ROTATIONAL DYNAMICS IN THE SYSTEM OF TWO COUPLED PENDULUMS


Cite this article as:

Smirnov L. А., Kryukov А. К., Osipov G. V. ROTATIONAL DYNAMICS IN THE SYSTEM OF TWO COUPLED PENDULUMS. Izvestiya VUZ, Applied Nonlinear Dynamics, 2015, vol. 23, iss. 5, pp. 41-61 DOI: 10.18500/0869-6632-2015-23-5-41-61


We consider dynamics in a pair of nonlinearly coupled pendulums. With existence of dissipation and constant torque such system can demonstrate in-phase periodical rotation in addition to the stable state. We have shown in numerical simulations that such in-
phase rotation becomes unstable at certain values of coupling strength. In the limit of small dissipation we have created an asymptotic theory that explains instability of the in-phase cycle. Found analytical equations for coupling strength values corresponding to the boundaries of the instability area. Numerical simulations show that there is a coupling strength interval where the system can have a pair of stable and unstable non in-phase cycles in addition to the stable in-phase motion. Therefore, we demonstrated that nonlinearly coupled pendulums have a bi-stability of the limit cycles. Analysed bifurcations which lead to originating and disappearing of non in-phase cycles.

 

Download full version

DOI: 
10.18500/0869-6632-2015-23-5-41-61
Literature

1. Pikovsky A., Rosenblum M., Kurths J. Synchronization. A Universal Concept in Nonlinear Sciences. Cambridge University Press, 2001.

2. Braun O., Kivshar Yu.S. The Frenkel–Kontorova model: Concepts, Methods, and Applications. Springer, 2004.

3. Yakushevich L.V. Nonlinear Physics of DNA. 2nd Edition. Wiley-Vch, 2004.

4. Afraimovich V.S., Nekorkin V.I., Osipov G.V., Shalfeev V.D. Stability, structures and chaos in nonlinear synchronization network. Singapore: World Scientific, 1994.

5. Astakhov V.V., Bezruckho B.P., Kuznetsov S.P., Seleznyov E.P. // ZhETF Letters. 1988. Vol. 14, No 1. P. 37.

6. Leeman C., Lereh P., Racine G. A., Martinoli P. // Phys. Rev. Lett. 1986. Vol. 56, No 12. P. 1291.

7. Ryu S., Yu W., Stroud D. // Phys. Rev. E. 1996. Vol. 53, No 3. P. 2190.

8. Kim B. J., Kim S., Lee S. J. // Phys. Rev. B. 1995. Vol. 51, No 13. P. 8462.

9. Kim J., Choe W. G., KimS., Lee H. J. // Phys. Rev. B. 1994. Vol. 49, No 1. P. 459.

10. Denniston C., Tang C. // Phys. Rev. Lett. 1995. Vol. 75, No 21. P. 3930.

11. Qjan M., Weng J.-Z. // Annals of Physics. 2008. Vol. 323. P. 1956.

12. Fishman R. S., Stroud D. // Phys. Rev. B. 1988. Vol. 38, No 1. P. 290.

13. Yakushevich L.V., Gapa S., Awrejcewicz J. Mechanical analog of the DNA base pair oscillations // Dynamical Systems. Theory and Applications / Eds. by J. Awrejcewicz et al. Lodz: Left Grupa, 2009. P. 879.

14. Yakushevich L.V. // Computer Research and Modeling. 2011. Vol. 3, No 3. P. 319.

15. Avreytsevich Y., Mlynarska S., Yakushevich L.V. // Journal of Applied Mathematics and Mechanics. 2013. Vol. 77, No 4. P. 1.

16. Krueger A., Protozanova E., Frank-Kamenetskii M. // Biophys. J. 2006. Vol. 90. P. 3091.

17. Takeno S., Peyrard M. // Physica D. 1996. Vol. 92. P. 140.

18. Zhang F. // Physica D. 1997. Vol. 110. P. 51.

19. Kosterlitz J.M., Thouless D.J. // J. Phys. C: Solid State Phys. 1973. Vol. 6. P. 1181.

20. Antoni M., Ruffo S. // Phys. Rev. E. 1995. Vol. 52, No 3. P. 2361.

21. Wang X.Y., Taylor P.L. // Phys. Rev. Lett. 1996. Vol. 76, No 4. P. 640.

22. Fillaux F., Carlile C.J. // Phys. Rev. B. 1990. Vol. 42, No 10. P. 5990.

23. Fillaux F., Carlile C. J., Kearley G. J. // Phys. Rev. B. 1991. Vol. 44, No 22. P. 12280.

24. Zhang F., Collins M. A., Kivshar Yu. S. // Phys. Rev. E. 1995. V. 51, No 4. P. 3774.

25. Acebron J. A., Bonilla L. L., P  ́ erez Vicente C. J., Ritort F., Spigler R.  ́ // Rev. Mod. Phys. 2005. Vol. 77, No 1. P. 137.

26. Tanaka H.-A., Lichtenberg A. J., Oishi S. // Phys. Rev. Lett. 1997. Vol. 78, No 11. P. 2104.

27. Tanaka H.-A., Lichtenberg A. J., Oishi S. // Physica D: Nonlin. Phenom. 1997. Vol. 100, No 3–4. P. 279.

28. Rohden M., Sorge A., Timme M., Witthaut D. // Phys. Rev. Lett. Vol. 109, No 6. P. 064101(1).

29. Rohden M., Sorge A., Witthaut D., Timme M.// Chaos. 2014. Vol.24, No1.P. 013123(1).

30. Olmi S., Navas A., Boccaletti S., Torcini A. //

31. Olmi S., Martens E. A., Thutupalli S., Torcini A. // Phys. Rev. E. 2015. Vol. 92, No 3. P. 030901(1).

32. Ha S.-Y., Kim Y., Li Z. // SIAM J. Appl. Dyn. Syst. 2014. Vol. 13, No 1. P. 466.

33. Gupta S., Campa A., Ruffo S. // Phys. Rev. E. 2014. Vol. 89, No 2. P. 022123(1).

34. Komarov M., Gupta S., Pikovsky A. // Europhysics Letters. 2014. Vol. 106, No 4. P. 40003(1).

35. Ji P., Peron T.K. DM., Menck P.J., Rodrigues F.A., Kurths J. // Phys. Rev. Lett. 2013. Vol. 110, No 21. P. 218701(1).

36. Ji P., Peron T.K. DM., Menck P.J., Rodrigues F.A., Kurths J. // Phys. Rev. E. 2014. Vol. 90, No 6. P. 062810(1).

37. Peron T.K. DM., Ji P., Rodrigues F. A., Kurths J. // Phys. Rev. E. 2015. Vol. 91, No 5. P. 052805(1).

38. Goldstein G. Classical Mechanics. 3rd Edition. Addison–Wesley, 2001.

39. Landau L.D., Lifshitz E.M. Mechanics. 5rd Edition. Moscow: Physmatlit, 2004.

40. Andronov A.A., Vitt A.A., Khaikin S.E. Theory of Oscillations. Moscow: Science, 1981.

41. Belykh V.N., Pedersen N.F., Soerensen O.H. // Phys. Rev. B. 1977. Vol. 16, No 11. P. 4853.

42. Tricomi F. Integrazioni di unequazione differenziale presentatasi in elettrotecnica //  ́ Annalidella Scuola Normale Superiore di Pisa-Classe di Scienze. 1933. Vol. 2, No 1.P. 1. 4 (2014).

43. Yakubovich V.A., Starzhinskiy V.M. Parametric resonance in linear systems. Moscow: Science, 1987.

44. Bogolyubov N.N., Mitropolsky Y.A. Asymptotic methods in the theory of non-linear oscillations. Moscow: Science, 1974.

Status: 
одобрено к публикации
Short Text (PDF): 
Full Text (PDF): 

BibTeX

@article{Смирнов-IzvVUZ_AND-23-5-41,
author = {L. А. Smirnov and А. К. Kryukov and G. V. Osipov },
title = {ROTATIONAL DYNAMICS IN THE SYSTEM OF TWO COUPLED PENDULUMS},
year = {2015},
journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
volume = {23},number = {5},
url = {http://andjournal.sgu.ru/en/articles/rotational-dynamics-in-the-system-of-two-coupled-pendulums},
address = {Саратов},
language = {russian},
doi = {10.18500/0869-6632-2015-23-5-41-61},pages = {41--61},issn = {0869-6632},
keywords = {synchronization,oscillator,nonlinear dynamics},
abstract = {We consider dynamics in a pair of nonlinearly coupled pendulums. With existence of dissipation and constant torque such system can demonstrate in-phase periodical rotation in addition to the stable state. We have shown in numerical simulations that such in- phase rotation becomes unstable at certain values of coupling strength. In the limit of small dissipation we have created an asymptotic theory that explains instability of the in-phase cycle. Found analytical equations for coupling strength values corresponding to the boundaries of the instability area. Numerical simulations show that there is a coupling strength interval where the system can have a pair of stable and unstable non in-phase cycles in addition to the stable in-phase motion. Therefore, we demonstrated that nonlinearly coupled pendulums have a bi-stability of the limit cycles. Analysed bifurcations which lead to originating and disappearing of non in-phase cycles.   Download full version }}