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June  2022, 27(6): 3435-3453. doi: 10.3934/dcdsb.2021191

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

 1 V. Hetman Kyiv National Economic University, Department of Computer Mathematics and Information Security, Kyiv 03068, Peremogy 54/1, Ukraine 2 University of Białystok, Faculty of Mathematics, K. Ciołkowskiego 1M, 15-245 Białystok, Poland

* Corresponding author: Miroslava Růžičková

Received  November 2020 Revised  June 2021 Published  June 2022 Early access  July 2021

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.

Citation: Irada Dzhalladova, Miroslava Růžičková. Simplification of weakly nonlinear systems and analysis of cardiac activity using them. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3435-3453. doi: 10.3934/dcdsb.2021191
##### References:
 [1] H. M. Ahmed and Q. Zhu, The averaging principle of Hilfer fractional stochastic delay differential equations with Poisson jumps, Appl. Math. Lett., 112 (2021), 106755, 7 pp. doi: 10.1016/j.aml.2020.106755. [2] M. Bendahmane, F. Mroue, M. Saad and R. Talhouk, Mathematical analysis of cardiac electromechanics with physiological ionic model, Discrete Continuous Dynam. Systems - B, 24 (2019), 4863-4897.  doi: 10.3934/dcdsb.2019035. [3] G. D. Bifkhoff, Dynamical Systems, American Mathematical Society, Providence, R.I., IX, 1966. [4] N. N. Bogolyubov, On Certain Statistical Methods in Mathematical Physics, (in Russian), Kiev, 1935. [5] N. N. Bogolyubov and Y. A. Mitropolskiy, Asymptotic Methods in the Theory of Nonlinear Oscillations, (Translated from Russian), Gordon and Breach, New York, 1961. [6] A. D. Bryuno, The normal form of differential equations, Dokl. Akad. Nauk SSSR, 157 (1964), 1276-1279. [7] A. D. Bryuno, A Local Method of Nonlinear Analysis of Differential Equations, Nauka, Moscow, 1979. [8] A. D. Bryuno, Power Geometry in Algebraic and Differential Equations, Fizmatlit, Moscow, 1998. [9] G. Chen and J. Della Dora, Further reductions of normal forms for dynamical systems, J. Differential Equations, 166 (2000), 79-106.  doi: 10.1006/jdeq.2000.3783. [10] A. Deprit, Canonical transformations depending on a small parameter, Celest. Mech., 1 (1969), 12-30.  doi: 10.1007/BF01230629. [11] A. Deprit, J. Henrard, J. F. Price and A. Rom, Birkhoff's normalization, Celest. Mech., 1 (1969), 222-251.  doi: 10.1007/BF01228842. [12] S. P. Diliberto, New results on periodic surfaces and the averaging principle, Proc. U.S.-Japan Seminar on Differential and Functional Equations, Minneapolis, Minn., Benjamin, New York, (1967), 49–87. [13] P. Fatou, Sur le mouvement d'un système soumis à des forces à courte période, Bulletin de la Société Mathématique de France, 56 (1928), 98-139.  doi: 10.24033/bsmf.1131. [14] J. K. Hale, Oscillations in Non-Linear Systems, McGraw-Hill, New York, 1963. [15] M. Han, Y. Xu and B. Pei, Mixed stochastic differential equations: Averaging principle result, Applied Mathematics Letters, 112 (2021), 106705, 7 pp. doi: 10.1016/j.aml.2020.106705. [16] G. Hori, Theory of general perturbations with unspecified canonical variables, Publ. Astron. Soc. Japan, 18 (1966), 287-296. [17] M. Kesmia, S. Boughaba and S. Jacquir, New approach of controlling cardiac alternans, Discrete Continuous Dynam. Systyms - B, 23 (2018), 975-989.  doi: 10.3934/dcdsb.2018051. [18] N. M. Krylov and N. N. Bogolyubov, Introduction to Non-Linear Mechanics, Princeton Univ. Press, Princeton, 1947. (Translated from Russian, Izd-vo AN SSSR, Kiev, 1937) [19] P. Kügler, Modelling and simulation for preclinical cardiac safety assessment of drugs with Human iPSC-derived cardiomyocytes, Jahresber Dtsch Math-Ver., 122 (2020), 209-257.  doi: 10.1365/s13291-020-00218-w. [20] J. L. Lagrange, Mécanique Céleste $(2$ vols.$)$, {Edition Albert Blanchard}, Paris, 1788. [21] A. K. Lopatin, Averaging, Normal forms and Symmetry in Non-Linear Mechanics, Preprint Inst. Mat. Nat. Acad. Ukrainy, Kiev, 1994, (in Russian) [22] D. Luo, Q. Zhu and Z. Luo, An averaging principle for stochastic fractional differential equations with time-delays, Applied Mathematics Letters, 105 (2020), 106290, 8 pp. doi: 10.1016/j.aml.2020.106290. [23] L. I. Mandelshtam ana N. D. Papaleksi, On justification of a method of approximate solving differential equations, J. Exp. Theor. Physik, 4 (1934), 117–121. (in Russian). [24] W. Mao, L. Hu, S. You and X. Mao, The averaging method for multivalued SDEs with jumps and non-Lipschitz coefficients, Discrete Continuous Dynam. Systems - B, 24 (2019), 4937-4954.  doi: 10.3934/dcdsb.2019039. [25] J. A. Mitropolskiy and A. M. Samoilenko, To the problem on asymptotic decompositions of non-linear mechanics, Ukr. Mat. Zhurn., 31 (1979), 42–53. (in Russian). [26] Y. A. Mitropolskiy, Basic lines of research in the theory of nonlinear oscillations and the progress achieved, Proceedings of the International Symposium on Non-linear Oscillations, Kiev, I (1963), 15–22. [27] Y. A. Mitropolskiy and A. K. Lopatin, Group Theory, Approach in Asymptotic Methods of Non-Linear Mechanics, Naukova Dumka, Kiev, 1988. (in Russian). [28] Y. A. Mitropolskiy and N. Van Dao, Averaging method, In: Applied Asymptotic Methods in Nonlinear Oscillations, Solid Mechanics and Its Applications, Vol 55, Springer, Dordrecht, (1997), 282–326. doi: 10.1007/978-94-015-8847-8. [29] A. M Molchanov, Separation of motions and asymptotic methods in the theory of linear oscillations, DAN SSSR, 5, (1961), 1030–1033. (in Russian). [30] A. Poincaré, New Methods of Celestial Mechanics, Gauthiers-Villars, Paris, 1892. (Translated to Russian, Nauka, Moscow, 1971.) [31] M. I. Rabinovich and D. I. Trubetskov, Oscillations and Waves in Linear and Nonlinear Systems, Kluwer Academic Publishers, Dordrecht, 1989. (Translated from the Russian by R. N. Hainsworth, "Vvedenie v teoriyu kolebanij i voln, " Nauka, Moscow, 1984.) doi: 10.1007/978-94-009-1033-1. [32] J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4757-4575-7. [33] T. G. Strizhak, Averaging Method in Problems of Mechanics, Vishcha Shkola, Kiev-Donetsk, 1982. (in Russian). [34] T. G. Strizhak, An Asymptotic Normalization Method, Vishcha Shkola, Glavnoe Izd., Kiev, 1984. [35] B. van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Rev., 1 (1920), 701–710. [36] B. van der Pol, On "Relaxation Oscillations", Philos. Mag., 2 (1926), 978-992.  doi: 10.1080/14786442608564127. [37] B. van der Pol, The nonlinear theory of electric oscillations, Proceedings of the Institute of Radio Engineers, 22 (1934), 1051-1086.  doi: 10.1109/JRPROC.1934.226781.

show all references

##### References:
 [1] H. M. Ahmed and Q. Zhu, The averaging principle of Hilfer fractional stochastic delay differential equations with Poisson jumps, Appl. Math. Lett., 112 (2021), 106755, 7 pp. doi: 10.1016/j.aml.2020.106755. [2] M. Bendahmane, F. Mroue, M. Saad and R. Talhouk, Mathematical analysis of cardiac electromechanics with physiological ionic model, Discrete Continuous Dynam. Systems - B, 24 (2019), 4863-4897.  doi: 10.3934/dcdsb.2019035. [3] G. D. Bifkhoff, Dynamical Systems, American Mathematical Society, Providence, R.I., IX, 1966. [4] N. N. Bogolyubov, On Certain Statistical Methods in Mathematical Physics, (in Russian), Kiev, 1935. [5] N. N. Bogolyubov and Y. A. Mitropolskiy, Asymptotic Methods in the Theory of Nonlinear Oscillations, (Translated from Russian), Gordon and Breach, New York, 1961. [6] A. D. Bryuno, The normal form of differential equations, Dokl. Akad. Nauk SSSR, 157 (1964), 1276-1279. [7] A. D. Bryuno, A Local Method of Nonlinear Analysis of Differential Equations, Nauka, Moscow, 1979. [8] A. D. Bryuno, Power Geometry in Algebraic and Differential Equations, Fizmatlit, Moscow, 1998. [9] G. Chen and J. Della Dora, Further reductions of normal forms for dynamical systems, J. Differential Equations, 166 (2000), 79-106.  doi: 10.1006/jdeq.2000.3783. [10] A. Deprit, Canonical transformations depending on a small parameter, Celest. Mech., 1 (1969), 12-30.  doi: 10.1007/BF01230629. [11] A. Deprit, J. Henrard, J. F. Price and A. Rom, Birkhoff's normalization, Celest. Mech., 1 (1969), 222-251.  doi: 10.1007/BF01228842. [12] S. P. Diliberto, New results on periodic surfaces and the averaging principle, Proc. U.S.-Japan Seminar on Differential and Functional Equations, Minneapolis, Minn., Benjamin, New York, (1967), 49–87. [13] P. Fatou, Sur le mouvement d'un système soumis à des forces à courte période, Bulletin de la Société Mathématique de France, 56 (1928), 98-139.  doi: 10.24033/bsmf.1131. [14] J. K. Hale, Oscillations in Non-Linear Systems, McGraw-Hill, New York, 1963. [15] M. Han, Y. Xu and B. Pei, Mixed stochastic differential equations: Averaging principle result, Applied Mathematics Letters, 112 (2021), 106705, 7 pp. doi: 10.1016/j.aml.2020.106705. [16] G. Hori, Theory of general perturbations with unspecified canonical variables, Publ. Astron. Soc. Japan, 18 (1966), 287-296. [17] M. Kesmia, S. Boughaba and S. Jacquir, New approach of controlling cardiac alternans, Discrete Continuous Dynam. Systyms - B, 23 (2018), 975-989.  doi: 10.3934/dcdsb.2018051. [18] N. M. Krylov and N. N. Bogolyubov, Introduction to Non-Linear Mechanics, Princeton Univ. Press, Princeton, 1947. (Translated from Russian, Izd-vo AN SSSR, Kiev, 1937) [19] P. Kügler, Modelling and simulation for preclinical cardiac safety assessment of drugs with Human iPSC-derived cardiomyocytes, Jahresber Dtsch Math-Ver., 122 (2020), 209-257.  doi: 10.1365/s13291-020-00218-w. [20] J. L. Lagrange, Mécanique Céleste $(2$ vols.$)$, {Edition Albert Blanchard}, Paris, 1788. [21] A. K. Lopatin, Averaging, Normal forms and Symmetry in Non-Linear Mechanics, Preprint Inst. Mat. Nat. Acad. Ukrainy, Kiev, 1994, (in Russian) [22] D. Luo, Q. Zhu and Z. Luo, An averaging principle for stochastic fractional differential equations with time-delays, Applied Mathematics Letters, 105 (2020), 106290, 8 pp. doi: 10.1016/j.aml.2020.106290. [23] L. I. Mandelshtam ana N. D. Papaleksi, On justification of a method of approximate solving differential equations, J. Exp. Theor. Physik, 4 (1934), 117–121. (in Russian). [24] W. Mao, L. Hu, S. You and X. Mao, The averaging method for multivalued SDEs with jumps and non-Lipschitz coefficients, Discrete Continuous Dynam. Systems - B, 24 (2019), 4937-4954.  doi: 10.3934/dcdsb.2019039. [25] J. A. Mitropolskiy and A. M. Samoilenko, To the problem on asymptotic decompositions of non-linear mechanics, Ukr. Mat. Zhurn., 31 (1979), 42–53. (in Russian). [26] Y. A. Mitropolskiy, Basic lines of research in the theory of nonlinear oscillations and the progress achieved, Proceedings of the International Symposium on Non-linear Oscillations, Kiev, I (1963), 15–22. [27] Y. A. Mitropolskiy and A. K. Lopatin, Group Theory, Approach in Asymptotic Methods of Non-Linear Mechanics, Naukova Dumka, Kiev, 1988. (in Russian). [28] Y. A. Mitropolskiy and N. Van Dao, Averaging method, In: Applied Asymptotic Methods in Nonlinear Oscillations, Solid Mechanics and Its Applications, Vol 55, Springer, Dordrecht, (1997), 282–326. doi: 10.1007/978-94-015-8847-8. [29] A. M Molchanov, Separation of motions and asymptotic methods in the theory of linear oscillations, DAN SSSR, 5, (1961), 1030–1033. (in Russian). [30] A. Poincaré, New Methods of Celestial Mechanics, Gauthiers-Villars, Paris, 1892. (Translated to Russian, Nauka, Moscow, 1971.) [31] M. I. Rabinovich and D. I. Trubetskov, Oscillations and Waves in Linear and Nonlinear Systems, Kluwer Academic Publishers, Dordrecht, 1989. (Translated from the Russian by R. N. Hainsworth, "Vvedenie v teoriyu kolebanij i voln, " Nauka, Moscow, 1984.) doi: 10.1007/978-94-009-1033-1. [32] J. A. Sanders and F. Verhulst, Averaging Methods in Nonlinear Dynamical Systems, Springer-Verlag, New York, 1985. doi: 10.1007/978-1-4757-4575-7. [33] T. G. Strizhak, Averaging Method in Problems of Mechanics, Vishcha Shkola, Kiev-Donetsk, 1982. (in Russian). [34] T. G. Strizhak, An Asymptotic Normalization Method, Vishcha Shkola, Glavnoe Izd., Kiev, 1984. [35] B. van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Rev., 1 (1920), 701–710. [36] B. van der Pol, On "Relaxation Oscillations", Philos. Mag., 2 (1926), 978-992.  doi: 10.1080/14786442608564127. [37] B. van der Pol, The nonlinear theory of electric oscillations, Proceedings of the Institute of Radio Engineers, 22 (1934), 1051-1086.  doi: 10.1109/JRPROC.1934.226781.
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 $2$
The limit cycle $x^2(t) +\frac{1}{\omega} \dot x^2(t) = a^2$ and some trajectories to van der Pol equation if $a_0<2$
If the initial amplitude value is close to zero, the amplitude exponentially increases to $2$ with increasing $t$
The area of the viability of the heart. The intensity of energy replenishment depends on $\mu$
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