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Invariant Coordinate Subspaces of Normal Form of a System of Ordinary Differential Equations

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Abstract

For systems of ordinary differential equations (ODEs) with a nondegenerate linear part in the general and Hamiltonian cases, the problem of finding invariant coordinate subspaces in the coordinates of the normal form calculated in the vicinity of the equilibrium is stated. Conditions for the existence of such invariant subspaces in terms of the resonant relations between the eigenvalues of the linear part of the system are obtained. An algorithm for finding the resonant relations between the eigenvalues without their explicit calculation is described; this algorithm substantially uses computer algebra methods and the q-analog of the polynomial subresultants. The implementation of this algorithm in three popular computer algebra systems—Mathematica, Maple, and SymPy—is discussed. Interesting model examples are provided.

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REFERENCES

  1. Bruno, A.D., Analytical form of differential equations (I), Trans. Moscow Math. Soc., 1971, 25, pp. 131–288.

  2. Bruno, A.D., The Restricted 3-Body Problem: Plane Periodic Orbits, Moscow: Nauka, 1990; New York: de Gruyter, 1994.

  3. Markeev, A.P., Libration Points in Celestial Mechanics and Astrodynamics, Moscow: Nauka, 1978 [in Russian].

    MATH  Google Scholar 

  4. Bruno, A.D., Local Methods in Nonlinear Differential Equations, Berlin: Springer, 1989.

    Book  Google Scholar 

  5. Bruno, A.D., Enderal, V.F., and Romanovski, V.G., On new integrals of the Algaba–Gamero–Garcia system, in Computer Algebra in Scientific Computing, Gerdt, V.P., Koepf, W, Mayr, E.W., and Vorozhtsov, E.V., Eds., Berlin: Springer, 2017; Lect. Notes Comput. Sci., vol. 10490, pp. 40–50.

  6. Bruno, A.D., Normal Form of a Hamiltonian System with a Periodic Perturbation, Comput. Math. Math. Phys., 2020, vol. 60, no. 1, pp. 36–52.

    Article  MathSciNet  Google Scholar 

  7. Bruno, A.D., Normalization of a Periodic Hamiltonian System, Program. Comput. Software, 2020, vol. 46, no. 2, pp. 76–83.

    Article  MathSciNet  Google Scholar 

  8. Zhuravlev, V.F., Petrov, A.G., and Shunderyuk, M.M., Selected Problems of Hamiltonian Mechanics, Moscow: LENAND, 2015 [in Russian].

    Google Scholar 

  9. Wolfram, S., The Mathematica Book, Wolfram Media, Inc., 2003.

    MATH  Google Scholar 

  10. Thompson, I., Understanding Maple, Cambridge Univ., 2016.

    MATH  Google Scholar 

  11. Meurer, A., Smith, C.P., Paprocki, M., et al., SymPy: Symbolic computing in Python, Peer J. Comput. Sci., 2017, vol. 3, P. 103. https://doi.org/10.7717/peerjcs.103

    Article  Google Scholar 

  12. Bruno, A.D., Analytical form of differential equations (II), Trans. Moscow Math. Soc., 1972, 26, pp. 199–239.

  13. Birkhoff, G.D., Dynamical Systems, New York: American mathematical society, 1927.

    MATH  Google Scholar 

  14. Lyapunov, A.M., Problème général de la stabilité du mouvement, Ann. Fac. Sci. Toulouse Math., 1892, vol. 9; Stability of Motion, New York: Academic, 1966.

    MATH  Google Scholar 

  15. Siegel, C.L. and Moser, J.K., Lectures on Celestial Mechanics Berlin: Springer, 1971.

    Book  Google Scholar 

  16. Kalinina, E.A. and Uteshev, A.Yu., Elimination Theory, St. Petersburg: Naucho-Issledovatel’skii Inst. Khimii, St. Petersburg Univ., 2002 [in Russian].

    Google Scholar 

  17. Basu, S., Pollack, R., and Roy, M.-F., Algorithms in Real Algebraic Geometry, in Algorithms and Computations in Mathematics, Vol. 10, Berlin: Springer, 2006.

    Google Scholar 

  18. Sushkevich, A.K., Foundation of Higher Algebra, 4 ed., Moscow: ONTI, 1941 [in Russian].

    Google Scholar 

  19. von zur Gathen, J. and Lücking, T., Subresultants revisited, Theor. Comput. Sci., 2003, vol. 297, pp. 199–239.

    Article  MathSciNet  Google Scholar 

  20. Batkhin, A.B., Parameterization of the discriminant set of a polynomial, Program. Comput. Software, 2016, vol. 42, no. 2, pp. 67–76.

    Article  MathSciNet  Google Scholar 

  21. Kac, V. and Cheung, P., Quantum Calculus, New York, Heidelber: Springer, 2002.

  22. Batkhin, A.B., Parameterization of the discriminant set of a polynomial, Program. Comput. Software, 2016, vol. 42, no. 2, pp. 65–76.

    Article  MathSciNet  Google Scholar 

  23. Batkhin, A.B., Parameterization of a set determined by the generalized discriminant of a polynomial, Program. Comput. Software, 2018, vol. 44, no. 2, pp. 75–85.

    Article  MathSciNet  Google Scholar 

  24. Batkhin, A.B., Computation of the generalized discriminant of a real polynomial, Preprint of the Keldysh Institute of Applied Mathematics, Moscow, 2017, no. 88. http://www.keldysh.ru/papers/2016/prep2017_88.pdf.

  25. Jury, E.I., Inners and Stability of Dynamic Systems, New York: Wiley, 1974.

    MATH  Google Scholar 

  26. Akritas, A.G., Elements of Computer Algebra with Applications, New York: Wiley, 1989.

    MATH  Google Scholar 

  27. Batkhin, A.B., Bruno, A. D., and Varin, V.P., Stability sets of multiparameter Hamiltonian systems, J. Appl. Math. Mech., 2012, vol. 76, no. 1, pp. 56–92.

    Article  MathSciNet  Google Scholar 

  28. Markeev, A.P., Linear Hamiltonian Systems and Some Problems of a Satellite Motion Stability Relative to the Center of Mass, Moscow: Regulyarnay i Chaoticheskaya Dinamika, 2009 [in Russian].

    Google Scholar 

  29. Markeev, A.P., Nonlinear oscillations of sympathetic pendulums, Rus. J. Nonlin. Dyn., 2010, vol. 6, no. 3, pp. 605–621.

    Google Scholar 

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ACKNOWLEDGMENTS

I am grateful to Prof. A. D. Bruno for his support and fruitful discussion of this work.

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  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 125047, Moscow, Russia

    A. B. Batkhin

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  1. A. B. Batkhin
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Correspondence to A. B. Batkhin.

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Translated by A. Klimontovich

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Batkhin, A.B. Invariant Coordinate Subspaces of Normal Form of a System of Ordinary Differential Equations. Program Comput Soft 47, 99–107 (2021). https://doi.org/10.1134/S0361768821020031

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