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Simplification of weakly nonlinear systems and analysis of cardiac activity using them

  • * Corresponding author: Miroslava Růžičková

    * Corresponding author: Miroslava Růžičková
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  • The paper deals with the transformation of a weakly nonlinear system of differential equations in a special form into a simplified form and its relation to the normal form and averaging. An original method of simplification is proposed, that is, a way to determine the coefficients of a given nonlinear system in order to simplify it. We call this established method the degree equalization method, it does not require integration and is simpler and more efficient than the classical Krylov-Bogolyubov method of normalization. The method is illustrated with several examples and provides an application to the analysis of cardiac activity modelled using van der Pol equation.

    Mathematics Subject Classification: Primary: 34C29, 34C20, 34C60; Secondary: 34B30, 34C15, 34A34.


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  • Figure 1.  The amplitude of any solution to van der Pol equation increases if its initial value is from the interval $ (0, 2) $, and decreases if the initial value is greater than two. In both cases it converges to the value $ 2 $

    Figure 2.  The limit cycle $ x^2(t) +\frac{1}{\omega} \dot x^2(t) = a^2 $ and some trajectories to van der Pol equation if $ a_0<2 $

    Figure 3.  If the initial amplitude value is close to zero, the amplitude exponentially increases to $ 2 $ with increasing $ t $

    Figure 4.  The area of the viability of the heart. The intensity of energy replenishment depends on $ \mu $

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