On an application of dynamical systems theory to determine all the zeros of a vector function

AbstractA continuous-time dynamical system is constructed and analyzed in this paper to help locate all the zeros of nonlinear vector functions. Lyapunov stability technique is used to show that the zeros of the vector function become asymptotically stable equilibria for this dynamical system. A strict Lyapunov function is constructed to indicate local stability and to estimate the domains of attraction of all equilibria. Manifolds on which the Jacobian of the vector field is singular play a significant role in characterizing the global behavior of the system. Examples are provided to illustrate the extent of the theory. New computational techniques to determine all the zeros of a vector function can be developed based on the dynamical aspects of the theory given in this paper