On Nonlinear and Nonideal Dynamics of Three Coupled Van der Pol Oscillators Applied to Hemodynamics

Abstract

In this paper, we examined the nonlinear dynamic behavior of the coupling interactions between three oscillators that describe the re- gions of the human heart responsible for electrical impulses. Specif- ically, we focused on the sinoatrial node, the atrioventricular node, and the Purkinje complex. The dynamics of cardiac behavior can be effectively represented by a nonlinear dynamic model, such as the Van der Pol Oscillator. However, we added an external force described by Bessel functions applied to oscillators with non-ideal characteristics to explore the system's behavior further. Consequently, we analyzed the range of frequencies that alter the dynamic behavior of the system. To diagnose the nonlinear dynamic behavior, we employed classic tools such as the Lyapunov exponent, which established the convergence of trajectories, and the bifurcation diagram that confirms the periodic windows of the system and the phase space. Our findings enabled us to reconstruct an approximation of the Electrocardiogram (ECG) for a set of parameters, owing to the applicability of the mathematical model analyzed in the field of biomedical engineering. ©2025 L&H Scientific Publishing, LLC. All rights reserved.