Estimation of Parameter in Chaotic Systems Using Invariant Features of Recurrence Plots
Abstract
Chaotic dynamical systems, known for their extreme sensitivity to initial conditions, present a significant challenge for accurate parameter estimation. This paper introduces a robust Objective Function (OF) that uses the invariant topological properties of the system’s attractor, derived from Recurrence Plots (RPs), to address this challenge. The OF quantifies similarity by combining the Pearson correlation coefficient of the normalized probability distributions of the reference and model time series with the Euclidean distance among three main Recurrence Quantification Analysis (RQA) properties, which are recurrence rate, determinism, and mean diagonal line length. These RQA metrics capture the system’s deterministic structure, regardless of the original trajectory. The proposed RQA-based method accurately estimates parameters for various chaotic models, including the one-dimensional Baghdadi map, the two-dimensional Lozi map, and the three-dimensional continuous Rössler flow. The OF demonstrates high accuracy and robustness when evaluated against the Baghdadi map with additive white Gaussian noise, indicating its appropriateness for real-world applications. The results consistently reveal unique maxima at the true parameter values, validating the method’s power for reliable parameter identification in complex nonlinear dynamics.