Stabilization of a steady state in network oscillators by using diffusive connections with two long time delays

Abstract

The present study shows that diffusive connections with two long-time delays can induce the stabilization of a steady state in network oscillators. A linear stability analysis shows that, if the two delay times retain a proportional relation with a certain bias, the stabilization can be achieved independent of the delay times. Furthermore, a simple systematic procedure for designing the coupling strength and the delay times in the connections is proposed. The procedure has the following two advantages: one can employ time delays as long as one wants and the stabilization can be achieved independently of its network topology. Our analytical results are applied to the well-known double-scroll circuit model on a small-world network.

Amplitude death: The emergence of stationarity in coupled nonlinear systems

“…One application is to the stabilization of fixed points [60,61,62]. Velocity delayed coupling has been also explored in the context of amplitude death [63,64] (see Sec.…”

“…The occurrence of AD in networks of coupled oscillators [29,30,31,32,33,34,35] has been investigated for a variety of topologies ranging from small-world connections [38] to scale-free networks [39] and the ring topology [36,37]. AD as an emergent phenomenon has been seen in networks of coupled maps [55] and in chaotic coupled map lattices [61].…”

“…A systematic procedure for achieving amplitude death involves adjusting the coupling strength and delay times in the connections [61]. Such a procedure has the advantage that stabilization can be achieved independent of the network topology.…”

Amplitude death in nonlinear oscillators with mixed time-delayed coupling

“…In this connection, investigations of delay-induced AD in coupled nonlinear oscillators have been an active area of research for over a decade. For example, it has been proposed that different kinds of delayed interaction among coupled nonlinear oscillators successfully generate AD, such as distributed time delays [39], partial time delays [40], integrative time delays [41], a time-varying delay [42], and multiple time delays [43,44]. Some works addressed the effects of the nature of the coupling on delay-induced AD such asthe coupling gradient [45], the coupling phase [46,47], and the coupling asymmetry [48].…”

Delayed feedback control based on the act-and-wait concept

“…Short delay times cannot be obtained in fast dynamical systems due to the finite operating speed of these electronic devices. To overcome this problem, the multiple DFC [22][23][24] and time-varying DFC [25][26][27] methods were proposed; however, these methods incur the implementation cost of multiple delay times or a high-speed time-varying delay.…”