Period doublings in coupled parametrically forced damped pendulums

Sang-Yoon Kim; Kijin Lee

doi:10.1103/PhysRevE.54.1237

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Period doublings in coupled parametrically forced damped pendulums

Sang-Yoon Kim

Kijin Lee

School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332-0430

Department of Physics, Kangwon National University, Chunchon, Kangwon-Do 200-701, Korea

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Phys. Rev. E 54, 1237 – Published 1 August, 1996

DOI: https://doi.org/10.1103/PhysRevE.54.1237

Abstract

We study the critical behaviors of period doublings in N (N=2,3,4,...) coupled parametrically forced damped pendulums by varying A (the amplitude of the external driving force) and c (the strength of coupling). The two-coupled case with N=2 is first investigated. As A is increased, the coupled parametrically forced damped pendulums exhibit multiple period-doubling transitions to chaos. For each period-doubling transition to chaos, the zero-coupling critical point and an infinity of critical line segments constitute the critical set in the A-c plane. Three kinds of critical behaviors are found on the critical set. Note that the structure of the critical set and the critical behaviors are the same as those for the abstract system of the coupled one-dimensional maps. We also extend the results of the N=2 case to many-coupled cases with N⩾3, in which the critical behaviors depend on the range of coupling. © 1996 The American Physical Society.

References (25)

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P. Hadley and M.R. Beasley, Appl. Phys. Lett. 50, 621 (1987); P. Hadley, M.R. Beasley and K. Wiesenfeld, Phys. Rev. B 38, 8712 (1988).
S.H. Strogatz, C.M. Marcus, R.M. Westervelt and R.E. Mirollo, Physica D 36, 23 (1989).
Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer-Verlag, New York, 1984).
A.T. Winfree, The Geometry of Biological Time (Springer-Verlag, Berlin, 1980).
L.D. Landau and E.M. Lifshitz, Mechanics (Pergamon Press, New York, 1976), p. 80.
V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1978), p. 113; Ordinary Differential Equations (MIT Press, Cambridge, 1973), p. 203.
J.B. McLaughlin, J. Stat. Phys. 24, 375 (1981).
R.W. Leven and B.P. Koch, Phys. Lett. A 86, 71 (1981); B.P. Koch, R.W. Leven, B. Pompe and C. Wilke, ibid. 96, 219 (1983); B.P. Koch and R.W. Leven, Physica D 16, 1 (1985); R.W. Leven, B. Pompe, C. Wilke and B.P. Koch, ibid. 16, 371 (1985).
A. Arneodo, P. Coullet, C. Tresser, A. Libchaber, J. Maurer and D. d'Humières, Physica D 6, 385 (1983).
S.-Y. Kim and K. Lee, Phys. Rev. E 53, 1579 (1996).
M.J. Feigenbaum, J. Stat. Phys. 19, 25 (1978); ibid. 21, 669 (1979).
S. Kuznetsov, Radiophys. Quantum Electron. 28, 681 (1985); H. Kook, F.H. Ling and G. Schmidt, Phys. Rev. A 43, 2700 (1991).
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F.H. Ling, G. Schmidt, and H. Kook [Int. J. Bifurc. Chaos 1, 363 (1991)] attempted to study the critical behaviors of PDB's in coupled oscillators. However, only the critical behaviors near the zero-coupling critical point were considered, because the existence of an infinity of additional critical line segments in the coupled 1D maps was not known at that time. Moreover, without giving any explicit numerical values of the scaling factors, they insisted that the critical behaviors for the zero-coupling case are the same as those of the coupled 1D maps, based on the stability diagram of orbits with only low period 2 and 4 (e.g., see Fig. 1 in their paper).
S. Lefschetz, Differential Equations: Geometric Theory (Dover Publications, Inc., New York, 1977), Sec. 3.5.
S. Lefschetz, Differential Equations: Geometric Theory (Ref. 16 ), p. 60.
S.-Y. Kim and B. Hu, Phys. Rev. A 44, 934 (1991); S.-Y. Kim and D.-S. Lee, ibid. 45, 5480 (1992).
J. Gukenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983), Sec. 3.5.
A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983), Sec. 5.3.
S.-Y. Kim, Phys. Rev. E 49, 1745 (1994).
S.-Y. Kim, Phys. Rev. E 50, 4237 (1994).
J.-m. Mao and B. Hu, J. Stat. Phys. 46, 111 (1987); Int. J. Mod. Phys. B 2, 65 (1988) C. Reick, Phys. Rev. A 45, 777 (1992).
Omitted end note.
I. Waller and R. Kapral, Phys. Rev. A 30, 2047 (1984).

[c1] R.V. Buskirk and C. Jeffries, Phys. Rev. A 31, 3332 (1985).

[c2] P. Hadley and M.R. Beasley, Appl. Phys. Lett. 50, 621 (1987); P. Hadley, M.R. Beasley and K. Wiesenfeld, Phys. Rev. B 38, 8712 (1988).

[c3] S.H. Strogatz, C.M. Marcus, R.M. Westervelt and R.E. Mirollo, Physica D 36, 23 (1989).

[c4] Y. Kuramoto, Chemical Oscillations, Waves and Turbulence (Springer-Verlag, New York, 1984).

[c5] A.T. Winfree, The Geometry of Biological Time (Springer-Verlag, Berlin, 1980).

[c6] L.D. Landau and E.M. Lifshitz, Mechanics (Pergamon Press, New York, 1976), p. 80.

[c7] V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer-Verlag, New York, 1978), p. 113; Ordinary Differential Equations (MIT Press, Cambridge, 1973), p. 203.

[c8] J.B. McLaughlin, J. Stat. Phys. 24, 375 (1981).

[c9] R.W. Leven and B.P. Koch, Phys. Lett. A 86, 71 (1981); B.P. Koch, R.W. Leven, B. Pompe and C. Wilke, ibid. 96, 219 (1983); B.P. Koch and R.W. Leven, Physica D 16, 1 (1985); R.W. Leven, B. Pompe, C. Wilke and B.P. Koch, ibid. 16, 371 (1985).

[c10] A. Arneodo, P. Coullet, C. Tresser, A. Libchaber, J. Maurer and D. d'Humières, Physica D 6, 385 (1983).

[c11] S.-Y. Kim and K. Lee, Phys. Rev. E 53, 1579 (1996).

[c12] M.J. Feigenbaum, J. Stat. Phys. 19, 25 (1978); ibid. 21, 669 (1979).

[c13] S. Kuznetsov, Radiophys. Quantum Electron. 28, 681 (1985); H. Kook, F.H. Ling and G. Schmidt, Phys. Rev. A 43, 2700 (1991).

[c14] S.-Y. Kim and H. Kook, Phys. Rev. A 46, R4467 (1992); Phys. Lett. A 178, 258 (1993) Phys. Rev. E 48, 785 (1993) in the Proceeding of the First International Workshop on Nonlinear Dynamics and Chaos, edited by H. Lee (Pohang Institute of Science and Technology, Pohang, Korea, 1993), pp. 49–90.

[c15] F.H. Ling, G. Schmidt, and H. Kook [Int. J. Bifurc. Chaos 1, 363 (1991)] attempted to study the critical behaviors of PDB's in coupled oscillators. However, only the critical behaviors near the zero-coupling critical point were considered, because the existence of an infinity of additional critical line segments in the coupled 1D maps was not known at that time. Moreover, without giving any explicit numerical values of the scaling factors, they insisted that the critical behaviors for the zero-coupling case are the same as those of the coupled 1D maps, based on the stability diagram of orbits with only low period 2 and 4 (e.g., see Fig. 1 in their paper).

[c16] S. Lefschetz, Differential Equations: Geometric Theory (Dover Publications, Inc., New York, 1977), Sec. 3.5.

[c17] S. Lefschetz, Differential Equations: Geometric Theory (Ref. 16 ), p. 60.

[c18] S.-Y. Kim and B. Hu, Phys. Rev. A 44, 934 (1991); S.-Y. Kim and D.-S. Lee, ibid. 45, 5480 (1992).

[c19] J. Gukenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer-Verlag, New York, 1983), Sec. 3.5.

[c20] A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic Motion (Springer-Verlag, New York, 1983), Sec. 5.3.

[c21] S.-Y. Kim, Phys. Rev. E 49, 1745 (1994).

[c22] S.-Y. Kim, Phys. Rev. E 50, 4237 (1994).

[c23] J.-m. Mao and B. Hu, J. Stat. Phys. 46, 111 (1987); Int. J. Mod. Phys. B 2, 65 (1988) C. Reick, Phys. Rev. A 45, 777 (1992).

[c24] Omitted end note.

[c25] I. Waller and R. Kapral, Phys. Rev. A 30, 2047 (1984).

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