强噪声作用下双稳态Van der Pol系统的随机分岔

郜心茹; 吴志强; 陈胜利

doi:10.21656/1000-0887.440375

强噪声作用下双稳态Van der Pol系统的随机分岔

doi: 10.21656/1000-0887.440375

天津大学力学系，天津 300350

详细信息

作者简介:
郜心茹(1999―)，女，硕士(E-mail: rgaoxin@163.com)

陈胜利(1993―)，男，博士(E-mail: shengli_c@163.com)

通讯作者:
吴志强(1968―)，男，教授，博士，博士生导师(通讯作者. E-mail: zhiqwu@tju.edu.cn)

中图分类号: O324
计量
- 文章访问数: 43
- HTML全文浏览量: 22
- PDF下载量: 10
- 被引次数: 0
出版历程
- 收稿日期: 2023-12-29
- 修回日期: 2024-04-10
- 刊出日期: 2024-12-01

Stochastic Bifurcations of Bi-Stable Van der Pol Systems Under Strong Noise

Department of Mechanics, Tianjin University, Tianjin 300350, P.R.China

摘要

摘要: 从联合概率密度的角度分析随机非线性系统的随机分岔行为，现有研究通常需要人为判断概率密度特征有无本质变化，并且此过程无法自动化. 该文提出了一种新的计算方法，能够实现随机分岔点的自动计算. 以强噪声激励下的双稳态Van der Pol系统为例，分析了阻尼系数变化对随机动力学响应的影响. 研究结果表明，随着阻尼系数的增加，系统的联合概率密度会发生三次分岔，呈现四种不同类型的几何特征. 该文提出的方法有望应用于其他随机非线性系统的随机分岔行为研究.
- 随机P分岔 /
- 分岔点 /
- 联合概率密度 /
- 自动计算
Abstract: The analysis of the stochastic bifurcation behaviors of stochastic nonlinear systems often requires artificial judgment based on the joint probability density and cannot be automated. A new calculation method for automatic calculation of random bifurcation points was proposed. The bi-stable Van der Pol system under strong noise excitation was taken as an example, the influences of damping coefficient changes on stochastic dynamic responses were analyzed. The research results show that, the joint probability density of the system bifurcates for 3 times with the increase of the damping coefficient, exhibiting 4 different types of geometric features. The proposed method can hopefully be applied to the study of stochastic bifurcation behaviors of other stochastic nonlinear systems.
- stochastic P-bifurcation /
- bifurcation point /
- joint probability density /
- automatic calculation

HTML全文

图 1 转迁集图

Figure 1. Diagram of the transition set

下载: 全尺寸图片幻灯片

图 2 联合概率密度验证

注为了解释图中的颜色，读者可以参考本文的电子网页版本，后同.

Figure 2. The joint probability density validation

下载: 全尺寸图片幻灯片

图 3 D=0.06时，系统的分岔图

Figure 3. The bifurcation diagram of the system for D=0.06

下载: 全尺寸图片幻灯片

图 4 D=0.06时，系统联合概率密度曲面图及其几何特征图

Figure 4. System joint probability density surface graphs and geometric feature graphs for D=0.06

下载: 全尺寸图片幻灯片

表 1 激励强度对随机分岔点位置影响

Table 1. The influences of the stimulation intensity on the positions of stochastic bifurcation points

D	the disappearance of the limit cycle ε	the appearance of the origin peak ε	the disappearance of the peak on a limit cycle ε
0.06	0.061	0.072	0.180
0.07	0.053	0.073	0.193
0.08	0.043	0.073	0.207
0.09	0.031	0.072	0.218
0.1	0.022	0.072	0.232

下载: 导出CSV

参考文献(32)

[1]	陈予恕. 非线性振动系统的分叉和混沌理论[M]. 北京: 高等教育出版社, 1993. CHEN Yushu. Bifurcation and Chaos Theory of Nonlinear Vibration Systems[M]. Beijing: Higher Education Press, 1993. (in Chinese)
[2]	朱位秋. 非线性随机动力学与控制: Hamilton理论体系框架[M]. 北京: 科学出版社, 2003. ZHU Weiqiu. Nonlinear Stochastic Dynamics and Control: Hamilton Theory System Frame[M]. Beijing: Science Press, 2003. (in Chinese)
[3]	朱位秋, 蔡国强. 随机动力学引论[M]. 北京: 科学出版社, 2017. ZHU Weiqiu, CAI Guoqiang. Introduction to Stochastic Dynamics[M]. Beijing: Science Press, 2017. (in Chinese)
[4]	VAN DER POL B. Forced oscillations in a circuit with non-linear resistance (reception with reactive triode)[J]. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 1927, 3(13): 65-80. doi: 10.1080/14786440108564176
[5]	刘坤峰, 靳艳飞. 相关白噪声激励下双稳态Duffing-Van der Pol系统的随机分岔[J]. 动力学与控制学报, 2020, 18 (4): 12-18. LIU Kunfeng, JIN Yanfei. Stochastic bifurcation in bistable duffing-Van der Pol system driven by correlated white noises[J]. Journal of Dynamics and Control, 2020, 18(4): 12-18. (in Chinese)
[6]	顾仁财, 许勇, 郝孟丽, 等. Lévy稳定噪声激励下的Duffing-Van der Pol振子的随机分岔[J]. 物理学报, 2011, 60 (6): 157-161. GU Rencai, XU Yong, HAO Mengli, et al. Stochastic bifurcations in Duffing-Van der Pol oscillator with Lévy stable noise[J]. Acta Physica Sinica, 2011, 60(6): 157-161. (in Chinese)
[7]	LI Y, WU Z, ZHANG G, et al. Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation[J]. Advances in Difference Equations, 2019, 2019(1): 448. doi: 10.1186/s13662-019-2356-1
[8]	宋凯令, 吴志强. 加性噪声激励对双稳态Van der Pol系统联合概率密度的影响[J]. 应用力学学报, 2020, 37 (6): 2395-2403. SONG Kailing, WU Zhiqiang. Influence of additive noise excitation on joint probability density of bi-stable Van der Pol system[J]. Chinese Journal of Applied Mechanics, 2020, 37(6): 2395-2403. (in Chinese)
[9]	SUN Z, FU J, XIAO Y, et al. Delay-induced stochastic bifurcations in a bistable system under white noise[J]. Chaos: an Interdisciplinary Journal of Nonlinear Science, 2015, 25(8): 083102. doi: 10.1063/1.4927646
[10]	CHAMGOUÉ A C, YAMAPI R, WOAFO P. Bifurcations in a birhythmic biological system with time-delayed noise[J]. Nonlinear Dynamics, 2013, 73(4): 2157-2173. doi: 10.1007/s11071-013-0931-7
[11]	GUO Q, SUN Z, XU W. Stochastic bifurcations in a birhythmic biological model with time-delayed feedbacks[J]. International Journal of Bifurcation and Chaos, 2018, 28(4): 1850048. doi: 10.1142/S0218127418500487
[12]	GUIMFACK B A, MBAKOB Y R, TABI C B, et al. On stochastic response of fractional-order generalized birhythmic Van der Pol oscillator subjected to delayed feedback displacement and Gaussian white noise excitation[J]. Chaos, Solitons and Fractals: the Interdisciplinary Journal of Nonlinear Science, and Nonequilibrium and Complex Phenomena, 2022, 157. DOI: 10.1016/j.chaos.2022.111936.
[13]	YONKEU R M, YAMAPI R, FILATRELLA G, et al. Stochastic bifurcations induced by correlated noise in a birhythmic Van der Pol system[J]. Communications in Nonlinear Science and Numerical Simulation, 2016, 33: 70-84. doi: 10.1016/j.cnsns.2015.09.005
[14]	LI Y, WU Z. Stochastic P-bifurcation in a tri-stable Van der Pol system with fractional derivative under Gaussian white noise[J]. Journal of Vibroengineering, 2019, 21(3): 803-815. doi: 10.21595/jve.2019.20118
[15]	吴志强, 王文博, 张祥云. 三稳态Van der Pol系统随机P分岔电路实验研究[J]. 振动与冲击, 2018, 37 (13): 111-116. WU Zhiqiang, WANG Wenbo, ZHANG Xiangyun. A tri-stable Van der Pol system's stochastic P-bifurcation circuit experiment[J]. Journal of Vibration and Shock, 2018, 37(13): 111-116. (in Chinese)
[16]	李亚杰, 吴志强, 兰奇逊, 等. 联合噪声激励下分数阶三稳Van der Pol振子的随机P分岔[J]. 振动与冲击, 2021, 40 (16): 275-280. LI Yajie, WU Zhiqiang, LAN Qixun, et al. Stochastic P bifurcation in a tri-stable Van der Pol oscillator with fractional derivative excited by combined Gaussian white noises[J]. Journal of Vibration and Shock, 2021, 40(16): 275-280. (in Chinese)
[17]	郝颖, 吴志强. 三稳态Van der Pol-Duffng振子的随机P分岔[J]. 力学学报, 2013, 45 (2): 257-264. HAO Ying, WU Zhiqiang. Stochastic P-bifurcation of tri-stable Van der Pol-Duffing oscillator[J]. Chinese Journal of Theoretical and Applied Mechanics, 2013, 45(2): 257-264. (in Chinese)
[18]	吴志强, 王耀光, 张祥云, 等. 一类三稳态系统确定性及随机分岔现象分析[J]. 天津大学学报(自然科学与工程技术版), 2018, 51 (9): 895-902. WU Zhiqiang, WANG Yaoguang, ZHANG Xiangyun, et al. Deterministic and stochastic bifurcations of a tri-stable system[J]. Journal of Tianjin University (Science and Technology), 2018, 51 (9): 895-902. (in Chinese)
[19]	吴志强, 郝颖. 乘性色噪声激励下三稳态Van der Pol-Duffing振子随机P-分岔[J]. 物理学报, 2015, 64 (6): 53-58. WU Zhiqiang, HAO Ying. Stochastic P-bifurcations in tri-stable Van der Pol-Duffing oscillator with multiplicative colored noise[J]. Acta Physica Sinica, 2015, 64(6): 53-58. (in Chinese)
[20]	吴志强, 郝颖. 随机激励Van der Pol-Duffing方程三峰P-分岔[J]. 中国科学: 物理学力学天文学, 2013, 43 (4): 524-529. WU Zhiqiang, HAO Ying. Three-peak P-bifurcations in stochastically excited Van der Pol-Duffing oscillator[J]. Scientia Sinica: Physica, Mechanica & Astronomica, 2013, 43(4): 524-529. (in Chinese)
[21]	CHEN L C, LIU J, SUN J Q. Stationary response probability distribution of SDOF nonlinear stochastic systems[J]. Journal of Applied Mechanics, 2017, 84(5): 051006. doi: 10.1115/1.4036307
[22]	TIAN Y, WANG Y, JIANG H, et al. Stationary response probability density of nonlinear random vibrating systems: a data-driven method[J]. Nonlinear Dynamics, 2020, 100(3): 2337-2352. doi: 10.1007/s11071-020-05632-4
[23]	CHEN S, WU Z. Method for extracting geometrical characteristics of joint probability density based on contour lines[J]. Acta Mechanica Sinica, 2022, 38(2): 521029. doi: 10.1007/s10409-021-09017-x
[24]	沈壕. 强噪声学[M]. 北京: 科学出版社, 1996. SHEN Hao. Strong Noise Science[M]. Beijing: Science Press, 1996. (in Chinese)
[25]	LI P, YAN Y, LIN H. Numerical simulation and experimental researches on the vibration-acoustic coupled property of an aircraft model under strong reverberation noise[J]. Journal of Vibration and Control, 2017, 23(17): 2757-2766. doi: 10.1177/1077546315621417
[26]	MA Q, CAO S, GONG T, et al. Weak fault feature extraction of rolling bearing under strong Poisson noise and variable speed conditions[J]. Journal of Mechanical Science and Technology, 2022, 36(11): 5341-5351. doi: 10.1007/s12206-022-1001-1
[27]	LI H, WANG G Z, LI L J, et al. Design of the swimming system of a bionic jellyfish robot for seabed exploration[J]. Applied Ocean Research, 2023, 134: 103498. doi: 10.1016/j.apor.2023.103498
[28]	马乐, 刘杰, 马志丽, 等. 强噪声下全驱空中作业平台自抗扰反演控制[J]. 中国惯性技术学报, 2022, 30 (3): 395-402. MA Le, LIU Jie, MA Zhili, et al. Active disturbance rejection backstepping control for fully-actuated flight platform of aerial manipulation with strong noise[J]. Journal of Chinese Inertial Technology, 2022, 30(3): 395-402. (in Chinese)
[29]	雷春丽, 史佳硕, 马淑珍, 等. 强噪声环境下基于MSDCNN的滚动轴承故障诊断方法[J/OL]. 北京航空航天大学学报[2024-04-10]. https://doi.org/10.13700/j.bh.1001-5965.2023.0456. LEI Chunli, SHI Jiashuo, MA Shuzhen, et al. Based on MSDCNN in strong noise environment rolling bearing fault diagnosis method[J/OL]. Journal of Beijing University of Aeronautics and Astronautics[2024-04-10]. https://doi.org/10.13700/j.bh.1001-5965.2023.0456s. (in Chinese)
[30]	叶正伟, 邓生文, 梁相玲. Gauss白噪声激励下的永磁同步电动机模型的分岔分析[J]. 应用数学和力学, 2023, 44 (7): 884-894. doi: 10.21656/1000-0887.430285 YE Zhengwei, DENG Shengwen, LIANG Xiangling. Bifurcation analysis of the permanent magnet synchronous motor model under white Gaussian noises[J]. Applied Mathematics and Mechanics, 2023, 44(7): 884-894. (in Chinese) doi: 10.21656/1000-0887.430285
[31]	郝颖. 多稳态系统随机P-分岔及其在高维机翼颤振系统中的应用[D]. 天津: 天津大学, 2014. HAO Ying. Stochastic P-bifurcation in multi-stable system and its application in high dimensional wing flutter system[D]. Tianjin: Tianjin University, 2014. (in Chinese)
[32]	ALEXANDROV D V, BASHKIRTSEVA I A, CRUCIFIX M, et al. Nonlinear climate dynamics: from deterministic behaviour to stochastic excitability and chaos[J]. Physics Reports, 2021, 902: 1-60. doi: 10.1016/j.physrep.2020.11.002