• A Topological Framework for Detecting Chaos in Erbium-Doped Fiber Lasers

Understanding dynamical transitions between periodic and chaotic regimes in nonlinear fiber lasers is essential for stability control and performance optimization. In this study, we investigate regime transitions in an erbium-doped fiber laser (EDFL) by integrating nonlinear dynamical analysis with Topological Data Analysis (TDA). The governing differential equations are solved numerically to generate time series across varying control parameters, and Lyapunov exponents are computed to identify stability boundaries and chaotic behavior. To capture the global geometric structure of the underlying attractors, we apply persistent homology to the time series and compute persistence diagrams and Betti curves as quantitative topological descriptors. In addition, the Mapper algorithm is used to construct graph representations of the attractors, enabling the extraction of topological and statistical features that characterize structural organization. Our results reveal clear topological transitions between periodic and chaotic regimes, with chaotic dynamics exhibiting significantly higher geometric complexity and distinct structural patterns in the corresponding graphs. Importantly, the evolution of topological invariants shows strong agreement with Lyapunov-based chaos indicators while simultaneously capturing global attractor reorganization that is not accessible through classical local stability measures alone. These findings demonstrate that TDA provides a robust and complementary geometric framework for detecting, classifying, and understanding dynamical regime transitions in nonlinear fiber laser systems, with potential applications in stability monitoring and control of advanced photonic devices.