Advanced Noise Reduction in Chaotic Maps: Cost-Function Strategy by Representation of Phase-Space Distance

Chaotic systems are deterministic, yet measurement noise rapidly obscures the geometric structure of their trajectories and hampers state reconstruction and analysis. This paper presents a noise-reduction method for chaotic time series based on a manifold-driven cost function defined through distances in the reconstructed phase space. The method employs a sliding-window optimization framework in which only the initial state of each window is optimized, with the remaining samples generated by forward simulation of the known map. Symmetry-induced branch-selection errors are corrected by a backward repair pass. For the logistic map as the primary case study, the method preserves the system’s underlying dynamics and fine-scale attractor structure while effectively reducing noise contamination. Numerical results demonstrate a substantial improvement in Signal-to-Noise Ratio (SNR), from 14.07[Formula: see text]dB to 22.20[Formula: see text]dB (a gain of 8.13[Formula: see text]dB) under additive white Gaussian noise at approximately 20% of the signal RMS amplitude. Robustness is further assessed under a single hyper-parameter setting on five additive noise models: additive white Gaussian, AR(1)-correlated, periodic hum, symmetric [Formula: see text]-stable, and bursty impulsive. An impulsive salt-and-pepper sweep is tested as well. The method is benchmarked against Kalman-type nonlinear filters and smoothers as well as particle-filter baselines. Applicability to multistable chaotic maps is also demonstrated: the desired dynamical branch is reliably reconstructed even when initial states belonging to a neighboring attractor lie inside the search domain. The framework suppresses noise while preserving the fine structure of the chaotic attractor, which is a prerequisite for chaos-based communication and control applications.