Low-dimensional chaos induced by frustration in a non- monotonic system

– We report a novel mechanism for the occurrence of chaos at the macroscopic level induced by the frustration of interaction, namely frustration-induced chaos, in a nonmonotonic sequential associative memory model. We succeed in deriving exact macroscopic dynamical equations from the microscopic dynamics in the case of the thermodynamic limit and prove that two order parameters dominate this large-degree-of-freedom system. Two-parameter bifurcation diagrams are obtained from the order-parameter equations. Then we analytically show that the chaos is low-dimensional at the macroscopic level when the system has some degree of frustration, but that the chaos definitely does not occur without the frustration. The occurrence of chaotic behaviors at the macroscopic level has been extensively investigated, and thus its mechanisms can be classified into several classes according to their typical systems. Examples of the first class are the globally coupled map (GCM) and chaotic neural networks whose processing units are themselves chaotic, and the macroscopic behaviors are also chaotic [1, 4, 2, 3]. Since the processing units are chaotic, the chaos at the macroscopic level seems to be high-dimensional. However, large-scale computer simulations show that the dimension of the chaos in the GCM can be lower than the degree of freedom [4]. In the other class of mechanisms, the processing units are not chaotic themselves, and thus resultant macroscopic chaotic behaviors would seem to be induced by other properties. A typical example of this class is that of neural networks. Since their processing units are simple binary units, their chaotic behaviors should be induced by additional mechanisms, that is, synaptic pruning, synaptic delay, thermal noise, sparse connections and/or so on [5, 6, 7]. Frustration is worth notice as inducing chaos. For continuous systems, chaotic behaviors can be analyzed in some small networks with frustration of interaction at the microscopic level [8, 9, 10] and random neural networks at the macroscopic level [11]. Here, we report a novel mechanism of the occurrence of chaos at the macroscopic level induced by the frustration of interaction, namely frustration-induced chaos, for discrete system.