Skip to main content
SpringerLink
Log in

On the Motions of One Near-Autonomous Hamiltonian System at a 1:1:1 Resonance

  • Published:
Regular and Chaotic Dynamics Aims and scope Submit manuscript

Abstract

We consider the motion of a 2π-periodic in time two-degree-of-freedom Hamiltonian system in a neighborhood of the equilibrium position. It is assumed that the system depends on a small parameter e and other parameters and is autonomous at e = 0. It is also assumed that in the autonomous case there is a set of parameter values for which a 1:1 resonance occurs, and the matrix of the linearized equations of perturbed motion is reduced to a diagonal form. The study is carried out using an example of the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on an elliptical orbit with small eccentricity in the neighborhood of the cylindrical precession. The character of the motions of the reduced two-degree-of-freedom system in the vicinity of the resonance point in the three-dimensional parameter space is studied. Stability regions of the unperturbed motion (the cylindrical precession) and two types of parametric resonance regions corresponding to the case of zero frequency and the case of equal frequencies in the transformed approximate system of the linearized equations of perturbed motion are considered. The problem of the existence, number and stability of 2π-periodic motions of the satellite is solved, and conclusions on the existence of two- and three-frequency conditionally periodic motions are obtained.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Korteweg, D.J., Sur certaines vibrations d’ordre supérieur et d’intensité anormale — vibrations de relations, — dans les mécanismes à plusieurs degrés de liberté, Arch. Néerl. sci. exactes et natur. Sér. 2, 1898, vol. 1, pp. 229–260.

    MATH  Google Scholar 

  2. Beth, H. I. E., Les oscillations autour d’une position dans le cas d’existence d’une relation linéaire simple entre les nombres vibratoires, Arch. Néerl. sci. exactes et natur. Sér. 2, 1910, vol. 15, pp. 246–283.

    Google Scholar 

  3. Beth, H. I. E., Les oscillations autour d’une position dans le cas d’existence d’une relation linéaire simple entre les nombres vibratoires (suite), Arch. Néerl. sci. exactes et natur. Sér. 3A, 1912, vol. 1, pp. 185–213.

    MATH  Google Scholar 

  4. Markeev, A. P., Stability of a Canonical System with Two Degrees of Freedom in the Presence of Resonance, J. Appl. Math. Mech., 1968, vol. 32, no. 4, pp. 738–744; see also: Prikl. Mat. Mekh., 1968, vol. 32, no. 4, pp. 766–772.

    Article  MathSciNet  MATH  Google Scholar 

  5. Markeev, A. P., Libration Points in Celestial Mechanics and Space Dynamics, Moscow: Nauka, 1978 (Russian).

    Google Scholar 

  6. Henrard, J., Periodic Orbits Emanating from a Resonant Equilibrium, Celestial Mech., 1969/1970, vol. 1, nos. 3–4, pp. 437–466.

    MathSciNet  MATH  Google Scholar 

  7. Henrard, J., Lyapunov’s Center Theorem for Resonant Equilibrium, J. Differential Equations, 1973, vol. 14, no. 3, pp. 431–441.

    Article  MathSciNet  MATH  Google Scholar 

  8. Sweet, D., Periodic Solutions for Dynamical Systems Possessing a First Integral in the Resonance Case, J. Differential Equations, 1973, vol. 14, no. 1, pp. 171–183.

    Article  MathSciNet  MATH  Google Scholar 

  9. Schmidt, D. S., Periodic Solutions near a Resonant Equilibrium of a Hamiltonian System, Celestial Mech., 1974, vol. 9, no. 1, pp. 81–103.

    Article  MathSciNet  MATH  Google Scholar 

  10. Duistermaat, J. J., Bifurcation of Periodic Solutions near Equilibrium Points of Hamiltonian Systems, in Bifurcation Theory and Applications (Montecatini, 1983), L. Salvadori (Ed.), Lecture Notes in Math., vol. 1057, Berlin: Springer, 1984, pp. 57–105.

    Chapter  Google Scholar 

  11. Arnol’d, V.I., Kozlov, V. V., and Neĭshtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, 3rd ed., Encyclopaedia Math. Sci., vol. 3, Berlin: Springer, 2006.

    Book  Google Scholar 

  12. Joyeux, M., Classical Dynamics of the 1: 1, 1: 2 and 1: 3 Resonance Hamiltonians, Chem. Phys., 1996, vol. 203, no. 3, pp. 281–307.

    Article  Google Scholar 

  13. Markeev, A. P., On Stability and Non-Linear Oscillations of Hamiltonian System in a Resonant Case, Mech. Solids, 1998, vol. 33, no. 4, pp. 32–41; see also: Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 1998, no. 4, pp. 38–49.

    Google Scholar 

  14. Markeyev, A. P., The Critical Case of a Pair of Zero Roots in a Two-Degree-of-Freedom Hamiltonian System, J. Appl. Math. Mech., 1998, vol. 62, no. 3, pp. 341–349; see also: Prikl. Mat. Mekh., 1998, vol. 62, no. 3, pp. 372–382.

    Article  MathSciNet  Google Scholar 

  15. Markeyev, A. P., Non-Linear Oscillations of a Hamiltonian System with 2: 1 Resonance, J. Appl. Math. Mech., 1999, vol. 63, no. 5, pp. 715–726; see also: Prikl. Mat. Mekh., 1999, vol. 63, no. 5, pp. 757–769.

    Article  MathSciNet  Google Scholar 

  16. Markeyev, A. P., The Problem of the Stability of the Equilibrium Position of a Hamiltonian System at Resonance 3: 1, J. Appl. Math. Mech., 2001, vol. 65, no. 4, pp. 639–645; see also: Prikl. Mat. Mekh., 2001, vol. 65, no. 4, pp. 653–660.

    Article  MathSciNet  Google Scholar 

  17. Bardin, B. S., On Nonlinear Motions of Hamiltonian System in Case of Fourth Order Resonance, Regul. Chaotic Dyn., 2007, vol. 12, no. 1, pp. 86–100.

    Article  MathSciNet  MATH  Google Scholar 

  18. Bardin, B. S., On the Orbital Stability of Periodic Motions of a Hamiltonian System with Two Degrees of Freedom in the case of 3: 1 Resonance, J. Appl. Math. Mech., 2007, vol. 71, no. 6, pp. 880–891; see also: Prikl. Mat. Mekh., 2007, vol. 71, no. 6, pp. 976–988.

    Article  MathSciNet  MATH  Google Scholar 

  19. Bardin, B. S. and Chekin, A. M., Non-Linear Oscillations of a Hamiltonian System in the Case of 3: 1 Resonance, J. Appl. Math. Mech., 2009, vol. 73, no. 3, pp. 249–258; see also: Prikl. Mat. Mekh., 2009, vol. 73, no. 3, pp. 353–367.

    Article  MathSciNet  Google Scholar 

  20. Markeev, A. P., On a Multiple Resonance in Linear Hamiltonian Systems, Dokl. Phys., 2005, vol. 50, no. 5, pp. 278–282; see also: Dokl. Akad. Nauk, 2005, vol. 402, no. 3, pp. 339–343.

    Article  MathSciNet  Google Scholar 

  21. Markeyev, A. P., Multiple Parametric Resonance in Hamiltonian Systems, J. Appl. Math. Mech., 2006, vol. 70, no. 2, pp. 176–194; see also: Prikl. Mat. Mekh., 2006, vol. 70, no. 2, pp. 200–220.

    Article  MathSciNet  Google Scholar 

  22. Markeev, A. P., Linear Hamiltonian Systems and Some Problems on Stability of Motion of a Satellite about Its Center of Mass, Izhevsk: R&C Dynamics, Institute of Computer Science, 2009 (Russian).

    Google Scholar 

  23. Markeev, A. P., On One Special Case of Parametric Resonance in Problems of Celestial Mechanics, Astron. Lett., 2005, vol. 31, no. 5, pp. 350–356; see also: Pis’ma v Astron. Zh., 2005, vol. 31, no. 5, pp. 388–394.

    Article  MathSciNet  Google Scholar 

  24. Markeev, A. P., Multiple Resonance in One Problem of the Stability of the Motion of a Satellite Relative to the Center of Mass, Astron. Lett., 2005, vol. 31, no. 9, pp. 627–633; see also: Pis’ma v Astron. Zh., 2005, vol. 31, no. 9, pp. 701–708.

    Article  Google Scholar 

  25. Kholostova, O. V., On Periodic Motions of a Nonautonomous Hamiltonian System in One Case of Multiple Parametric Resonance, Nelin. Dinam., 2017, vol. 13, no. 4, pp. 477–504 (Russian).

    Article  MathSciNet  MATH  Google Scholar 

  26. Kholostova, O. V., Motions of a Two-Degree-of-Freedom Hamiltonian System in the Presence of Multiple Third-Order Resonances, Nelin. Dinam., 2012, vol. 8, no. 2, pp. 267–288 (Russian).

    Article  Google Scholar 

  27. Kholostova, O. V., The Interaction of Resonances of the Third and Fourth Orders in a Hamiltonian Two-Degree-of-Freedom System, Nelin. Dinam., 2015, vol. 11, no. 4, pp. 671–683 (Russian).

    Article  MathSciNet  MATH  Google Scholar 

  28. Kholostova, O. V., Stability of Triangular Libration Points in a Planar Restricted Elliptic Three Body Problem in Cases of Double Resonances, Int. J. Non Linear Mech., 2015, vol. 73, pp. 64–68.

    Article  Google Scholar 

  29. Safonov, A. I. and Kholostova, O. V., On the Periodic Motions of a Hamiltonian System in the Neighborhood of Unstable Equilibrium in the Presence of a Double Three-Order Resonance, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2016, vol. 26, no. 3, pp. 418–438 (Russian).

    Article  MATH  Google Scholar 

  30. Schmidt, D. S. and Sweet, D., A Unifying Theory in Determining Periodic Families for Hamiltonian Systems at Resonance, J. Differential Equations, 1973, vol. 14, no. 3, pp. 597–609.

    Article  MathSciNet  MATH  Google Scholar 

  31. van der Meer, J.-C., Nonsemisimple 1: 1 Resonance at an Equilibrium, Celestial Mech., 1982, vol. 27, no. 2, pp. 131–149.

    Article  MathSciNet  MATH  Google Scholar 

  32. Treshchev, D. V., Loss of Stability in Hamiltonian Systems That Depend on Parameters, J. Appl. Math. Mech., 1992, vol. 56, no. 4, pp. 492–500; see also: Prikl. Mat. Mekh., 1992, vol. 56, no. 4, pp. 587–596.

    Article  MathSciNet  MATH  Google Scholar 

  33. Bardin, B. S., On Motions near the Lagrange Equilibrium Point L 4 in the Case of Routh’s Critical Mass Ratio, Celest. Mech. Dynam. Astronom., 2002, vol. 82, no. 2, pp. 163–177.

    Article  MathSciNet  MATH  Google Scholar 

  34. Meyer, K. R. and Schmidt, D. S., Periodic Orbits near ℒ4 for Mass Ratios near the Critical Mass Ratio of Routh, Celestial Mech., 1971, vol. 4, no. 1, pp. 99–109.

    Article  MathSciNet  MATH  Google Scholar 

  35. Schmidt, D., Versal Normal Form of the Hamiltonian Function of the Restricted Problem of Three Bodies near L 4. Oscillations in Nonlinear Systems: Applications and Numerical Aspects, J. Comput. Appl. Math., 1994, vol. 52, nos. 1–3, pp. 155–176.

    Article  MathSciNet  Google Scholar 

  36. Sokolsky, A. G., To the Problem on Stability of Regular Precessions of a Symmetrical Satellite, Kosmicheskie Issledovaniya, 1980, vol. 18, no. 5, pp. 698–706 (Russian).

    Google Scholar 

  37. Hénon, M. and Heiles, C., The Applicability of the Third Integral of Motion: Some Numerical Experiments, Astronom. J., 1964, vol. 69, no. 1, pp. 73–79.

    Article  MathSciNet  Google Scholar 

  38. Roels, J., An Extension to Resonant Cases of Liapunov’s Theorem Concerning the Periodic Solutions near a Hamiltonian Equilibrium, J. Differential Equations, 1971, vol. 9, no. 2, pp. 300–324.

    Article  MathSciNet  MATH  Google Scholar 

  39. Braun, M., On the Applicability of the Third Integral of Motion, J. Differential Equations, 1973, vol. 13, no. 2, pp. 300–318.

    Article  MathSciNet  MATH  Google Scholar 

  40. Breiter, S. and Elipe, A., Pseudo-Oscillator with a Quartic Perturbation, Mech. Res. Comm., 2001, vol. 28, no. 2, pp. 119–126.

    Article  MathSciNet  MATH  Google Scholar 

  41. Markeyev, A. P., Non-Linear Oscillations of a 1: 1 Resonance Hamiltonian System, J. Appl. Math. Mech., 2011, vol. 75, no. 6, pp. 631–646; see also: Prikl. Mat. Mekh., 2011, vol. 75, no. 6, pp. 901–922.

    Article  MathSciNet  Google Scholar 

  42. Kholostova, O. V. and Safonov, A. I., Investigation of the Motions of an Autonomous Hamiltonian System at a 1: 1 Resonance, Regul. Chaotic Dyn., 2017, vol. 22, no. 7, pp. 792–807.

    Article  MathSciNet  MATH  Google Scholar 

  43. Beletskii, V. V., Motion of an Artificial Satellite about Its Center of Mass, Jerusalem: Israel Program for Scientific Translations, 1966.

    Google Scholar 

  44. Beletskii, V. V., Satellite’s Motion about Center of Mass in a Gravitational Field, Moscow: MGU, 1975 (Russian).

    Google Scholar 

  45. Markeev, A. P., On Rotational Motion of a Dynamically Symmetrical Satellite in an Elliptic Orbit, Kosmicheskie Issledovaniya, 1967, vol. 5, no. 4, pp. 530–539 (Russian).

    Google Scholar 

  46. Giacaglia, G. E. O., Perturbation Methods in Non-Linear Systems, Appl. Math. Sci., vol. 8, New York: Springer, 1972.

    Book  MATH  Google Scholar 

  47. Moser, J., Convergent Series Expansions for Quasi-Periodic Motions, Math. Ann., 1976, vol. 169, no. 1, pp. 136–176.

    Article  MathSciNet  MATH  Google Scholar 

  48. Mel’nikov, V. K., On Certain Cases of Conservation of Almost Periodic Motions with a Small Change of the Hamiltonian Function, Dokl. Akad. Nauk SSSR, 1965, vol. 165, no. 6, pp. 1245–1248 (Russian).

    MathSciNet  Google Scholar 

  49. Byrd, P. F. and Friedman, M. D., Handbook of Elliptic Integrals for Engineers and Scientists, 2nd ed., rev., Grundlehren Math. Wiss., vol. 67, Berlin: Springer, 1971.

    Book  MATH  Google Scholar 

  50. Markeev, A. P., Parametric Resonance and Nonlinear Oscillations of a Heavy Rigid Body in the Neighborhood of Its Planar Rotations, Izv. Ross. Akad. Nauk. Mekh. Tverd. Tela, 1995, no. 5, pp. 34–44 (Russian).

  51. Malkin, I. G., Some Problems in the Theory of Nonlinear Oscillations: In 2 Vols.: Vol. 1, German-town, Md.: United States Atomic Energy Commission, Technical Information Service, 1959.

    Google Scholar 

  52. Glimm, J., Formal Stability of Hamiltonian Systems, Comm. Pure Appl. Math., 1964, vol. 17, no. 4, pp. 509–526.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Moscow Aviation Institute, National Research University, Volokolamskoe sh. 4, GSP-3, A-80, Moscow, 125993, Russia

    Olga V. Kholostova

  2. Moscow Institute of Physics and Technology, National Research University, Institutskiy per. 9, Dolgoprudny, 141701, Russia

    Olga V. Kholostova

Authors
  1. Olga V. Kholostova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olga V. Kholostova.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kholostova, O.V. On the Motions of One Near-Autonomous Hamiltonian System at a 1:1:1 Resonance. Regul. Chaot. Dyn. 24, 235–265 (2019). https://doi.org/10.1134/S1560354719030018

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1560354719030018

Keywords

MSC2010 numbers

Search

Navigation