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Limit cycles by FEM for a one-parameter dynamical system associated to the Luo-Rudy I model

Cătălin Liviu Bichir, Adelina Georgescu, Bogdan Amuzescu, Gheorghe Nistor, Marin Popescu, Maria-Luiza Flonta, Alexandru Dan Corlan, Istvan Svab

Abstract

An one-parameter dynamical system is associated to the mathematical problem governing the membrane excitability of a ventricular cardiomyocyte, according to the Luo-Rudy I model. Limit cycles are described by the solutions of an extended system. A finite element method time approximation (FEM) is used in order to formulate the approximate problem. Starting from a Hopf bifurcation point, approximate limit cycles are obtained, step by step, using an arc-length-continuation method and Newton's method. Some numerical results are presented.

Limit cycles by FEM for a one-parameter dynamical system associated to the Luo-Rudy I model

Abstract

An one-parameter dynamical system is associated to the mathematical problem governing the membrane excitability of a ventricular cardiomyocyte, according to the Luo-Rudy I model. Limit cycles are described by the solutions of an extended system. A finite element method time approximation (FEM) is used in order to formulate the approximate problem. Starting from a Hopf bifurcation point, approximate limit cycles are obtained, step by step, using an arc-length-continuation method and Newton's method. Some numerical results are presented.

Paper Structure

This paper contains 8 sections, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Luo-Rudy I model
  3. The one - parameter dynamical system associated to the Luo - Rudy I model
  4. Extended system method for limit cycles
  5. Arc-length-continuation method for (\ref{['BichirCL_e25']})
  6. Newton's method for the steps of the algorithm from the end of section \ref{['BichirCL_sect_V']}
  7. Approximation of problem (\ref{['BichirCL_e36']}), (\ref{['BichirCL_e37']}), (\ref{['BichirCL_e38']}) by finite element method time approximation
  8. Numerical results

Figures (2)

  • Figure 1: Two projections of limit cycles and of a part of the equilibrium curve (marked by "$\blacktriangle$"). The Hopf bifurcation point is marked by "$\bullet$".
  • Figure 2: Projections of two limit cycles calculated for $I_{st}$$=$$-1.2000465026$ and for $I_{st}$$=$$-1.2000183729$ (marked by "x") (20 elements, 41 nodes).