Conformable Fractional-order Analysis of King Cobra Chaotic System and ADRC-Based Synchronization Design

This paper investigates the conformable fractional-order King Cobra chaotic system and proposes a robust synchronization scheme based on multi-state Proportional–Integral Active Disturbance Rejection Control (PI-ADRC). The conformable fractional derivative is introduced to generalize the classical King Cobra model which offers a simple and numerically efficient way to capture fractional-order dynamics. A detailed parametric study is carried out to explore how the fractional order and system parameters affect the onset of chaos, supported by phase portraits, Lyapunov exponents, and bifurcation analyses. To synchronize two identical chaotic systems with different initial conditions, a multi-state PI-ADRC controller equipped with an Extended State Observer (ESO) is developed. The ESO estimates and compensates for internal uncertainties and external disturbances and, allows the controller to maintain stability and accuracy even under stochastic noise. Numerical simulations show that an efficient synchronization is achieved in about one second, with smooth control signals and minimal error. The results demonstrate that the proposed approach provides a practical and robust framework for the control and synchronization of chaotic systems with conformable fractional-order behavior.