College of Computer Science and Mathematics

Discussion of a master’s thesis in the College of Computer Science and Mathematics – Department of Mathematics Sciences entitled:

“ Study Chaotic Dynamics with Application on Biological Systems “

It was discussed in the discussion hall of Computer Science and Mathematics at the University of Mosul on Thursday 28/8/2025

master’s thesis by Maha Waleed Khaleel Mahmood

under the supervision of Prof. Dr. Maysoon Malallah Aziz

The periodontium is a complex biological system composed of different structures that support the teeth including the gingiva, cementum of periodontium, periodontal-ligament and the alveolar-bone. In light of its complexity, this thesis as Case One analyzes the dynamics of the periodontium using chaos theory and nonlinear system.

We proposed a biomathematical model of the periodontium system as a unique prey-predator system, incorporating functional responses. The Iotka-Volterra and Holling types are among the most widely used functional responses to model a predator’s average feeding rate. The systems behavior is investigated through equilibrium points, stability analysis, dissipativity and bifurcation. Numerical simulations reveal that the periodontium model exhibits chaotic behavior with Lyapunov dimension based on Lyapunov exponents  .

Chaos control was successfully achieved using active and adaptive control strategies. This highlights the potential of mathematical modeling in understanding and managing periodontal diseases such as periodontitis. In parallel study, we examine Enzyme-kinetics another-complex biological system using a system of ordinary differential equations underlying how well it represents the complicated system dynamics. We deal with 2D Enzyme-kinetics model as Case Two and 3D Enzyme-kinetics model as finally case three represents interactions between the excited enzyme (N), unexcited enzyme (R) and substrate molecules (S) for lipase activity in pathological conditions. Real data from the (Biostudies/Array Express) repository were used to estimate parameters for Enzyme-kinetics models. The Stability of the two systems is analyzed using roots of characteristic equation, Routh criteria, Hurwitz criteria and Lyapunov function. The Enzymes systems exhibit dissipativity and undergoes bifurcation, ultimately displaying hyper chaotic behavior with Lyapunov dimension for 2D Enzyme-kinetics system and for lipase Enzyme-kinetic system.

Adaptive and active control methods effectively stabilized both systems. These findings emphasize how useful mathematical model is for capturing the complex dynamics of biological processes and suggest pathways for controlling pathological states. The proposed models not only enhance our understanding of periodontium and Enzyme systems but also confirm how chaos theory can be widely applied in biomedical research.

The scientific committee included the following members:

• Dr. Omar Saber Qasim-Chairman
• Prof. Dr. Thair Younis Thanoon– Member
• Zahraa Abdulaziz Taha – Member
• Dr. . Maysoon Malallah Aziz – Member and supervisor.