Mathematical Periodontium model with Holling Type I: Chaos and Control

Maha W.  Khaleel; Dr. Maysoon M.  Aziz

Maha W. Khaleel College of computer science and mathematics, Mosul University, Mosul, Iraq
Dr. Maysoon M. Aziz College of computer science and mathematics, Mosul University, Mosul, Iraq

Keywords: Chaos, Control techniques, Equilibrium points, Holling type, Periodontium system, stability analysis.

Abstract

The periodontium is a assisting structure that surrounds and helps the teeth, it consists of various tissues consisting of the gingiva, the cementum, the periodontal ligament and alveolar helping bone. Knowing the reality that the periodontium is a complex system in the frame, this paper demonstrates viable mirrored image of chaos idea and the sector of periodontology, and most generally used practical responses to describe the average feeding rate of a predator are Lotka-Volterra type and Holling type practical reaction characteristic’s.System features are discussed by its equilibrium points, stability, dissipativity and bifurcation analysis. Graphical representation through numerical simulations are presented. Our study has shown that the periodontium system (1) is bifurcate and unstable system. It has dissipative equilibrium point and conservative equilibrium point, in addition to all of this the periodontium system (1) indicates a state of extreme chaos with Lyapunov dimension where the Lyapunov exponents are , after that we applied active control technique and adaptive control technique. By understanding the periodontium system (1) very well we estimated the rate of periodontitis in mathematical periodontium model ‘b’ as control parameter and successfully controlled the chaos. This brings us to the purpose of this research paper, whereas understanding the periodontium system’s structure and function may prove valuable in managing illness.

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