Enhanced stability and root detection in a derivative-free Steffensen algorithm for nonlinear dynamical systems

We present a stabilized, derivative-free root-finding algorithm inspired by Steffensen's method, tailored to uncover fixed points in high-dimensional nonlinear systems. The method retains second-order convergence while significantly enhancing stability across diverse dynamical regimes. We demonstrate that a carefully chosen nonlinearity in the divided difference estimate extends convergence basins and reduces iteration counts. Applications include systems from theoretical neuroscience, where the proposed method recovers exponentially more fixed points, revealing complex attractor landscapes. This algorithm offers an efficient, memory-lean alternative for stability-focused root finding in chaotic and multistable systems.