Generation of cyclic/toroidal chaos by Hopfield neural networks

We discuss the appearance of cyclic and toroidal chaos in Hopfield neural networks. The theoretical results may strongly relate to investigations of brain activities performed by neurobiologists. As new phenomena, extension of chaos by entrainment of several limit cycles as well as the attraction of cyclic chaos by an equilibrium are discussed. Appropriate simulations that support the theoretical results are depicted. Stabilization of tori in a chaotic attractor is realized not only for neural networks, but also for differential equations theory, and this phenomenon has never been reported before in the literature. It is demonstrated that the proposed chaos generation technique cannot be considered as generalized synchronization.