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A ball dropped on a vertically vibrating table exhibits intricate dynamical behaviors including period-doubling bifurcations and chaos. If the collision between the ball and the table is completely inelastic, the motion of the ball is always periodic, and the plateaus caused by saddle-node instability and clumping structures for periodic trajectories occur in the bifurcation diagram. Here the effect of air damping on the dynamics of the ball with zero elasticity is analyzed. The air damping is treated as linear viscous one. It is shown that a weak air damping does not change the sequence of bifurcations, but makes the bifurcation points shift to larger values and broadens the transverse dimensions of the plateaus and the clumping zones in the diagrams. However, when the air damping becomes larger, overlapping between the plateaus and clumping zones takes place. In the overlapping section, the mechanism originally leading to periodic motion is destroyed, and chaos is introduced.
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Keywords:
- period-doubling bifurcation /
- bouncing ball /
- chaos /
- air-damping
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[2] Lichtenberg A J, Lieberman M A 1983 Regular and Stochastic Motion (New York: Springer-Verlag) p190
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[26] Pak H K, Van Doorn E, Behringer R P 1995 Phys. Rev. Lett. 74 4643
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[28] Paster J M, Maza D, Zuriguel I, Garcimartíin A, Boudet J F 2007 Physica D 232 128
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[31] Yan X, Shi Q, Hou M, Lu K, Chan C K 2003 Phys. Rev. Lett. 91 14302
[32] Klein M, Tsai L L, Rosen M S, Pavlin T, Candela D Walsworth R L 2006 Phys. Rev. E 74 010301
[33] Liu C, Wang L, Wu P, Jia M 2010 Phys. Rev. Lett. 104 188001
[34] Naylor M A, Sanchez P, Swift M R 2002 Phys. Rev. E 66 057201
[35] Zhou Z G, Shi Y Y, Liu C B, Wang G H, Yang H J 2012 Acta. Phys. Sin. 61 200501 (in Chinese) [周志刚, 石玉仁, 刘丛波, 王光辉, 杨红娟 2012 物理学报 61 200501]
[36] Jiang Z H, Guo B, Zhang F, Wang F L 2010 Acta. Phys. Sin. 59 8444 (in Chinese) [姜泽辉, 郭波, 张峰, 王福力 2010 物理学报 59 8444]
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[1] Fermi E 1949 Phys. Rev. 15 1169
[2] Lichtenberg A J, Lieberman M A 1983 Regular and Stochastic Motion (New York: Springer-Verlag) p190
[3] Pustylnikov L D 1978 Trans. Moscow Math. Society 2 1
[4] Tufillaro N B, Albano A M 1986 Am. J. Phys. 54 939
[5] Tufillaro N B, Abbott T, Reilly J 1992 An experimental approach to nonlinear dynamics and chaos (Redwood: Addison-Wesley)
[6] Pierański P, Kowalik Z, Franaszek M 1985 J. Phys. 46 681
[7] Celaschi S, Zimmerman R L 1987 Phys. Lett. A 120 447
[8] Holmes P 1981 J. Sound Vibration 84 173
[9] Bapat C, Sankar S, Popplewell N 1986 J. Sound Vibration 108 99
[10] Tufillaro N B 1994 Phys. Rev. E 50 4509
[11] Luo A C J, Han R P S 1996 Nonlinear Dynamics 1996 10 1
[12] Barroso J J, Carneiro M V, Macau E E N 2009 Phys. Rev. E 79 026206
[13] Lichtenberg A J, Lieberman M A, Cohen R H 1980 Physica D 1 291
[14] Mehta A, Luck J M 1990 Phys. Rev. Lett. 65 393
[15] Luck J M, Mehta A 1993 Phys. Rev. E 48 3988
[16] Giusepponi S, Marchesoni F 2003 Europhys. Lett. 64 36
[17] Giusepponi S, Marchesoni F, Borromeo M 2005 Physica A 351 142
[18] Jiang Z H, Zhao H F, Zheng R H 2009 Acta. Phys. Sin. 58 7579 (in Chinese) [姜泽辉, 赵海发, 郑瑞华 2009 物理学报 58 7579]
[19] Gilet T, Vandewalle N, Dorbolo S 2009 Phys. Rev. E 79 1539
[20] Wassgren C R, Brennen C E, Hunt M L 1996 J. Appl. Mech. 63 712
[21] Jiang Z H, Liu X Y, Peng Y J, Li J W 2005 Acta. Phys. Sin. 54 5692 (in Chinese) [姜泽辉, 刘新影, 彭亚晶, 李建伟 2005 物理学报 54 5692]
[22] Jiang Z H, Wang Y Y, Wu J 2006 Europhys. Lett. 43 417
[23] Jiang Z H, Zheng R H, Zhao H F, Wu J 2007 Acta. Phys. Sin. 56 3727 (in Chinese) [姜泽辉, 郑瑞华, 赵海发, 吴晶 2007 物理学报 56 3727]
[24] Melo F, Umbanhowar P B, Swinney H L 1995 Phys. Rev. Lett. 75 3838
[25] Moon S J, Shattuck M D, Bizon C, Goldman D I, Swift J B, Swinney H L 2001 Phys. Rev. E 65 011301
[26] Pak H K, Van Doorn E, Behringer R P 1995 Phys. Rev. Lett. 74 4643
[27] Aoki K M, Akiyama T, Yamamoto K, Yoshikawa T 1997 Europhys. Lett. 40 159
[28] Paster J M, Maza D, Zuriguel I, Garcimartíin A, Boudet J F 2007 Physica D 232 128
[29] Möbius M E, Lauderdale B E, Nagel S R, Jaeger H M 2001 Nature 414 270
[30] Naylor M A, Swift M R, King P J 2003 Phys. Rev. E 68 012301
[31] Yan X, Shi Q, Hou M, Lu K, Chan C K 2003 Phys. Rev. Lett. 91 14302
[32] Klein M, Tsai L L, Rosen M S, Pavlin T, Candela D Walsworth R L 2006 Phys. Rev. E 74 010301
[33] Liu C, Wang L, Wu P, Jia M 2010 Phys. Rev. Lett. 104 188001
[34] Naylor M A, Sanchez P, Swift M R 2002 Phys. Rev. E 66 057201
[35] Zhou Z G, Shi Y Y, Liu C B, Wang G H, Yang H J 2012 Acta. Phys. Sin. 61 200501 (in Chinese) [周志刚, 石玉仁, 刘丛波, 王光辉, 杨红娟 2012 物理学报 61 200501]
[36] Jiang Z H, Guo B, Zhang F, Wang F L 2010 Acta. Phys. Sin. 59 8444 (in Chinese) [姜泽辉, 郭波, 张峰, 王福力 2010 物理学报 59 8444]
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