TITLE:
Lyapunov-Based Stability and Numerical Convergence in a Nonlinear Cardiac Oscillator Model
AUTHORS:
Ifeoma Deborah Omoko, Philip Danso, Clement Tayie, Samuel Mensah, Michael Fosu
KEYWORDS:
Coupled Oscillator Model, Cardiac Rhythms, Nonlinear Differential Equations, Stability, Ultimate Boundedness, Lyapunov’s Direct Method, Convergence, Runge-Kutta Method
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.14 No.1,
January
27,
2026
ABSTRACT: A precise understanding of cardiac rhythm dynamics is fundamental for uncovering the mechanisms underlying arrhythmias and for guiding effective therapeutic strategies. In this study, we introduce a nonlinear coupled oscillator model that captures the physiological interactions among the sinoatrial (SA) node, atrioventricular (AV) node, and Purkinje fibers. The model is formulated as a system of second-order differential equations incorporating inertial effects, damping, inter-nodal coupling, and a nonlinear perturbation term to represent pathological deviations. To uphold analytical precision, global stability and boundedness are examined using Lyapunov’s direct method. In parallel, the numerical performance of the system is assessed through the classical fourth-order Runge-Kutta (RK4) scheme, with convergence and stability analyzed under varying parameter regimes. Simulation results demonstrate that adjustments in damping coefficients, coupling strengths, and nonlinear contributions can induce transitions between normal cardiac rhythms and arrhythmic states, including conduction blocks and chaotic oscillations. All computational experiments were performed in both MATLAB and Wolfram Mathematica to ensure reproducibility and cross-validation. By integrating physiological relevance with analytical precision, the proposed framework provides a robust foundation for investigating rhythm disorders, supporting predictive diagnostics, and informing the development of control-oriented therapeutic interventions.