A Novel 4D Nonlinear Interaction Football Model: Stability Analysis
I. C. Nwokike *
Department of Mathematics, Federal University of Technology, Owerri, Nigeria and Centre of Excellence in Sustainable Procurement, Environmental & Social Standards, Federal University of Technology, Owerri, Nigeria.
W. I. Osuji
Department of Mathematics, Federal University of Technology, Owerri, Nigeria.
G. O. Nwafor
Centre of Excellence in Sustainable Procurement, Environmental & Social Standards, Federal University of Technology, Owerri, Nigeria and Department of Statistics, Federal University of Technology, Owerri, Nigeria.
I. C. Obinwanne
Department of Statistics, Federal University of Technology, Owerri, Nigeria.
S. Musa
Department of Mathematics, Sule Lamido University Kafin Hausa, Jigawa, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
A novel nonlinear mathematical model is formulated to study the effects of defense, midfield, and attack on an overall football team performance. Bilinear inhibitory effects are considered for match-based factors such as defensive breakdown under attacking pressure, midfield congestion, and failure to convert attacking opportunities into match outcomes. Qualitative analysis is carried out to confirm positivity, boundedness, and well-posedness of solutions. Equilibrium points corresponding to stable tactical configurations are obtained and analyzed. Local stability analysis is carried out through analysis of the Jacobian, while global stability is established with the use of a Lyapunov function and LaSalle’s Invariance Principle. Analytical results indicate that nonlinear inhibitory factors are a major force in maintaining control over football tactics, as they ensure that none of the considered aspects are able to grow unbounded. In fact, it is demonstrated that synergy between effective defense, midfield, and attack leads to more sustainable tactics and higher rates of team performance.
Keywords: Dynamical systems, nonlinear interaction model, global and local stability analysis, lyapunov functions, sport modeling, football, soccer