Enhanced stability and root detection in a derivative-free Steffensen algorithm for nonlinear dynamical systems
Alexandre Wagemakers1, Vipul Periwal2
1Nonlinear Dynamics, Chaos and Complex Systems Group, Departamento de Biología y Geología, Física aplicada y Química inorgánica, Universidad Rey Juan Carlos, Tulipán, Móstoles, 28933 Madrid, Spain.
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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.
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