Abstract
Bio-electric activity of the heart is modeled by the Barrio–Varea–Aragon–Maini (BVAM) model that covers normal rhythm and several arrhythmia that lie in the chaotic region and exhibits several bifurcations. In this chapter, we develop the analytic solution of the BVAM model and identify the Hopf bifurcation. The center manifold reduction is applied to the governing equations to reduce the order of the system. The method of multiple scales is employed to develop the normal form of the Hopf bifurcation for the center manifolds. These are then transformed back into original coordinates where the analytical solution is compared with the numerical solution.
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References
Sinha, S.(auth.), In, V., Longhini, P., Palacios, A., Ditto, W.L., Murali, K. (eds.): Applications of Nonlinear Dynamics: Model and Design of Complex Systems, 1st edn. In: Understanding Complex Systems. Springer, Berlin/Heidelberg (2009)
Matsushita, H., Miyata, T., Christiansen, L.E., Lehn-Schioler, T., Mosekilde, E.: Nonlinear dynamics approach of modeling the bifurcation for aircraft wing flutter in transonic speed. In: Proceedings of the 41st SICE Annual Conference. SICE 2002., vol. 2, pp. 695–700 (2002)
Quiroz-Juárez, M.A., Jiménez-Ramírez, O., Vázquez-Medina, R., Breña-Medina, V., Aragón, J.L., Barrio, R.A.: Generation of ECG signals from a reaction-diffusion model spatially discretized. Sci. Rep. 9(1), 1–10 (2019)
Gray, R.A., Pertsov, A.M., Jalife, J.: Spatial and temporal organization during cardiac fibrillation. Nature 392(6671), 75–78 (1998)
Nayfeh, A.H.: Introduction to Perturbation Techniques. John Wiley & Sons, New York (2011)
Keener, J.P., Sneyd, J.: Mathematical Physiology, vol. 1. Springer, New York (1998)
Qu, Z., Hu, G., Garfinkel, A., Weiss, J.N.: Nonlinear and stochastic dynamics in the heart. Phys. Rep. 543(2), 61–162 (2014)
Newhouse, S., Ruelle, D., Takens, F.: Occurrence of strange axiom a attractors near quasi periodic flows on T\({ }^m\), \(m \geqq 3\). Commun. Math. Phys. 64(1), 35–40 (1978)
Quiroz-Juarez, M.A., Vázquez-Medina, R., Ryzhii, E., Ryzhii, M., Aragón, J.L.: Quasiperiodicity route to chaos in cardiac conduction model. Commun. Nonlinear Sci. Numer. Simul. 42, 370–378 (2017)
Ryzhii, E., Ryzhii, M.: A heterogeneous coupled oscillator model for simulation of ECG signals. Comput. Methods Prog. Biomed. 117(1), 40–49 (2014)
Barrio, R.A., Varea, C., Aragón, J.L., Maini, P.K.: A two-dimensional numerical study of spatial pattern formation in interacting Turing systems. Bull. Math. Biol. 61(3), 483–505 (1999)
Toole, G., Hurdal, M.K.: Growth in a Turing model of cortical folding. Biomath 1(1), ID–1209252 (2012)
Nayfeh, A.H., Balachandran, B.: Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods. John Wiley & Sons, New York (2008)
Cole, J.D.: Perturbation Methods (Ali Hasan Nayfeh). Wiley (1976); WILEY-VCH Verlag GmbH & CO. KGaA, Weinheim (2004)
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Naseer, A., Akhtar, I., Hajj, M.R. (2024). Hopf Bifurcation Analysis of the BVAM Model for Electrocardiogram. In: Lacarbonara, W. (eds) Advances in Nonlinear Dynamics, Volume III. ICNDA 2023. NODYCON Conference Proceedings Series. Springer, Cham. https://doi.org/10.1007/978-3-031-50635-2_55
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DOI: https://doi.org/10.1007/978-3-031-50635-2_55
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