doi: 10.3934/dcdsb.2021191
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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 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 & Continuous Dynamical Systems - B, 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.  Google Scholar

[2]

M. BendahmaneF. MroueM. 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.  Google Scholar

[3]

G. D. Bifkhoff, Dynamical Systems, American Mathematical Society, Providence, R.I., IX, 1966.  Google Scholar

[4]

N. N. Bogolyubov, On Certain Statistical Methods in Mathematical Physics, (in Russian), Kiev, 1935. Google Scholar

[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.  Google Scholar

[6]

A. D. Bryuno, The normal form of differential equations, Dokl. Akad. Nauk SSSR, 157 (1964), 1276-1279.   Google Scholar

[7]

A. D. Bryuno, A Local Method of Nonlinear Analysis of Differential Equations, Nauka, Moscow, 1979.  Google Scholar

[8]

A. D. Bryuno, Power Geometry in Algebraic and Differential Equations, Fizmatlit, Moscow, 1998.  Google Scholar

[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.  Google Scholar

[10]

A. Deprit, Canonical transformations depending on a small parameter, Celest. Mech., 1 (1969), 12-30.  doi: 10.1007/BF01230629.  Google Scholar

[11]

A. DepritJ. HenrardJ. F. Price and A. Rom, Birkhoff's normalization, Celest. Mech., 1 (1969), 222-251.  doi: 10.1007/BF01228842.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. K. Hale, Oscillations in Non-Linear Systems, McGraw-Hill, New York, 1963.  Google Scholar

[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.  Google Scholar

[16]

G. Hori, Theory of general perturbations with unspecified canonical variables, Publ. Astron. Soc. Japan, 18 (1966), 287-296.   Google Scholar

[17]

M. KesmiaS. Boughaba and S. Jacquir, New approach of controlling cardiac alternans, Discrete Continuous Dynam. Systyms - B, 23 (2018), 975-989.  doi: 10.3934/dcdsb.2018051.  Google Scholar

[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) Google Scholar

[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.  Google Scholar

[20]

J. L. Lagrange, Mécanique Céleste $(2$ vols.$)$, {Edition Albert Blanchard}, Paris, 1788. Google Scholar

[21]

A. K. Lopatin, Averaging, Normal forms and Symmetry in Non-Linear Mechanics, Preprint Inst. Mat. Nat. Acad. Ukrainy, Kiev, 1994, (in Russian) Google Scholar

[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.  Google Scholar

[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). Google Scholar

[24]

W. MaoL. HuS. 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.  Google Scholar

[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).  Google Scholar

[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. Google Scholar

[27]

Y. A. Mitropolskiy and A. K. Lopatin, Group Theory, Approach in Asymptotic Methods of Non-Linear Mechanics, Naukova Dumka, Kiev, 1988. (in Russian).  Google Scholar

[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.  Google Scholar

[29]

A. M Molchanov, Separation of motions and asymptotic methods in the theory of linear oscillations, DAN SSSR, 5, (1961), 1030–1033. (in Russian). Google Scholar

[30]

A. Poincaré, New Methods of Celestial Mechanics, Gauthiers-Villars, Paris, 1892. (Translated to Russian, Nauka, Moscow, 1971.) Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

T. G. Strizhak, Averaging Method in Problems of Mechanics, Vishcha Shkola, Kiev-Donetsk, 1982. (in Russian). Google Scholar

[34]

T. G. Strizhak, An Asymptotic Normalization Method, Vishcha Shkola, Glavnoe Izd., Kiev, 1984.  Google Scholar

[35]

B. van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Rev., 1 (1920), 701–710. Google Scholar

[36]

B. van der Pol, On "Relaxation Oscillations", Philos. Mag., 2 (1926), 978-992.  doi: 10.1080/14786442608564127.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

M. BendahmaneF. MroueM. 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.  Google Scholar

[3]

G. D. Bifkhoff, Dynamical Systems, American Mathematical Society, Providence, R.I., IX, 1966.  Google Scholar

[4]

N. N. Bogolyubov, On Certain Statistical Methods in Mathematical Physics, (in Russian), Kiev, 1935. Google Scholar

[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.  Google Scholar

[6]

A. D. Bryuno, The normal form of differential equations, Dokl. Akad. Nauk SSSR, 157 (1964), 1276-1279.   Google Scholar

[7]

A. D. Bryuno, A Local Method of Nonlinear Analysis of Differential Equations, Nauka, Moscow, 1979.  Google Scholar

[8]

A. D. Bryuno, Power Geometry in Algebraic and Differential Equations, Fizmatlit, Moscow, 1998.  Google Scholar

[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.  Google Scholar

[10]

A. Deprit, Canonical transformations depending on a small parameter, Celest. Mech., 1 (1969), 12-30.  doi: 10.1007/BF01230629.  Google Scholar

[11]

A. DepritJ. HenrardJ. F. Price and A. Rom, Birkhoff's normalization, Celest. Mech., 1 (1969), 222-251.  doi: 10.1007/BF01228842.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

J. K. Hale, Oscillations in Non-Linear Systems, McGraw-Hill, New York, 1963.  Google Scholar

[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.  Google Scholar

[16]

G. Hori, Theory of general perturbations with unspecified canonical variables, Publ. Astron. Soc. Japan, 18 (1966), 287-296.   Google Scholar

[17]

M. KesmiaS. Boughaba and S. Jacquir, New approach of controlling cardiac alternans, Discrete Continuous Dynam. Systyms - B, 23 (2018), 975-989.  doi: 10.3934/dcdsb.2018051.  Google Scholar

[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) Google Scholar

[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.  Google Scholar

[20]

J. L. Lagrange, Mécanique Céleste $(2$ vols.$)$, {Edition Albert Blanchard}, Paris, 1788. Google Scholar

[21]

A. K. Lopatin, Averaging, Normal forms and Symmetry in Non-Linear Mechanics, Preprint Inst. Mat. Nat. Acad. Ukrainy, Kiev, 1994, (in Russian) Google Scholar

[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.  Google Scholar

[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). Google Scholar

[24]

W. MaoL. HuS. 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.  Google Scholar

[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).  Google Scholar

[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. Google Scholar

[27]

Y. A. Mitropolskiy and A. K. Lopatin, Group Theory, Approach in Asymptotic Methods of Non-Linear Mechanics, Naukova Dumka, Kiev, 1988. (in Russian).  Google Scholar

[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.  Google Scholar

[29]

A. M Molchanov, Separation of motions and asymptotic methods in the theory of linear oscillations, DAN SSSR, 5, (1961), 1030–1033. (in Russian). Google Scholar

[30]

A. Poincaré, New Methods of Celestial Mechanics, Gauthiers-Villars, Paris, 1892. (Translated to Russian, Nauka, Moscow, 1971.) Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

T. G. Strizhak, Averaging Method in Problems of Mechanics, Vishcha Shkola, Kiev-Donetsk, 1982. (in Russian). Google Scholar

[34]

T. G. Strizhak, An Asymptotic Normalization Method, Vishcha Shkola, Glavnoe Izd., Kiev, 1984.  Google Scholar

[35]

B. van der Pol, A theory of the amplitude of free and forced triode vibrations, Radio Rev., 1 (1920), 701–710. Google Scholar

[36]

B. van der Pol, On "Relaxation Oscillations", Philos. Mag., 2 (1926), 978-992.  doi: 10.1080/14786442608564127.  Google Scholar

[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.  Google Scholar

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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