Appearance of an Invariant Manifold and On-Off Intermittency in Coupled-Maps: Application to a 3-Valued Neuron

Abstract

Yamada and Fujisaka's three· logistic-maps model, whose maps are slightly different from each other, is studied theoretically and numerically_ It is found that motion of the model exhibits a nearly-synchronized state whose time series manifests on-off intermittency, and a one-dimensional invariant manifold appears at a critical coupling constant. Association in a network of 3-valued neurons, where the model acts the role of the neuron, is also studied. 387 Recently, a considerable number of studies have been made of chaos synchroniza­ tion 1 )-20) and on-off intermittency8),21)-29) in chaotic systems. However, only a few such attempts 10 ),14),30) have so far been made at nearly-synchronized states which can be observed in a coupled-maps system whose maps are slightly different from each other. The nearly-synchronized state has been treated as synchronization with noise, and with coherency inherent in the noise. In a previous paper,31) we found a one­ dimensional invariant manifold of a coupled two-logistic-maps model (Yamada and Fujisaka's2) model of two maps). In the present paper, we study a nearly­ synchronized state inYamada and Fujisaka's three-logistic-maps model. The coupled system has been utilized for a model neuron, 32 1 where synchronous, partially synchronous and asynchronous are represented by 1, 0 and -1, respectively. A network of the neurons can act as Nakano's33) associatron. The equations of motion of the Yamada-Fujisaka model are expressed as