.01$ at $y_{\mathrm{tot}}=0.833n$, marked by the horizontal dashed line. Red circles indicate clusters of up to three components. (d) Equivalent circuits of the entire ensemble at the equilibria in (c). (e) Function $g$ for $u=u_3$ and function $f$ with $x$ fixed to the respective equilibrium values at $u=u_3$.

Figure 4

$y_k$ time series obtained from numerically integrating Eqs. (30) and (31) for (a) $n=3$ and (b) $n=20$. Other parameters [in (a) and (b)]: $a=0.05$, $b=0.01$, $y_{\mathrm{tot}}=0.833n$. Note that an offset of $k/20$ was added to $y_k$ for better visualization of the individual time series in (a) and (b).

Figure 5

Two-cluster limit sets of Eqs. (30) and (31) for $a=0.05,b=0.01$, for a cluster size ratio of $4:1$, valid for any $n$.