Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops

doi:10.1088/1674-1056/20/4/040505

Abstract A two-degree-of-freedom vibro-impact system having symmetrical rigid stops and subjected to periodic excitation is investigated in this paper. By introducing local maps between different stages of motion in the whole impact process, the Poincaré map of the system is constructed. Using the Poincaré map and the Gram-Schmidt orthonormalization, a method of calculating the spectrum of Lyapunov exponents of the above vibro-impact system is presented. Then the phase portraits of periodic and chaotic attractors for the system and the corresponding convergence diagrams of the spectrum of Lyapunov exponents are given out through the numerical simulations. To further identify the validity of the aforementioned computation method, the bifurcation diagram of the system with respect to the bifurcation parameter and the corresponding largest Lyapunov exponents are shown.

Keywords: vibro-impact system Poincaré map Gram--Schmidt orthonormalization Lyapunov exponent

Received: 14 August 2010
Revised: 16 November 2010
Accepted manuscript online:

PACS:	05.45.-a	(Nonlinear dynamics and chaos)
	05.10.-a	(Computational methods in statistical physics and nonlinear dynamics)

Fund: Project supported by the National Natural Science Foundation of China (Grant No. 10972059), the Natural Science Foundation of the Guangxi Zhuang Autonmous Region of China (Grant Nos. 0640002 and 2010GXNSFA013110), the Guangxi Youth Science Foundation of China (Grant No. 0832014) and the Project of Excellent Innovating Team of Guangxi University.

Cite this article:

Li Qun-Hong(李群宏) and Tan Jie-Yan(谭洁燕) Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops 2011 Chin. Phys. B 20 040505

[1]	Benettin G, Galgani L, Giorgilli A and Strelcyn J M 1980 Meccanica 15 9
[2]	Benettin G, Galgani L, Giorgilli A and Strelcyn J M 1980 Meccanica 15 21
[3]	Wolf A, Swift J B, Swinney H L and Vastano J A 1985 Physica D 16 285
[4]	Müller P C 1995 Chaos, Solitons and Fractals 5 1671
[5]	De Souza S L T and Caldas I L 2004 Chaos, Solitons and Fractals 19 569
[6]	Nordmark A B 1991 J. Sound Vib. 145 279
[7]	Jin L, Lu Q S and Twizell E H 2006 J. Sound Vib. 298 1019
[8]	Luo G W, Ma L and Lü X H 2009 Anal. Real. 10 756
[9]	Yue Y and Xie J H 2009 Phys. Lett. A 373 2041
[10]	Stefanski A 2000 Chaos, Solitons and Fractals 11 2443
[11]	Liu W D, Ren K F, Meunier-Guttin-Cluzel S and Gouesbet G 2003 Chin. Phys. 12 1366

[1]	Effect of pressure evolution on the formation enhancement in dual interacting vortex rings Jianing Dong(董佳宁), Yang Xiang(向阳), Hong Liu(刘洪), and Suyang Qin(秦苏洋). Chin. Phys. B, 2022, 31(8): 084701.
[2]	Stationary response of colored noise excited vibro-impact system Jian-Long Wang(王剑龙), Xiao-Lei Leng(冷小磊), and Xian-Bin Liu(刘先斌). Chin. Phys. B, 2021, 30(6): 060501.
[3]	Some new advance on the research of stochastic non-smooth systems Wei Xu(徐伟), Liang Wang(王亮), Jinqian Feng(冯进钤), Yan Qiao(乔艳), Ping Han(韩平). Chin. Phys. B, 2018, 27(11): 110503.
[4]	Coherent structures over riblets in turbulent boundary layer studied by combining time-resolved particle image velocimetry (TRPIV), proper orthogonal decomposition (POD), and finite-time Lyapunov exponent (FTLE) Shan Li(李山), Nan Jiang(姜楠), Shaoqiong Yang(杨绍琼), Yongxiang Huang(黄永祥), Yanhua Wu(吴彦华). Chin. Phys. B, 2018, 27(10): 104701.
[5]	Homoclinic and heteroclinic chaos in nonlinear systems driven by trichotomous noise You-Ming Lei(雷佑铭), Hong-Xia Zhang(张红霞). Chin. Phys. B, 2017, 26(3): 030502.
[6]	Multi-valued responses and dynamic stability of a nonlinear vibro-impact system with a unilateral non-zero offset barrier Wei Xu(徐伟), Dong-Mei Huang(黄冬梅), Wen-Xian Xie(谢文贤). Chin. Phys. B, 2016, 25(3): 030502.
[7]	A study of the early warning signals of abrupt change in the Pacific decadal oscillation Wu Hao (吴浩), Hou Wei (侯威), Yan Peng-Cheng (颜鹏程), Zhang Zhi-Sen (张志森), Wang Kuo (王阔). Chin. Phys. B, 2015, 24(8): 089201.
[8]	A circular zone counting method of identifying a Duffing oscillator state transition and determining the critical value in weak signal detection Li Meng-Ping (李梦平), Xu Xue-Mei (许雪梅), Yang Bing-Chu (杨兵初), Ding Jia-Feng (丁家峰). Chin. Phys. B, 2015, 24(6): 060504.
[9]	A perturbation method to the tent map based on Lyapunov exponent and its application Cao Lv-Chen (曹绿晨), Luo Yu-Ling (罗玉玲), Qiu Sen-Hui (丘森辉), Liu Jun-Xiu (刘俊秀). Chin. Phys. B, 2015, 24(10): 100501.
[10]	Effect of metal oxide arrester on the chaotic oscillations in the voltage transformer with nonlinear core loss model using chaos theory Hamid Reza Abbasi, Ahmad Gholami, Seyyed Hamid Fathi, Ataollah Abbasi. Chin. Phys. B, 2014, 23(1): 018201.
[11]	New predication of chaotic time series based on local Lyapunov exponent Zhang Yong (张勇). Chin. Phys. B, 2013, 22(5): 050502.
[12]	Stability of operation versus temperature of a three-phase clock-driven chaotic circuit Zhou Ji-Chao (周继超), Hyunsik Son, Namtae Kim, Han Jung Song. Chin. Phys. B, 2013, 22(12): 120506.
[13]	Stochastic responses of Duffing–Van der Pol vibro-impact system under additive colored noise excitation Li Chao (李超), Xu Wei (徐伟), Wang Liang (王亮), Li Dong-Xi (李东喜). Chin. Phys. B, 2013, 22(11): 110205.
[14]	Controlling and synchronization of a hyperchaotic system based on passive control Zhu Da-Rui (朱大锐), Liu Chong-Xin (刘崇新), Yan Bing-Nan (燕并男). Chin. Phys. B, 2012, 21(9): 090509.
[15]	Forming and implementing a hyperchaotic system with rich dynamics Liu Wen-Bo(刘文波), Wallace K. S. Tang(邓榤生), and Chen Guan-Rong(陈关荣) . Chin. Phys. B, 2011, 20(9): 090510.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Lyapunov exponent calculation of a two-degree-of-freedom vibro-impact system with symmetrical rigid stops

Cite this article:

Online attention