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ISSN: 2475-4811 (print)
ISSN: 2475-482X (online)
Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland

Email: pawel.olejnik@p.lodz.pl

C. Steve Suh (editor)

Texas A&M University, USA

Email: ssuh@tamu.edu

Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China

Email: tuoxianguo@suse.edu.cn

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

Journal of Vibration Testing and System Dynamics 9(2) (2025) 135--146 | DOI:10.5890/JVTSD.2025.06.004

Gabriella de O. M. Silva$^1$, Mauricio A. Ribeiro$^1$, Jose Manoel Balthazar$^{1,3}$, Jeferson J. de Lima$^1$, Cristhiane Gonçalves$^1$, Vinicius Piccirilo$^1$, Marcus Varanis$^{2}$, Clivaldo de Oliveira$^4$, Ângelo M. Tusset$^1$

$^{1}$ Departament of Electronics Engineering, University Federal Technology of Paraná - UTFPR, R. Doutor Washington Subtil Chueire, 330 - Jardim Carvalho, Ponta Grossa, Paraná, Brazil

$^{2}$ Physics Institute, Federal University of Mato Grosso – UFMG, Av. Costa e Silva, Bairro Universitário, Campo Grande, Mato Grosso do Sul, Brazil

$^{3}$ Senior Professor, Department of Mechanics Engineering, State University of São Paulo – UNESP, Av. Eng. Luís Edmundo Carrijo Coube, 2085 - Nucleo Res. Pres. Geisel, Bauru, São Paulo, Brazil

$^{4}$ Departament of Mechanical Engineering, University Federal of Dourados - UFGD, R. João Rosa Góes, Vila Progresso, Dourados, Mato Grosso do Sul, Brazil

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Abstract

In this paper, we examined the nonlinear dynamic behavior of the coupling interactions between three oscillators that describe the regions of the human heart responsible for electrical impulses. Specifically, 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.

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