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Chaotic Synchronization of memristive neurons: Lyapunov function versus Hamilton function

We study the dynamical behaviors of this improved memristive neuron model by changing external harmonic current and the magnetic gain parameters. The model shows rich dynamics including periodic and chaotic spiking and bursting, and remarkably, chaotic super-bursting, which has greater information encoding potentials than a standard bursting activity. Based on Krasovskii-Lyapunov stability theory, the sufficient conditions (on the synaptic strengths and magnetic gain parameters) for the chaotic synchronization of the improved 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 asymptotic 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 non-zero (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 clearly therefore paves an alternative way to determine the asymptotic 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.