一类碰撞振动系统在内伊马克沙克-音叉分岔点附近的局部两参数动力学

乐源

doi:10.6052/0459-1879-15-144

一类碰撞振动系统在内伊马克沙克-音叉分岔点附近的局部两参数动力学

doi: 10.6052/0459-1879-15-144

乐源^,

西南交通大学力学与工程学院应用力学与结构安全四川省重点实验室, 成都 610031

基金项目: 国家自然科学基金资助项目(11272268,11172246).

详细信息

通讯作者:
乐源,副教授,主要研究方向:动力系统的分岔与混沌、非光滑动力学.E-mail:yueyuan2011@home.swjtu.edu.cn

中图分类号: O313
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出版历程
- 收稿日期: 2015-04-23
- 修回日期: 2015-07-14
- 刊出日期: 2016-01-18

LOCAL DYNAMICAL BEHAVIOR OF TWO-PARAMETER FAMILY NEAR THE NEIMARK-SACKER-PITCHFORK BIFURCATION POINT IN A VIBRO-IMPACT SYSTEM

Yue Yuan^,

Applied Mechanics and Structure Safety Key Laboratory of Sichuan Province, School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu 610031, China

摘要

摘要: 考虑一类具有对称性的三自由度碰撞振动系统. 系统的庞加莱映射在一定条件下存在对称不动点,对应于系统的对称周期运动. 根据对称性导出庞加莱映射 P 是另外一个隐式虚拟映射 Q 的二次迭代. 推导了庞加莱映射对称不动点的解析表达式. 根据映射不动点的稳定性及分岔理论,映射 P 的对称不动点发生内伊马克沙克- 音叉(Neimark-Saker-pitchfork) 分岔对应于映射 Q 发生内伊马克沙克- 倍化(Neimark-Sakerflip)分岔. 利用隐式虚拟映射 Q ,通过对范式作两参数开折分析,研究了映射 P 的对称不动点在内伊马克沙克-音叉分岔点附近的局部动力学行为. 碰撞振动系统在这个余维二分岔点附近的局部动力学行为可能表现为投影后的庞加莱截面上的单一对称不动点、一对共轭不动点、单一对称拟周期吸引子以及一对共轭拟周期吸引子. 数值模拟得到了内伊马克沙克-音叉分岔点附近的各种可能情况. 内伊马克沙克-分岔和音叉分岔互相作用可能产生新的结果:对称不动点虽然首先分岔为两个共轭不动点,但是这两个共轭不动点是不稳定的,最终收敛到同一个对称拟周期吸引子.
- 碰撞振动系统 /
- 对称不动点 /
- 两参数开折 /
- 内伊马克沙克-音叉分岔 /
- 共轭拟周期吸引子
Abstract: A three-degree-of-freedom vibro-impact system with symmetry is considered. Due to the symmetry, the Poncar é map P is the second iteration of another virtual implicit map Q . It is shown that the symmetric period n-2 motion of the vibro-impact system corresponds to the symmetric fixed point of the Poncaré map. Then we can investigate bifurcations of the symmetric period n-2 motion by researching into bifurcations of the associated symmetric fixed point. Based on the symmetry of the system, it is shown that the Neimark-Saker-pitchfork bifurcation of the symmetric fixed point of the Poncaré map P corresponds to the Neimark-Saker-flip bifurcation of the map Q . By using the map Q , according to the two-parameter unfolding of the normal form, we reveal the possible local dynamical behaviors of the symmetric fixed point of the Poncaré map P near the Neimark-Saker-pitchfork bifurcation point in detail. Near this codimension two bifurcation point, the dynamic behaviors of the vibro-impact system can be expressed by a single symmetric fixed point, a pair of conjugate fixed points, a pair of conjugate quasi-periodic attractors or a single symmetric quasi-periodic attractor in the projected Poncaré section. The numerical simulation represents various possible cases near the Neimark- -Saker-pitchfork bifurcation point. It is shown that the interaction of the Neimark-Saker bifurcation and the pitchfork bifurcation may lead into the creation of some new results. The symmetric fixed point bifurcates into a pair of conjugate unstable fixed points firstly, and the two conjugate fixed points will bifurcate into the same symmetric quasi-periodic attractor finally.
- vibro-impact system /
- symmetric fixed point /
- two-parameter unfolding /
- Neimark-Sacker-pitchfork bifurcation /
- conjugate quasi-periodic attractors

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参考文献(38)

1 Guckenheimer J, Holmes P. Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields. New York: Springer, 1990

2 陆启韶. 分岔与奇异性. 上海:上海科技教育出版社, 1995 (Lu Qishao. Bifurcation and Singularity. Shanghai: Science, Technology and Education Press, 1995 (in Chinese))

3 Kuznetsov YA. Elements of Applied Bifurcation Theory, The Second Edition. New York: Springer-Verlag, 1998

4 Parker TS, Chua LO. Practical Numerical Algorithms for Chaotic Systems. Springer-Verlag, 1989

5 陈予恕, 唐云. 非线性动力学中的现代分析方法. 北京:科学出版社, 2000 (Chen Yushu, Tang Yun. Modern Analysis Methods of Nonlinear Dynamics. Beijing: Science Press, 2000 (in Chinese))

6 罗冠炜, 谢建华. 碰撞振动系统的周期运动和分岔. 北京:科学出版社, 2004 (Luo Guanwei, Xie Jianhua. The Periodic Motion and Bifurcations of Vibro-impact Systems. Beijing: Science Press, 2004 (in Chinese))

7 金栋平, 胡海岩. 碰撞振动与控制. 北京:科学出版社, 2005 (Jin Dongping, Hu haiyan. Vibration and Control of Collision. Beijing: Science Press, 2005 (in Chinese))

8 Holmes PJ. The dynamics of repeated impacts with a sinusoidally vibrating table. Journal of Sound and Vibration, 1982, 84 (2): 173-189

9 Shaw SW, Holmes PJ. Periodically forced linear oscillator with impacts: Chaos and long-period motions. Physical Review Letters,1983, 51 (8): 623-626

10 Shaw SW. A periodically forced piecewise linear oscillator. Journal of Sound and Vibration, 1983, 90(1): 129-155

11 Shaw SW. The dynamics of a harmonically excited system having rigid amplitude constraints, Part 1: Subharmonic motion and local bifurcation. Journal of Applied Mechanics, 1985, 52: 453-458

12 Shaw SW. The dynamics of a harmonically excited system having rigid amplitude constraints, Part 2: Chaotic motions and global bifurcations. Journal of Applied Mechanics, 1985, 52: 459-464

13 Shaw SW. Forced vibrations of a beam with one-sided amplitude constraint: Theory and experiment. Journal of Sound and Vibration,1985, 92(2): 199-212

14 Whiston GS. Global dynamics of a vibro-impacting linear oscillator. Journal of Sound and Vibration, 1987, 115(2): 303-319

15 Bapat CN, Sankar S. Single unit impact damper in free and forced vibration. Journal of Sound and Vibration, 1985, 99 (1): 85-94

16 Thompson JMT, Gha ari R. Chaos after period-doubling bifurcations in the resonance of an impact oscillator. Physics Letters A,1982, 91(1): 5-8

17 Jin L, Lu QS, Twizell EH. A method for calculating the spectrum of Lyapunov exponents by local maps in non-smooth impacting systems. Journal of Sound and Vibration, 2006, 298: 1019-1033

18 金俐, 陆启韶, 王琪. 双自由度非定点斜碰撞振动系统的动力学分析. 应用数学和力学, 2005, 26(7): 810-818 (Jin Li, Lu Qishao, Wang Qi. Dynamic analysis of two-degree-of-freedom oblique impact system with non-fixed impact positions. Applied Mathematics and Mechanics, 2005, 26(7): 810-818 (in Chinese))

19 Luo GW, Xie JH. Hopf bifurcation of a two-degree-of-freedom vibro-impact system. Journal of Sound and Vibration, 1998, 213(3):391-408

20 Luo GW, Xie JH. Bifurcation and chaos in a system with impacts. Physica D, 2001, 148: 183-200

21 Luo GW, Xie JH. Hopf bifurcation and chaos of a two-degree-offreedom vibro-impact system in two strong resonance cases. International Journal of Non-Linear Mechanics, 2002, 37(1): 19-34

22 Xie JH, Ding WC. Hopf-Hopf bifurcation and invariant torus T2 of a vibro-impact system. International Journal of Non-Linear Mechanics,2005, 40: 531-543

23 Ding WC, Xie JH. Torus T2 and its routes to chaos of a vibro-impact system. Physics Letters A, 2006, 349: 324-330

24 Ding WC, Xie JH. Dynamical analysis of a two-parameter family for a vibro-impact system in resonance cases. Journal of Sound and Vibration, 2005, 287: 101-115

25 Gan CB, Lei H. Stochastic dynamic analysis of a kind of vibroimpact system under multiple harmonic and random excitations. Journal of Sound and Vibrations, 2011, 330: 2174-2184

26 Zhai HM, Ding Q. Stability and nonlinear dynamics of a vibration system with oblique collisions. Journal of Sound and Vibrations,2013, 332: 3015-3031

27 Zhang YX, Luo GW. Torus-doubling bifurcations and strange nonchaotic attractors in a vibro-impact system. Journal of Sound and Vibrations, 2013, 332: 5462-5475

28 Ding WC, Li GF, Luo GW, et al. Torus T2 and its locking, doubling, chaos of a vibro-impact system. Journal of the Franklin Institute,2012, 349: 337-348

29 Kryzhevich S,Wiercigroch M. Topology of vibro-impact systems in the neighborhood of grazing. Physical D, 2012, 241: 1919-1931

30 Du ZD, Li YR, Shen J, et al. Impact oscillators with homoclinic orbit tangent to the wall. Physical D, 2013, 245: 19-33

31 Xu HD, Wen GL, Qin QX, et al. New explicit critical criterion of Hopf-Hopf bifurcation in a general discrete time system. Communication in Nolinear Science and Numerical Simulation, 2013, 18:2120-2128

32 冯进钤, 徐伟. 碰撞振动系统中周期轨擦边诱导的混沌激变. 力学学报. 2013, 45(1):30-36 (Feng Jinqian, Xu Wei. Grazing-induced chaostic crisis for periodic orbits in vibro-impact systems. Chinese Journal of theoretic and applied mechanics, 2013, 45(1):30-36 (in Chinese))

33 Yue Y, Xie JH. Symmetry of the Poncaré map and its influence on bifurcations in a vibro-impact system. Journal of Sound and Vibration,2009, 323: 292-312

34 Yue Y, Xie JH. Lyapunov exponents and coexistense of attractors in vibro-impact systems with symmetric two-sided constraints. Physics Letters A, 2009, 373: 2041-2046

35 Yue Y, Xie JH. Capturing the symmetry of attractors and the transition to symmetric chaos in a vibro-impact system. International Journal of Bifurcation and Chaos, 2012, 5(22): 1250109

36 Yue Y, Xie JH. Neimark-Sacker-pitchfork bifurcation of the symmetric period fixed point of the Poncaré map in a three-degree-offreedom vibro-impact system. International Journal of Nonlinear Mechanics, 2013, 48: 51-58

37 Luo GW, Zhang YL, Zhang JG. Dynamical behavior of a class of vibratory systems with symmetrical rigid stops near the point of codimension two bifurcation. Journal of Sound and Vibration, 2006,297: 17-36

38 Luo GW, Zhang YL, Chu YD, et al. Codimension two bifurcations of fixed points in a class of vibratory systems with symmetrical rigid stops. Nonlinear Analysis: RealWorld Applications, 2007, 8: 1272-1292

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