Research On The Qualitative Behavior Of Hopfield Neural Networks

In this paper, several key problems, which exist in the designation of Hopfield neural networks, are discussed: global and local stability, convergency speed, the domains of attraction of equilibrium points and the networks' capacity.First, by using the medium theorem of continuous functions, the existence of equilibrium points of continuous-time Hopfield neural networks (CTHN) is proved, then a sufficient and necessary condition of the uniqueness of equilibrium points is presented. To verify the global stability of a generalized CTHN, Lyapunov energy function and LaSalle invariant principle are introduced. By decomposing the CTHN into smaller interconnected subsystems, local stability of CTHN with asymmetric connected weight-matrix is analyzed, and then the convergency speed and attracting range are estimated.In order to simulating CTHN on computers, synchronized discrete-time Hopfield neural networks (DTHN) are discussed. We consider a kind of DTHN with saturated segmented linear functions or threshold functions as activation functions, for which a sufficient condition is given under respectively serial and parallel working modes. We derive the weight-matrix by Hebb learning principle, and then analyze the capacity of DTHN and the methods of reducing the counts of spurious states.From the discussion in this paper we can see that Hopfield neural networks have many dynamical behaviors, and the results can be applied in theoretical analysis and practical problems.