×

Analytically bounded and numerically unbounded compound pendulum chaos. (English) Zbl 1115.70308

Summary: A compound pendulum with deterministically periodic perturbation is treated. In the analytical approximation, chaotic solution initially near the homoclinic one is constructed and its boundedness conditions are established. It is shown that the chaotic solution is analytically bounded and numerically unbounded, which describes a non-periodical vibration around unstable equilibrium of the corresponding unperturbed system.

MSC:

70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Moon, F. C.; Holmes, P. J., J. Sound Vib., 65, 275 (1979)
[2] Sato, S.; Sano, M.; Sawada, Y., Phys. Rev. A, 28, 1654 (1993)
[3] Hikihara, T.; Kawagoshi, T., Phys. Lett. A, 211, 29 (1996)
[4] Liu, Z.; Chen, S., Chin. Phys. Lett., 14, 85 (1997)
[5] Leven, R. W.; Pompe, B.; Wilke, C.; Koch, B. P., Physica D, 16, 371 (1985)
[6] Leven, R. W.; Selent, M., Chaos, Solitons & Fractals, 4, 2217 (1994)
[7] Clifford, M. J.; Bishop, S. R., J. Sound Vib., 172, 572 (1994)
[8] Capecchi, D.; Bishop, S. R., Dyn. Stab. Syst., 9, 123 (1994)
[9] Bishop, S. R.; Clifford, M. J., J. Sound Vib., 189, 142 (1996)
[10] Hai, W.; Duan, Y.; Pan, L., Phys. Lett. A, 234, 198 (1997)
[11] Hai, W.; Duan, Y.; Zhu, X., Acta Phys. Sin., 46, 2117 (1997)
[12] Hai, W.; Duan, Y.; Zhu, X., J. Phys. A, 31, 2991 (1998)
[13] J. Ford, in: P. Davies (Ed.), The New Physics, Cambridge University Press, Cambridge, 1989, p. 348.; J. Ford, in: P. Davies (Ed.), The New Physics, Cambridge University Press, Cambridge, 1989, p. 348.
[14] Z. Lin, X. Yang, Stability Theory of Differential Equations(Fujian Science and Technology Press, Fuzhou, 1987) (in Chinese).; Z. Lin, X. Yang, Stability Theory of Differential Equations(Fujian Science and Technology Press, Fuzhou, 1987) (in Chinese).
[15] Z. Liu, Perturbation Criteria for Chaos, (Shanghai Scientific and Technological Education Press, Shanghai, 1994) (in Chinese).; Z. Liu, Perturbation Criteria for Chaos, (Shanghai Scientific and Technological Education Press, Shanghai, 1994) (in Chinese).
[16] Melnikov, V. K., Trans. Moscow Math. Soc., 12, 1 (1963)
[17] Bartuccelli, M.; Christiansen, P. L.; Pedersen, N. F.; Soerensen, M. P., Phys. Rev. B, 33, 4686 (1986)
[18] A. Barone, G. Paterno, Physics and Applications of the Josephson Effect, Wiley, New York, 1982.; A. Barone, G. Paterno, Physics and Applications of the Josephson Effect, Wiley, New York, 1982.
[19] Hai, W.; Xiao, Y.; Shufang, Jian; Huang, Weili; Zhang, Xili, Phys. Lett. A, 256, 128 (2000)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.