Transition from homogeneous to inhomogeneous steady states in oscillators under cyclic coupling

Abstract

We report a transition from homogeneous steady state to inhomogeneous steady state in coupled oscillators, both limit cycle and chaotic, under cyclic coupling and diffusive coupling as well when an asymmetry is introduced in terms of a negative parameter mismatch. Such a transition appears in limit cycle systems via pitchfork bifurcation as usual. Especially, when we focus on chaotic systems, the transition follows a transcritical bifurcation for cyclic coupling while it is a pitchfork bifurcation for the conventional diffusive coupling. We use the paradigmatic Van der Pol oscillator as the limit cycle system and a Sprott system as a chaotic system. We verified our results analytically for cyclic coupling and numerically check all results including diffusive coupling for both the limit cycle and chaotic systems.

Analytic Solution for a Complex Network of Chaotic Oscillators

“…This inhomogeneous steady state is an example of oscillator death [12]. A transition from amplitude to oscillator death was reported early on by Turing [13], but this phenomenon has received significant attention more recently [14][15][16][17]. Increasing even further to 0.9 brings about a revival of oscillation as shown in Figure 3e.…”

“…Yet another organized global state is shown in Figure 3c where all nodes, hub and leaves settle to the same steady state value with = 0.6. Equations (17) and (18) imply the steady state value of u h and u n for all n is 1 − 1/r. Accordingly, Equations (14) and (15) can be summed explicitly to show the steady state evolution of the continuous time network.…”

Synchronization of cyclic coupled hyperchaotic systems and its circuit implementation

“…Several types of synchronization [Rosenblum et al, 1997;Pecora and Carroll, 1990;Ojo et al, 2016a;Ojo et al, 2011;Ojo et al, 2013b;Gasri et al, 2018], methods of synchronization [Yang, 2012;Njah, 2011;Lu et al, 2013;Ojo et al, 2013a;Adegoke et al, 2013;Yang, 2012], and schemes [Ouannas, 2014;Zhang and Deng, 2014;Dongmo et al, 2018;Ojo et al, 2016b;Yu et al, 2013;Adelakun et al, 2017] have been developed in order to obtain most efficient applications of synchronization to natural and artificial systems. The search for the best coupling techniques to achieve excellent synchronization efficiency has led to the discovery of different coupling techniques such as unidirectional coupling [Adelakun et al, 2014;Rulkov et al, 1995;Ojo et al, 2013c], bidirectional coupling [Kumar et al, 2016;Khan and Poria, 2012], cross coupling [Zhang and Deng, 2014;Mengfei et al, 2015], cyclic coupling [Bera et al, 2016;Olusola et al, 2013;Egunjobi et al, 2018] and others [Liu et al, 2011;Iqbal et al, 2018;Belykh et al, 2005;Kandel et al, 2000]. The types of coupling technique and topology determine the level of stability or instability of synchronization of coupled nonlinear systems.…”

“…Despite several advantages of cyclic coupling synchronization, not many research works have reported in this direction. [Bera et al, 2016;Olusola et al, 2013;Egunjobi et al, 2018;Adelakun et al, 2018].…”

Restoration of oscillation in network of oscillators in presence of direct and indirect interactions

“…The mechanisms leading to these two oscillation quenching phenomena are mainly detuning of oscillators under strong coupling [3][4][5], conjugate coupling [6], mean field coupling [7], nonlinear coupling [8], additional repulsive link [9], environmental coupling [10,11] and also sufficient amount of delay in the coupling form [12]. The diverse routes of transition from AD to OD have also been reported [13][14][15][16][17][18].…”