A Canard Mechanism for Localization in Systems of Globally Coupled Oscillators

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.