Amplitude death in systems of coupled oscillators with distributed-delay coupling

Abstract

This paper studies the effects of coupling with distributed delay on the suppression of oscillations in a system of coupled Stuart-Landau oscillators. Conditions for amplitude death are obtained in terms of strength and phase of the coupling, as well as the mean time delay and the width of the delay distribution for uniform and gamma distributions. Analytical results are confirmed by numerical computation of the eigenvalues of the corresponding characteristic equations. These results indicate that larger widths of delay distribution increase the regions of amplitude death in the parameter space. In the case of a uniformly distributed delay kernel, for sufficiently large width of the delay distribution it is possible to achieve amplitude death for an arbitrary value of the average time delay, provided that the coupling strength has a value in the appropriate range. For a gamma distribution of delay, amplitude death is also possible for an arbitrary value of the average time delay, provided that it exceeds a certain value as determined by the coupling phase and the power law of the distribution. The coupling phase has a destabilizing effect and reduces the regions of amplitude death.

Controlling Synchronization Patterns in Complex Networks

“…In networks with broad distributions, spiking is not possible any longer and initial excitations die out fast, leading to global amplitude death. This behavior is similar to the case of only two coupled oscillators with distributed delay in the coupling, where the regime of amplitude death increases with the width of the delay kernel [Atay, 2003a;Kyrychko et al, 2011Kyrychko et al, , 2013. Large regular networks and to some extent small-world networks are less prone to undergo global amplitude death, since they allow more easily for stable subnetwork spiking or travelling disruptions.…”

Delayed-feedback control: arbitrary and distributed delay-time and noninvasive control of synchrony in networks with heterogeneous delays

“…This is common in a variety of systems where the delay times are given by a continuous distribution [48,[52][53][54]. The model can be written as follows:…”

“…Nevertheless, most studies have assumed that all the interactions occur with the same time delay and, up to now, little is known about stabilizing UPOs and controlling the synchrony patterns in networks coupled with heterogeneous delays. For instance, the dynamics of an array of chaotic logistic maps coupled with random delay times [47], the effects of heterogeneous delays in the coupling of two excitable neural systems [48,49] or a neural network [50], and amplitude death in the Stuart-Landau system coupled with distributed delays [51][52][53] or periodically modulated delay [54] were investigated.…”

Delayed feedback control of unstable steady states with high-frequency modulation of the delay

“…In the nonmodulated case, χ ( ,0) ≡ 1, we have the usual TDFC. For general distributed delay, the delay term exp(− τ 0 ) is replaced in the characteristic equation (13) by the Laplace transform of the delay kernel ρ(θ ) [47]. The function χ ( ,ε) also allows for a compact explanation of the mechanism of variable-delay feedback control (VDFC).…”

“…II, where it is shown that the method could be explored analytically in the domain of a high-frequency modulation of the delay. In this case, we use the formalism of distributed delays [47,48], where the average contribution of the time-varying delay is represented by an integral kernel describing a particular delay distribution. In Sec.…”