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    Area of Science:

    • Chaos theory
    • Geometric modeling
    • Time series analysis

    Background:

    • Studying chaotic systems requires understanding physical properties from discrete-time signals.
    • Recovering chaotic properties from limited data is challenging for existing models.
    • Most current chaotic models necessitate large datasets for property extraction.

    Purpose of the Study:

    • To address the limitations of existing models in analyzing chaotic systems with small datasets.
    • To introduce a novel geometric approach for modeling and recovering chaotic properties.
    • To validate the effectiveness of the proposed method in capturing chaotic characteristics.

    Main Methods:

    • Utilizing geometric theory to analyze chaotic systems.
    • Developing a series-nonuniform rational B-spline (S-NURBS) modeling method.
    • Constructing smooth trajectories from time series data to implicitly represent chaotic properties.

    Main Results:

    • The S-NURBS model successfully recovers chaotic properties from small datasets.
    • Validation confirms the model's effectiveness from both statistical and chaotic perspectives.
    • The model demonstrated high credibility when applied to the Musa standard dataset for software reliability analysis.

    Conclusions:

    • The S-NURBS modeling method provides a feasible and effective approach to studying chaotic systems.
    • Geometric perspectives offer a new avenue for chaotic system research.
    • This work opens new possibilities for analyzing chaotic properties in practical time series data.