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On Some Constructions of Regular Linear Extensions of Dynamical Systems on a Torus

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We study Lyapunov functions in the form of a linear combination of quadratic forms and investigate the regularity conditions for linear extensions of dynamical systems on a torus.

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References

  1. A. M. Samoilenko, “Preservation of an invariant torus under perturbation,” Izv. Akad. Nauk SSSR, Ser. Math., 34, No. 6, 1219–1240 (1970); English translation: Math. USSR Izv., 4, Issue 6, 1225–1249 (1970).

  2. A. M. Samoilenko, Elements of the Mathematical Theory of Multi-Frequency Oscillations [in Russian], Science, Moscow (1987); English translation: Springer Dordrecht (1991);https://doi.org/10.1007/978-94-011-3520-7

  3. A. M. Samoilenko, “On some problems in perturbation theory of smooth invariant tori of dynamical systems,” Ukr. Mat. Zh., 46, No. 12, 1665–1699 (1994); English translation: Ukr. Math. J., 46, No. 12, 1848–1889 (1994); https://doi.org/10.1007/BF01063172.

  4. A. M. Samoilenko, “On the existence of a unique Green function for the linear extension of a dynamical system on a torus,” Ukr. Mat. Zh., 53, No. 4, 637–641 (2001); English translation: Ukr. Math. J., 53, No. 4, 584–594 (2001); https://doi.org/10.1023/A:1012326604969.

  5. Yu. A. Mitropol’skii, A. M. Samoilenko, and V. L. Kulyk, Investigation of the Dichotomy of Linear Systems of Differential Equations by Using Lyapunov Functions [in Russian], Naukova Dumka, Kiev (1990).

  6. A. A. Boichuk, “A condition for the existence of a unique Green–Samoilenko function for the problem of invariant torus,” Ukr. Mat. Zh., 53, No. 4, 556–559 (2001); English translation: Ukr. Math. J., 53, No. 4, 637–641 (2001).

  7. V. L. Kulyk and N. V. Stepanenko, “Alternating Lyapunov functions in the theory of linear extensions of dynamical systems on a torus,” Ukr. Mat. Zh., 59, No. 4, 488–500 (2007); English translation: Ukr. Math. J., 59, No. 4, 546–562 (2007); https://doi.org/10.1007/s11253-007-0035-y.

  8. V. L. Kulyk, A. N. Kulyk, and N. V. Stepanenko, “Supplementing weakly regular linear extensions of dynamical systems to regular extensions,” Math. J. Almaty, 11, No. 1(39), 74–86 (2011).

  9. I. M. Hrod and V. L. Kulyk, “Relationship between the Green and Lyapunov functions in linear extensions of dynamical systems,” Ukr. Mat. Zh., 66, No. 4, 551–557 (2014); English translation: Ukr. Math. J., 66, No. 4, 617–624 (2014); https://doi.org/10.1007/s11253-014-0958-z.

  10. J. Kenneth and J. Palmer, “On the reducibility of almost periodic systems of linear differential systems,” J. Different. Equat., 36, No. 3, 374–390 (1980); https://doi.org/10.1016/0022-0396(80)90056-X.

    Article  ADS  Google Scholar 

  11. A. N. Kulik, “Approximations of continuous periodic functions that are differentiate along the trajectories of dynamical systems,” Ukr. Mat. Zh., 38, No. 1, 111–114 (1986); English translation: Ukr. Math. J., 38, No. 1, 100–103 (1986); https://doi.org/10.1007/BF01056770.

  12. V. A. Lahoda and I. O. Parasyuk, “Theorem on the existence of an invariant section over Rm for the indefinite monotone system in ℝm × ℝn,” Ukr. Mat. Zh., 65, No. 1, 103–118 (2013); English translation: Ukr. Math. J., 65, No. 1, 114–131 (2013); https://doi.org/10.1007/s11253-013-0768-8.

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Authors and Affiliations

  1. Silesian University of Technology (Silesia Polytechnic in Gliwice), Kashubska St., 23, 44-100, Gliwice, Poland

    V. L. Kulyk

  2. “I. Sikorsky Kyiv Polytechnic Institute” National Technical University of Ukraine, Peremohy Ave., 37, Kyiv, 03056, Ukraine

    H. M. Kulyk & N. V. Stepanenko

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  1. V. L. Kulyk
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  2. H. M. Kulyk
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  3. N. V. Stepanenko
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Correspondence to V. L. Kulyk.

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Translated from Neliniini Kolyvannya, Vol. 26, No. 1, pp. 77–94, January–March, 2023.

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Kulyk, V.L., Kulyk, H.M. & Stepanenko, N.V. On Some Constructions of Regular Linear Extensions of Dynamical Systems on a Torus. J Math Sci 278, 1013–1033 (2024). https://doi.org/10.1007/s10958-024-06982-9

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  • DOI: https://doi.org/10.1007/s10958-024-06982-9

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