Further studies on zhang neural-dynamics and gradient dynamics for online nonlinear equations solving

By following Zhang et al´s neural-network design-method, a special kind of neural dynamics is generalized, developed and investigated in this work for online solution of nonlinear equation f(x) = 0. Different from conventional gradient-based dynamics (or termed, gradient-dynamics, GD), the resultant Zhang neural-dynamics (or termed, Zhang dynamics, ZD) is designed based on the elimination of an indefinite error-function (rather than the elimination of a square-based positive energy-function usually associated with gradient-based approaches). For comparative purposes, the gradient dynamics is developed and exploited as well for solving online such nonlinear equations. Conventionally and geometrically speaking, the gradient dynamics evolves along the surface descent direction (specifically, the tangent direction) of the square-based energy-function curve; but, how does Zhang neural-dynamics evolve? Together with our previous studies on gradient dynamics and Zhang dynamics, in this paper we further analyze, investigate and compare the characteristics of such two dynamics. Computer simulation results via three illustrative examples might show us some interesting implications, in addition to the efficacy of Zhang dynamics on nonlinear equations solving.