Chaotic synchronization of memristive neurons: Lyapunov function versus Hamilton function

Abstract

AbstractIn this paper, we consider a 5-dimensional Hindmarsh–Rose neuron model. This improved version of the original model shows rich dynamical behaviors, including a chaotic super-bursting regime. This regime promises a greater information encoding capacity than the standard bursting activity. Based on the Krasovskii–Lyapunov stability theory, the sufficient conditions (on the synaptic strengths and magnetic gain parameters) for stable chaotic synchronization of the model are obtained. Based on Helmholtz’s theorem, the Hamilton function of the corresponding error dynamical system is also obtained. It is shown that the time variation of this Hamilton function along trajectories can play the role of the time variation of the Lyapunov function—in determining the stability of the synchronization manifold. Numerical computations indicate that as the synaptic strengths and the magnetic gain parameters change, the time variation of the Hamilton function is always nonzero (i.e., a relatively large positive or negative value) only when the time variation of the Lyapunov function is positive, and zero (or vanishingly small) only when the time variation of the Lyapunov function is also zero. This, therefore, paves an alternative way to determine the stability of synchronization manifolds and can be particularly useful for systems whose Lyapunov function is difficult to construct, but whose Hamilton function corresponding to the dynamic error system is easier to calculate.