Effect of positive feedback on Devil's staircase input-output relationship

Abstract

We consider emerging hysteresis behaviour in a closed loop systemthat includes a nonlinear link $f$ of the Devil's staircase (Cantorfunction) type and a positive feedback. This type of closed loopsarises naturally in analysis of networks where local ``negative''coupling of network elements is combined with ``positive'' couplingat the level of the mean-field interaction (in the limit case whenthe impact of each individual vertex is infinitesimal, while thenumber of vertices is growing). For the Cantor function $f$, takenas a model, and for a monotonically increasing input, we present thecorresponding output of the system explicitly, showing that theoutput is piecewise constant and has a finite number of equal jumps.We then discuss hysteresis loops of the system for genericnon-monotone inputs. The results are presented in the context of differential equations describingnonlinear control systems with almost immediate linear feedback, i.e., in the limit where the time of propagation of the signalthrough the feedback loop tends to zero.