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Asymptotic Invariant Surfaces for Non-Autonomous Pendulum-Type Systems

Abstract

Invariant surfaces play a crucial role in the dynamics of mechanical systems separating regions filled with chaotic behavior. Cases where such surfaces can be found are rare enough. Perhaps the most famous of these is the so-called Hess case in the mechanics of a heavy rigid body with a fixed point.

We consider here the motion of a non-autonomous mechanical pendulum-like system with one degree of freedom. The conditions of existence for invariant surfaces of such a system corresponding to non-split separatrices are investigated. In the case where an invariant surface exists, combination of regular and chaotic behavior is studied analytically via the Poincaré-Mel’nikov separatrix splitting method, and numerically using the Poincaré maps.

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References

  1. Hess, W., Über die Euler’schen Bewegungsgleichungen und über eine neue particuläre Lösung des Problems der Bewegung eines starren Körpers um einen festen Punkt, Math. Ann., 1890, vol. 37, no. 2, pp. 153–181.

    MathSciNet  Article  Google Scholar 

  2. Appel’rot, G. G., Concerning Section 1 of the Memoir of S. V. Kovalevskaya “Sur le problème de la rotation d’un corps solide autour d’un point fixe”, and the Appendix to This Paper, Mat. Sb., 1892, vol. 16, no. 3, pp. 483–507 (Russian).

    Google Scholar 

  3. Nekrassov, P. A., Zur Frage von der Bewegung eines schweren starren Körpers um einen festen Punkt, Mat. Sb., 1892, vol. 16, no. 3, pp. 508–517 (Russian).

    Google Scholar 

  4. Zhukovsky, N. E., Hess’ Loxodromic Pendulum, in Collected Works: Vol. 1, Moscow: Gostekhizdat, 1937, pp. 332–348 (Russian).

    Google Scholar 

  5. Mlodzieiowski, B. K. and Nekrasov, P. A., Conditions for the Existence of Asymptotic Periodic Motions in the Hess Problem, Tr. Otdel. Fiz. Nauk Obsch. Lyubit. Estestvozn., 1893, vol. 6, no. 1, pp. 43–52 (Russian).

    Google Scholar 

  6. Chaplygin, S. A., Concerning Hess’ Loxodromic Pendulum, in Collected Works: Vol. 1, Moscow: Gostekhizdat, 1948, pp. 133–135 (Russian).

    Google Scholar 

  7. Nekrassov, P. A., Étude analytique d’un cas du mouvement d’un corps pesant autour d’un point fixe, Mat. Sb., 1896, vol. 18, no. 2, pp. 161–274 (Russian).

    Google Scholar 

  8. Demin, V. G. and Stepanova, L. A., Construction and Analysis of Exact Solutions for the Equations of Rigid-Body Dynamics, Soviet Appl. Mech., 1976, vol. 12, no. 9, pp. 875–887; see also: Prikl. Mekh., 1976, vol. 12, no. 9, pp. 3–17.

    Article  Google Scholar 

  9. Kozlov, V. V., Splitting of the Separatrices in the Perturbed Euler - Poinsot Problem, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1976, vol. 31, no. 6, pp. 99–104 (Russian).

    MathSciNet  MATH  Google Scholar 

  10. Ziglin, S. L., Splitting of Separatrices, Branching of Solutions and Nonexistence of an Integral in the Dynamics of a Solid Body, Trans. Moscow Math. Soc., 1982, no. 1, pp. 283–298; Tr. Mosk. Mat. Obs., 1980, vol. 41, pp. 287–303.

  11. Dovbysh, S. A., The separatrix of an unstable position of equilibrium of a Hess-Appelrot gyroscope, J. Appl. Math. Mech., 1992, vol. 56, no. 4, pp. 534–545.

    MathSciNet  Article  Google Scholar 

  12. Emel’yanova, I. S., A Case of the Solution of the Hess Problem in Trigonometric Functions, Russian Math. (Iz. VUZ), 1998, vol. 42, no. 3, pp. 7–12; see also: Izv. Vyssh. Uchebn. Zaved. Mat., 1998, no. 3, pp. 10–15.

    MathSciNet  MATH  Google Scholar 

  13. Dragović, V. and Gajić, B., An L-A Pair for the Hess-Apel’rot System and a New Integrable Case for the Euler - Poisson Equations on so(4) x so(4), Proc. Roy. Soc. Edinburgh Sect. A, 2001, vol. 131, no. 4, pp. 845–855.

    MathSciNet  Article  Google Scholar 

  14. Belyaev, A. V., Analytic Properties of Solutions of the Euler -Poisson Equations in the Hess Case, Ukr. Math. Bull., 2005, vol. 2, no. 3, pp. 301–321; see also: Ukr. Mat. Visn., 2005, vol. 2, no. 3, pp. 297–317.

    MathSciNet  Google Scholar 

  15. Lubowiecki, P. and Żołądek, H., The Hess - Appelrot System: 1. Invariant Torus and Its Normal Hyperbolicity, J. Geom. Mech., 2012, vol. 4, no. 4, pp. 443–467.

    MathSciNet  Article  Google Scholar 

  16. Lubowiecki, P. and Żołądek, H., The Hess - Appelrot System: 2. Perturbation and Limit Cycles, J. Differential Equations, 2012, vol. 252, no. 2, pp. 1701–1722.

    MathSciNet  Article  Google Scholar 

  17. Belyaev, A. V., On the General Solution of the Problem of the Motion of a Heavy Rigid Body in the Hess Case, Sb. Math., 2015, vol. 206, nos. 5–6, pp. 621–649; see also: Mat. Sb., 2015, vol. 206, no. 5, pp. 5–34.

    MathSciNet  Article  Google Scholar 

  18. Belyaev, A. V., On the Representation of Solutions of the Problem of a Heavy Rigid Body’s Motion in the Kovalevskaya Case by ζ-and \(\wp \)-Weierstrass Functions and Nonintegrability in Quadratures of the Hess Case, Sb. Math., 2016, vol. 207, nos. 7–8, pp. 889–914; see also: Mat. Sb., 2016, vol. 207, no. 7, pp. 3–28.

    MathSciNet  Article  Google Scholar 

  19. Kurek, R., Lubowiecki, P. and Żołądek, H., The Hess - Appelrot System: 3. Splitting of Separatrices and Chaos, Discrete Contin. Dyn. Syst. A, 2018, vol. 38, no. 4, pp. 1955–1981.

    MathSciNet  Article  Google Scholar 

  20. Żołądek, H., Perturbations of the Hess - Appelrot and the Lagrange Cases in the Rigid Body Dynamics, J. Geom. Phys., 2019, vol. 142, pp. 121–136.

    MathSciNet  Article  Google Scholar 

  21. Kozlov, V. V., Integrability and Non-Integrability in Hamiltonian Mechanics, Russian Math. Surveys, 1983, vol. 38, no. 1, pp. 1–76; see also: Uspekhi Mat. Nauk, 1983, vol. 38, no. 1(229), pp. 3–67.

    MathSciNet  Article  Google Scholar 

  22. Borisov, A. V. and Mamaev, I. S., Rigid Body Dynamics: Hamiltonian Methods, Integrability, Chaos, Izhevsk: R&C Dynamics, Institute of Computer Science, 2005 (Russian).

    MATH  Google Scholar 

  23. Borisov A. V., Mamayev I. S. The Hess Case in Rigid-Body Dynamics, J. Appl. Math. Mech., 2003, vol. 67, no. 2, pp. 227–235; see also: Prikl. Mat. Mekh., 2003, vol. 67, no. 2, pp. 256–265.

    MathSciNet  Article  Google Scholar 

  24. Burov, A. A., Nonintegrability of the Equation of Plane Oscillations of a Satellite in an Elliptic Orbit, Mosc. Univ. Mech. Bull., 1984, vol. 39, no. 1, pp. 38–41; see also: Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1984, no. 1, pp. 71–73.

    MathSciNet  MATH  Google Scholar 

  25. Koch, B.-P. and Bruhn, B., Chaotic and Periodic Motions of Satellites in Elliptic Orbits, Z. Naturforsch. A, 1989, vol. 44, no. 12, pp. 1155–1162.

    MathSciNet  Article  Google Scholar 

  26. Maciejewski, A. J., Non-Integrability of the Planar Oscillations of a Satellite, Acta Astron., 1995, vol. 45, no. 1, pp. 333–344.

    Google Scholar 

  27. Teofilatto, P. and Graziani, F., On Librational Motion of Spacecraft, Chaos Solitons Fractals, 1996, vol. 7, no. 10, pp. 1721–1744.

    MathSciNet  Article  Google Scholar 

  28. Cherry, T. M., The Asymptotic Solutions of Analytic Hamiltonian Systems, J. Differential Equations, 1968, vol. 4, no. 2, pp. 142–159.

    MathSciNet  Article  Google Scholar 

  29. Kozlov, V. V., Oscillations of One-Dimensional Systems with Periodic Potential, Mosc. Univ. Mech. Bull., 1980, vol. 35, nos. 5–6, pp. 74–78; see also: Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1980, no. 6, pp. 104–107, 120.

    MATH  Google Scholar 

  30. Burov, A. and Kosenko, I., On Planar Oscillations of a Body with a Variable Mass Distribution in an Elliptic Orbit, Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci., 2011, vol. 225, no. 10, pp. 2288–2295.

    Article  Google Scholar 

  31. Zhuravskii, A. M., Handbook of Elliptical Functions, Moscow: Akad. Nauk SSSR, 1941 (Russian).

    Google Scholar 

  32. Sretenskii, L. N., Some Integrability Cases for the Equations of Gyrostat Motion, Dokl. Akad. Nauk SSSR, 1963, vol. 149, no. 2, pp. 292–294 (Russian).

    MathSciNet  Google Scholar 

  33. Sretensky, L. N., On Some Cases of Motion of a Heavy Rigid Body with a Gyroscope, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1963, no. 3, pp. 60–71 (Russian).

  34. Kozlov V. V., Onishchenko D. A. Nonintegrability of Kirchhoff’s Equations, Sov. Math. Dokl., 1982, vol. 26, pp. 495–498; see also: Dokl. Akad. Nauk SSSR, 1982, vol. 266, no. 6, pp. 1298–1300.

    MATH  Google Scholar 

  35. Burov, A. A., Partial Integrals in the Problem on the Motion of a Body Suspended from a String, Izv. Akad. Nauk. SSSR. Mekh. Tverd. Tela, 1987, no. 2, p. 84 (Russian).

  36. Gorr, G. V. and Rubanovskii, V. N., On a New Class of Motions of a System of Heavy Hinged Rigid Bodies, J. Appl. Math. Mech., 1988, vol. 52, no. 5, pp. 551–555; see also: Prikl. Mat. Mekh., 1988, vol. 52, no. 5, pp. 707–712.

    MathSciNet  Article  Google Scholar 

  37. Burov, A. A., Particular Integrals of the Equations of Motion of a Rigid Body over a Smooth Horizontal Plane, Mech. Solids, 1986, vol. 21, no. 5, pp. 75–76; see also: Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela., 1986, no. 5, pp. 72–73.

    Google Scholar 

  38. Burov, A. A., On the Motion of a Heavy Rigid Body with Points Contiguous to a Smooth Surface, in Problems of Analytical Mechanics and Stability Theory: Collection of Papers Dedicated to the Memory of Academician V. V. Rumyantsev, V. V. Kozlov et al. (Eds.), Moscow: Fizmatlit, 2009, pp. 42–48 (Russian).

    Google Scholar 

  39. Burov, A. A. and Karapetyan, A. V., On the Motion of a Rigid Body in a Particle Flow, J. Appl. Math. Mech., 1993, vol. 57, no. 2, pp. 295–299; see also: Prikl. Mat. Mekh., 1993, vol. 57, no. 2, pp. 77–81.

    MathSciNet  Article  Google Scholar 

  40. Vecheslavov, V. V., Dynamics of Hamiltonian Systems under Piecewise Linear Force, JETP, 2005, vol. 100, no. 4, pp. 811–819; see also: Zh. Eksper. Teoret. Fiz., 2005, vol. 127, no. 4, pp. 915–924.

    Article  Google Scholar 

  41. Bizyaev, I. A., Borisov, A. V., and Mamaev, I. S., The Hess-Appelrot System and Its Nonholonomic Analogs, Proc. Steklov Inst. Math., 2016, vol. 294, pp. 252–275; see also: Tr. Mat. Inst. Steklova, 2016, vol. 294, pp. 268–292.

    Article  Google Scholar 

  42. Bizyaev, I. A., Borisov, A. V., and Mamaev, I. S., The Hess - Appelrot Case and Quantization of the Rotation Number, Regul. Chaotic Dyn., 2017, vol. 22, no. 2, pp. 180–196.

    MathSciNet  Article  Google Scholar 

  43. Beletskii, V. V., Essays on the Motion of Celestial Bodies, Basel: Birkhäuser, 2001.

    Book  Google Scholar 

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Funding

This research is partially supported by RFBR, grants 18-01-00335, project EMaDeS (Centro-01-0145-FEDER-000017), and the Portuguese Foundation for Science and Technologies via the Centre for Mechanical and Aerospace Science and Technologies, C-MAST, POCI-01-0145-FEDER-007718.

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Correspondence to Alexander A. Burov, Anna D. Guerman or Vasily I. Nikonov.

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The authors declare that they have no conflicts of interest.

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Burov, A.A., Guerman, A.D. & Nikonov, V.I. Asymptotic Invariant Surfaces for Non-Autonomous Pendulum-Type Systems. Regul. Chaot. Dyn. 25, 121–130 (2020). https://doi.org/10.1134/S1560354720010104

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  • DOI: https://doi.org/10.1134/S1560354720010104

Keywords

  • separatrices splitting
  • chaotic dynamics
  • invariant surface

MSC2010 numbers

  • 70H07
  • 70K40
  • 70K55