Journal of Prime Research in Mathematics

Optimizing Synchronization: A Comparative Study of Four-Dimensional Forced SIR Systems Employing Multiple Control Techniques

Derbouche Assia\(^{a,∗}\), Hamri Nasr-Eddine\(^a\), Laouira Widad\(^a\)

\(^a\)Laboratory of Mathematics and their Interactions, Abdelhafid Boussouf University Center, Mila, Algeria.


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The forced SIR system’s synchronization is the main subject of this study. Through the use of several control techniques, including active control (AC), active backstepping control (ABC), adaptive control (AdC), and sliding mode control (SMC). We redesigned the three-dimensional system to be autonomous and made it four-dimensional to ease numerical computations. The system’s phase portrait, Lyapunov exponent graph, and bifurcation diagram are used to analyze its dynamic properties through numerical simulation. The effectiveness of the AC, SMC, ABC, and AdC approaches is examined using dynamical error and necessary control inputs. By contrasting the integral square error with the required control energy measurements, the best control for synchronization is found.


SIR model, chaotic system, active control, sliding mode control, adaptive control, backstepping control, synchronization