One of the problems arising in the analysis of periodic nonlinear dynamical systems is to determine multiple equilibria and periodic solutions. These may be formulated in terms of finding zeros of a nonlinear vector function. The iterative methods commonly employed for this purpose concentrate on locating a single zero from a given initial guess. The homotopic and continuation methods are also used for locating multiple zeros but they primarily have been used to follow a single solution branch with a parameter variation. The purpose of this paper is to explore a method based on a differential equation for finding multiple zeros. The equilibria of this differential equation correspond with the zeros of a given vector function. Since the equilibria is asymptotically stable, they can be obtained by examining a large number of trajectories. For this purpose, the cell-to-cell mapping is ideally suited for finding the equilibria, and hence the zeros of a nonlinear vector function. Preliminary results for finding multiple fixed points of a highly nonlinear map are provided in this paper.