Control and synchronization of Julia sets of a discrete three-dimensional fractional HCV model

This paper investigates the fractal dynamics of a discrete Caputo fractional-order hepatitis C virus (HCV) system. Due to the memory effect and the hereditary property, the discrete fractional model helps in the study of the fractal behavior of the HCV model. First, the three-dimensional integer-order HCV model is extended to the fractional-order one, and the corresponding Julia set is defined. Next, a fractional-order controller based on a coordinate transformation is designed to control the system’s Julia set. The stability interval of the controlled system is determined by calculating the spectral radius of the Jacobian matrix at the system’s fixed point, and numerical simulations are then presented to illustrate how the Julia set changes as the control parameters are increased within the stability interval. In addition, a nonlinear coupling controller is constructed and added to the three-dimensional model to achieve synchronization between two discrete fractional-order systems with different fractional orders and different parameters. Rigorous mathematical proofs are provided to establish the correctness and effectiveness of the proposed synchronization method. Moreover, numerical simulation figures are presented to illustrate the synchronization of the response system’s Julia sets toward the target system’s Julia sets as the synchronization parameters vary. These results contribute to a systematic characterization of fractal dynamical behavior of the fractional-order HCV system and provide useful insights fto inform HCV control strategies.