A novel approach regarding the fixed points of repelling nature

Abstract

Although there are various numerical techniques to find the stable equilibria of a nonlinear system in scientific computing, no widely-used computational approach has been encountered to discover the unstable equilibria of a system in the literature. This study aims at presenting a new approach to uncover the equilibrium positions of a dynamical system exhibiting a repelling nature. A newly developed algorithm called the reversed fixed point iteration method (RFPIM) is presented to find the unstable equilibrium positions of a nonlinear system. The current method, keeping the real features of a problem, is able to uncover the behavior of a nonlinear system near the unstable equilibria without facing any conventional drawbacks. In this respect, it is mathematically proven and numerically observed that the present approach has various superiorities against the conventional approach. Some illustrative examples regarding the root-finding problem, population dynamics, integral equations, control problems, and chaotic systems have been utilized to test the current method. The results reveal that in finding the unstable equilibria of a differential operator, the use of relatively larger step sizes together with the RFPIM leads to more accurate results as opposed to common expectations. The findings have clearly shown that less consumption of CPU time and storage space could be succeeded by using larger step sizes via the RFPIM.