[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Log in

Segregation of stability domains of the Hamilton nonlinear system

  • Topical Issue
  • Published:
Automation and Remote Control Aims and scope Submit manuscript

Abstract

Consideration was given to an algorithm for segregation of the parts of stability domains in the space of parameters of the Hamiltonian linear system to which the Bryuno theorem of formal stability of the nonlinear Hamiltonian system is applicable. The algorithm was illustrated by a five-parameter problem with four degrees of freedom.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Malkin, I.G., Teoriya ustoichivosti dvizheniya (Theory of Motion Stability), Moscow: Nauka, 1966.

    Google Scholar 

  2. Batkhin, A.B., Bryuno, A.D., and Varin, V.P., Stability Sets of the Multiparameter Hamiltonian Systems, Prikl. Mat. Mekh., 2012, vol. 76, no. 1, pp. 80–133.

    MathSciNet  Google Scholar 

  3. Bryuno, A.D., Batkhin, A.B., and Varin, V.P., Calculation of the Stability Sets in the Multiparametric Problems, Preprint of Keldysh Inst. of Applied Mathematics, Moscow, 2010, no. 23, http://www.keldysh.ru/papers/2010/source/prep2010_23.pdf.

    Google Scholar 

  4. Bryuno, A.D., Ogranichennaya zadacha trekh tel: Ploskie periodicheskie orbity (Bounded Problem of Three Bodies. Planar Periodic Orbits), Moscow: Nauka, 1990.

    MATH  Google Scholar 

  5. Jury, E.I., Innors and Stability of Dynamic Systems, New York: Wiley, 1974. Translated under the title Innory i ustoichivost’ dinamicheskikh sistem, Moscow: Nauka, 1979.

    Google Scholar 

  6. Batkhin, A.B., Stability of One Multiparameter Hamilton System, Preprint of Keldysh Inst. of Applied Mathematics, Moscow, 2011, no. 69, http://www.keldysh.ru/papers/2011/source/prep2011 69.pdf.

    Google Scholar 

  7. Uteshev, A.Y. and Cherkasov, T.M., The Search for the Maximum of a Polynomial, J. Symbolic Comput., 1998, vol. 25, no. 5, pp. 587–618.

    Article  MathSciNet  MATH  Google Scholar 

  8. Basu, S., Pollack, R., and Roy, M.-F., Algorithms in Real Algebraic Geometry, vol. 10: Algorithms and Computation in Mathematics, Berlin: Springer, 2006, 2nd ed.

    Google Scholar 

  9. Batkhin, A.B., Nonlinear Stability of the Hamilton System in Linear Approximation, Preprint of Keldysh Inst. of Applied Mathematics, Moscow, 2012, no. 33, http://www.keldysh.ru/papers/2012/prep2012_33.pdf.

    Google Scholar 

  10. Kurosh, A.G., Kurs vysshei algebry (Course of Higher Algebra), Moscow: Nauka, 1968, 9th ed.

    Google Scholar 

  11. Zigel’, K. and Mozer, Yu., Lektsii po nebesnoi mekhanike (Lectures on Celestial Mechanics), Izhevsk: NITs “Regulyarnaya i Khaoticheskaya Dinamika,” 2001.

    Google Scholar 

  12. Moser, J., New Aspects in the Theory of Stability of Hamiltonian Systems, Comm. Pure Appl. Math., 1958, vol. 11, no. 1, pp. 81–114.

    Article  MathSciNet  MATH  Google Scholar 

  13. Markeev, A.P., Tochki libratsii v nebesnoi mekhanike i kosmodinamike (Libration Points in Celestial Mechanics and Space Dynamics), Moscow: Nauka, 1978.

    Google Scholar 

  14. Birkhoff, G.D., Dynamical Systems, Providence: AMS, 1966. Translated under the title Dinamicheskie sistemy, Izhevsk: Udmurt. Univ., 1999, vol. 8 (Library “R&C Dynamics”).

    MATH  Google Scholar 

  15. Mozer, Yu., KAM-teoriya i problemy ustoichivosti (KAM Theory and Stability Problems), Izhevsk: NITs “Regulyarnaya i Khaoticheskaya Dinamika,” 2001.

    Google Scholar 

  16. Bryuno, A.D., On Formal Stability of the Hamiltonian Systems, Mat. Zametki, 1967, vol. 1, no. 3, pp. 325–330.

    MathSciNet  MATH  Google Scholar 

  17. Gantmakher, F.R., Teoriya matrits (Theory of Matrices), Moscow: Fuzmatlit, 2004, 5th ed. Translated into English under the title Theory of Matrices, New York: Chelsea, 1959.

    Google Scholar 

  18. Markeev, A.P., Lineinye gamil’tonovy sistemy i nekotorye zadachi ob ustoichivosti dvizheniya sputnika otnositel’no tsentra mass (Linear Hamiltonian Systems and Some Problems of Satellite Motional Stability about the Center of Mass), Izhevsk: NITs “Regulyarnaya i Khaoticheskaya Dinamika,” Inst. Comput. Issled., 2009.

    Google Scholar 

  19. Maciejewski, A.J. and Gozdziewski, K., Normalization Algorithms of Hamiltonian Near an Equilibrium Point, Astrophys. Space Sci., 1991, vol. 179, pp. 1–11.

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhuravlev, V.F., Petrov, A.G., and Shunderyuk, M.M., Asimptoticheskaya simmetrizatsiya gamil’tonovykh sistem (Asymptotic Symmetrization of Hamiltonian Systems), Moscow: Mosk. Fiz.-Tekh. Inst., 2010.

    Google Scholar 

  21. Burgoyne, N. and Cushman, R., Normal Forms for Real Linear Hamiltonian Systems with Purely Imaginary Eigenvalues, Celestial Mech., 1974, vol. 8, pp. 435–443.

    Article  MathSciNet  MATH  Google Scholar 

  22. Mailybaev, A.A. and Seiranyan, A.P., Mnogoparametricheskie zadachi ustoichivosti. Teoriya i prilozheniya v mekhanike (Multiparameter Problems of Stability. Theory and Applications in Mechanics), Moscow: Fizmatlit, 2009.

    Google Scholar 

  23. Bryuno, A.D., Batkhin, A.B., and Varin, V.P., Stability Set of One Gyroscopic Problem, Preprint of Keldysh Inst. of Applied Mathematics, Moscow, 2010, no. 4, http://www.keldysh.ru/papers/2010/source/prep2010_04.pdf.

    Google Scholar 

  24. Batkhin, A.B., Bryuno, A.D., and Varin, V.P., Stability Sets of Multiparaneter Hamiltonian Systems, Preprint of Keldysh Inst. of Applied Mathematics, Moscow, 2011, no. 42, http://www.keldysh.ru/papers/2011/source/prep2011_42.pdf.

    Google Scholar 

  25. Akhiezer, N.I., Elementy teorii ellipticheskikh funktsii (Elements of the Theory of Elliptical Functions), Moscow: Nauka, 1970, 2nd ed.

    Google Scholar 

  26. Bryuno, A.D., Stepennaya geometriya v algebraicheskikh i differentsial’nykh uravneniyakh (Power Geometry in Algebraic and Differential Equations), Moscow: Fizmatlit, 1998.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Original Russian Text © A.B. Batkhin, 2013, published in Avtomatika i Telemekhanika, 2013, No. 8, pp. 47–64.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Batkhin, A.B. Segregation of stability domains of the Hamilton nonlinear system. Autom Remote Control 74, 1269–1283 (2013). https://doi.org/10.1134/S0005117913080043

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0005117913080043

Keywords

Navigation