Fractional Neutrosophic Dynamics Entropy, Recurrence, and Applications in Chaotic Systems and Signal Analysis

Abstract

This paper introduces a novel framework for analyzing dynamical systems by integrating the theory of neutrosophic MR-metric spaces with generalized fractional calculus. We define neutrosophic dynamical systems and establish a Neutrosophic Entropy Theorem, proving that the classical topological entropy can be expressed purely in terms of the neutrosophic membership functions of truth, indeterminacy, and falsity. Furthermore, we present a Fractional Neutrosophic Recurrence Theorem, demonstrating that for systems generated by a generalized fractional derivative, almost every point exhibits positive lower fractional density of returns to any neutrosophic neighborhood. The theoretical findings are illustrated through detailed examples, including rotations on the circle, Bernoulli shifts, and fractional relaxation processes. The practical utility of the framework is demonstrated through diverse applications: entropy estimation from chaotic time series, anomaly detection in network security and ECG signals, regime identification in financial markets, and the analysis of climate dynamics. These applications highlight the robustness and versatility of the proposed neutrosophic-fractional approach in quantifying uncertainty, memory, and complex recurrence patterns in real-world data. Comparative analysis with classical methods demonstrates that the neutrosophic-fractional approach achieves superior performance in anomaly detection (AUC improvement of 8-12%) and entropy estimation (error reduction of 15-20%).