A Canard Mechanism for Localization in Systems of Globally Coupled Oscillators

Abstract

Abstract. Localization in a discrete system of oscillators refers to the partition of the population into a subset that oscillates at high amplitudes and another that oscillates at much lower amplitudes. Motivated by experimental results on the Belousov-Zhabotinsky reaction, which oscillates in the relaxation regime, we study a mechanism of localization in a discrete system of relaxation oscillators globally coupled via inhibition. The mechanism is based on the canard phenomenon for a single relaxation oscillator: a rapid explosion in the amplitude of the limit cycle as a parameter governing the relative position of the nullclines is varied. Starting from a parameter regime in which each uncoupled oscillator has a large amplitude and no other periodic or other stable solutions, we show that the canard phenomenon can be induced by increasing a global negative feedback parameter γ, with the network then partitioned into low and high amplitude oscillators. For the case in which the oscillators are synchronous within each of the two such populations, we can assign a canard-inducing critical value of γ separately to each of the two clusters; localization occurs when the value for the system is between the critical values of the two clusters. We show that the larger the cluster size, the smaller is the corresponding critical value of γ, implying that it is the smaller cluster that oscillates at large amplitude. The theory shows that the above results come from a kind of self-inhibition of each cluster induced by the local feedback. In the full system, there are also effects of interactions between the clusters, and we present simulations showing that these nonlocal interactions do not destroy the localization created by the self-inhibition.

Dynamic mechanisms of generation of oscillatory cluster patterns in a globally coupled chemical system

“…11,12,16,17,19,36,[52][53][54][55][56][57][58][59][60][61][62][63][64] The results presented in this paper and the methods we and other authors have developed 18,31,32,65,66 can be applicable to the understanding of the underlying mechanisms that govern the dynamics in these systems.…”

“…11,12 Oscillatory clusters are sets of oscillators, or domains, in which nearly all elements in a given domain oscillate with the same amplitude and phase. [14][15][16][17][18] The three most relevant cluster patterns observed in the globally coupled BZ reaction are two-phase, three-phase, and localized clusters. 11 The former two consist of two or three clusters oscillating synchronously out-of-phase.…”

“…In previous work, 16 we have investigated the mechanism of generation of localized clusters in a modified version of the FHN model with global coupling. More recently, we have investigated the mechanism of generation of phase locked clusters in a globally coupled FHN model 18 (see also the accompanying technical report in Ref.…”

“…In particular, the global coupling term in the FHN models investigated in Refs. 16 and 18 is linear while the global coupling term in the Oregonator is nonlinear and includes a chemical parameter responsible for this nonlinearity. The additional complexity of the Oregonator translates into more complex oscillatory patterns under the effects of global coupling.…”

Multistate network for loop searching system with self-recovery property

“…Thus, in this article, we employ the sigmoid FitzHugh-Nagumo system [15], which shows SN bifurcation and PIR, to define node states. SN bifurcation is a local bifurcation where two equilibrium points collide and annihilate.…”