Hopf Bifurcation in a Chaotic Associative Memory

This paper has two basic objectives: the first is to investigate Hopf Bifurcation in the internal state of a Chaotic Associative Memory (CAM). For a small network with three neurons, resulting in a six-dimensional Equation of State, the existence and stability of Hopf Bifurcation were verified analytically. The second objective is to study how the Hopf Bifurcation changed the external state (output) of CAM, since this network was trained to associate a dataset of input–output patterns. There were three main differences between this study and others: the bifurcation parameter was not a time delay, but a physical parameter of a CAM; the weights of interconnections between chaotic neurons were neither free parameters nor chosen arbitrarily, but determined in the training process of classical AM; the Hopf Bifurcation occurred in the internal state of CAM, and not in the external state (input–output network signal). We present three examples of Hopf Bifurcation: one neuron with supercritical bifurcation while the other two neurons do not bifurcate; two neurons bifurcating into a subcritical bifurcation and one neuron does not bifurcate; and the same example as before, but with a supercritical bifurcation. We show that the presence of a limit cycle in the internal state of CAM prevents output signals from the network converging towards a desired equilibrium state (desired memory), although the CAM is able to access this memory.