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    This study introduces a proportional integral (PI) controller using DNA strand displacement (DSD) to achieve hybrid projective synchronization in chaotic systems. This novel approach enables synchronization between different four-variable chaotic systems.

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    Area of Science:

    • Biomolecular Engineering
    • Chaos Theory
    • Control Systems

    Background:

    • Previous studies achieved chaotic system synchronization using DNA strand displacement (DSD).
    • Existing methods focused on three-variable systems.

    Purpose of the Study:

    • To introduce a proportional integral (PI) controller for chaotic systems.
    • To achieve hybrid projective synchronization between different four-variable chaotic systems using DSD.

    Main Methods:

    • Utilizing DSD reactions as the foundational mechanism.
    • Designing DSD-based chaotic systems with catalysis, annihilation, and degradation modules.
    • Integrating PI controllers composed of catalysis, annihilation, and adjust DSD modules.

    Main Results:

    • Successfully demonstrated hybrid projective synchronization for different four-variable chaotic systems.
    • Developed a modular PI controller for enhanced adaptability and modification.
    • Established a framework for constructing chaotic attractors via DSD modules.

    Conclusions:

    • The proposed DSD-based PI controller effectively achieves hybrid projective synchronization.
    • The modular design offers flexibility for future chaos synchronization research.
    • This work provides a valuable reference for investigating advanced chaos synchronization techniques.