Bifurcation characteristics analysis of a class of nonlinear dynamical systems based on singularity theory

doi:10.1007/s10483-017-2234-8

Applied Mathematics and Mechanics (English Edition) ›› 2017, Vol. 38 ›› Issue (9): 1233-1246.doi: https://doi.org/10.1007/s10483-017-2234-8

Bifurcation characteristics analysis of a class of nonlinear dynamical systems based on singularity theory

Kuan LU^1,2, Yushu CHEN¹, Lei HOU^1,3

1. School of Astronautics, Harbin Institute of Technology, Harbin 150001, China;
2. College of Engineering, The University of Iowa, Iowa City, IA 52242, U. S. A.;
3. School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China

收稿日期:2016-10-16 修回日期:2017-02-15 出版日期:2017-09-01 发布日期:2017-09-01
通讯作者: Kuan LU,E-mail:lukuanyyzb@163.com E-mail:lukuanyyzb@163.com
基金资助:
Project supported by the National Basic Research Program of China (973 Program)(No.2015CB057400),the National Natural Science Foundation of China (No.11602070),and the China Postdoctoral Science Foundation (No.2016M590277)

Bifurcation characteristics analysis of a class of nonlinear dynamical systems based on singularity theory

Kuan LU^1,2, Yushu CHEN¹, Lei HOU^1,3

1. School of Astronautics, Harbin Institute of Technology, Harbin 150001, China;
2. College of Engineering, The University of Iowa, Iowa City, IA 52242, U. S. A.;
3. School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150001, China

Received:2016-10-16 Revised:2017-02-15 Online:2017-09-01 Published:2017-09-01
Contact: Kuan LU E-mail:lukuanyyzb@163.com
Supported by:
Project supported by the National Basic Research Program of China (973 Program)(No.2015CB057400),the National Natural Science Foundation of China (No.11602070),and the China Postdoctoral Science Foundation (No.2016M590277)

摘要/Abstract

摘要：

A method for seeking main bifurcation parameters of a class of nonlinear dynamical systems is proposed. The method is based on the effects of parametric variation of dynamical systems on eigenvalues of the Frechet matrix. The singularity theory is used to study the engineering unfolding (EU) and the universal unfolding (UU) of an arch structure model, respectively. Unfolding parameters of EU are combination of concerned physical parameters in actual engineering, and equivalence of unfolding parameters and physical parameters is verified. Transient sets and bifurcation behaviors of EU and UU are compared to illustrate that EU can reflect main bifurcation characteristics of nonlinear systems in engineering. The results improve the understanding and the scope of applicability of EU in actual engineering systems when UU is difficult to be obtained.

关键词: rigid line inclusion, elastic circular inclusion, stress intensity factor, interface stress, universal unfolding (UU), bifurcation, engineering unfolding (EU), nonlinear dynamics, singularity

Abstract:

Key words: singularity, rigid line inclusion, elastic circular inclusion, stress intensity factor, interface stress, nonlinear dynamics, bifurcation, universal unfolding (UU), engineering unfolding (EU)

中图分类号:

74H45

Kuan LU, Yushu CHEN, Lei HOU. Bifurcation characteristics analysis of a class of nonlinear dynamical systems based on singularity theory[J]. Applied Mathematics and Mechanics (English Edition), 2017, 38(9): 1233-1246.

参考文献

[1] Chen, Y. S. and Andrew Y. T. L. Bifurcation and Chaos in Engineering, Springer-Verlag, London (1998)
[2] Lu, K., Jin, Y. L., Chen, Y. S., Cao, Q. J., and Zhang, Z. Y. Stability analysis of reduced rotor pedestal looseness fault model. Nonlinear Dynamics, 82, 1611-1622(2015)
[3] Arnold, V. I. Singular Theory (London Mathematical Society Lecture Notes Series), Cambridge University Press, Cambridge (1981)
[4] Golubistky, M. and Schaeffer, D. G. Singularities and Groups in Bifurcation Theory I, SpringVerlag, New York (1985)
[5] Golubistky, M. and Schaeffer, D. G. Singularities and Groups in Bifurcation Theory/, SpringVerlag, New York (1988)
[6] Keyfitz, B. L. Classification of one state variable bifurcation problem up to codimension seven. Dynamical Systems, 1, 1-42(1986)
[7] Golubitsky, M. and Guillemin, V. Stable Mapping and Their Singularities, Spring-Verlag, New York (1973)
[8] Martinet, J. Singularities of Smooth Functions and Maps, Cambridge University Press, London (1982)
[9] Futer, J. E., Sitta, A. M., and Stewart, I. Singularity theory and equivariant bifurcation problems with parameter symmetry. Mathematical Proceedings of the Cambridge Philosophical Society, 120, 547-578(1996)
[10] Lari-Lavassani, A. and Lu, Y. C. Equivariant multi-parameter bifurcation via singularity theory. Journal of Dynamics and Differential Equations, 5, 189-218(1993)
[11] Uppal, A., Ray, W. H., and Poore, A. B. The classification of the dynamic behavior of continuous stirred tank reactors——influence of reactor residence time. Chemical Engineering Science, 31, 205-214(1976)
[12] Jin, J. D. and Matsuzaki, Y. Bifurcation analysis of double pendulum with a follower force. Journal of Sound and Vibration, 154, 191-204(1992)
[13] Jin, J. D. and Zou, G. S. Bifurcations and chaotic motions in the autonomous system of a restrained pipe conveying fluid. Journal of Sound and Vibration, 260, 783-805(2003)
[14] Bogoliubov, N. N. and Mitropolsky, Y. A. Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York (1961)
[15] Nayfeh, A. H. and Mook, D. T. Nonlinear Oscillations, John Wiley and Sons, New York (1979)
[16] Chen, Y. S. and Langford, W. F. The subharmonic bifurcation solution of nonlinear Mathieu's equation and Euler dynamically buckling problem. Acta Mechanica Sinica, 4, 350-362(1988)
[17] Langford, W. F. and Zhan, K. Dynamics of strong 1:1 resonance in vortex-induced vibration. Fundamental Aspects of Fluid-Structure Interactions, The American Society of Mechanical Engineers, Fairfield, NJ (1992)
[18] Golubisky, M. and Langford, W. F. Classification and unfoldings of degenerate Hopf bifurcations. Journal of Differential Equations, 41, 375-415(1981)
[19] Murdock, J. Asymptotic unfoldings of dynamical systems by normalizing beyond the normal form. Journal of Differential Equations, 143, 151-190(1998)
[20] Armbruster, D. and Kredel, H. Constructing universal unfoldings using Gröbner bases. Journal of Symbolic Computation, 2, 383-388(1986)
[21] Chen, F. Q., Liang, J. S., Chen, Y. S., Liu, X. J., and Ma, H. C. Bifurcation analysis of an arch structure with parametric and forced exciation. Mechanics Research Communications, 34, 213-221(2007)
[22] Zhang. W., Wang, F. X., and Zu, J. W. Local bifurcations and codimension-3 degenerate bifurcations of a quintic nonlinear beam under parametric excitation. Chaos, Solitons and Fractals, 24, 977-998(2005)
[23] Jones, M. Universal unfoldings of group invariant equations which model second and third harmonic resonant capillary-gravity waves. Computers and Mathematics with Applications, 32, 59-89(1996)
[24] Zhou, T. S., Chen, G. R., and Tang, Y. A. Universal unfolding of the Lorenz system. Chaos, Solitons and Fractals, 20, 979-993(2004)
[25] Lu, K., Chen, Y. S., Cao, Q. J., Hou L., and Jin, Y. L. Bifurcation analysis of reduced rotor model based on nonlinear transient POD method. International Journal of Nonlinear Mechanics, 89, 83-92(2017)
[26] Qin, Z. H. Singularity Method for Nonlinear Dynamical Analysis of Systems with Two Parameters and Its Application in Engineering (in Chinese), Ph. D. dissertation, Harbin Institute of Technology (2011)
[27] Hou, L. and Chen, Y. S. Bifurcation analysis of an aero-engine's rotor system under constant maneuver load. Applied Mathematics and Mechanics (English Edition), 36, 1417-1426(2015) DOI 10.1007/s10483-015-1992-7
[28] Qin, Z. H. and Chen, Y. S. Singular analysis of bifurcation systems with two parameters. Acta Mechanica Sinica, 26, 501-507(2010)
[29] Li, J. and Chen, Y. S. Transition sets of bifurcations of dynamical systems with two state variables with constraints. Applied Mathematics and Mechanics (English Edition), 33, 139-154(2012) DOI 10.1007/s10483-012-1539-7
[30] Zhang, H. B., Chen, Y. S., and Li, J. Bifurcation on the synchronous full annular rub of a rigidrotor elastic-support system. Applied Mathematics and Mechanics (English Edition), 33, 812-827(2012) DOI 10.1007/s10483-012-1591-7
[31] Seyranian, A. P. and Mailybaev, A. A. Multi-Parameter Stability Theory with Mechanical Application, Word Scientific, Singapore (2003)

Bifurcation characteristics analysis of a class of nonlinear dynamical systems based on singularity theory

Bifurcation characteristics analysis of a class of nonlinear dynamical systems based on singularity theory

PDF

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

编辑推荐

Metrics

本文评价

[1]	Runqing CAO, Zilong GUO, Wei CHEN, Huliang DAI, Lin WANG. Nonlinear dynamics of a circular curved cantilevered pipe conveying pulsating fluid based on the geometrically exact model[J]. Applied Mathematics and Mechanics (English Edition), 2024, 45(2): 261-276.
[2]	Huayang DANG, Dongpei QI, Minghao ZHAO, Cuiying FAN, C.S. LU. Thermal-induced interfacial behavior of a thin one-dimensional hexagonal quasicrystal film[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(5): 841-856.
[3]	Yansong WANG, Baolin WANG, Youjiang CUI, Kaifa WANG. Anti-plane pull-out of a rigid line inclusion from an elastic medium[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(5): 809-822.
[4]	Wei CHEN, Guozhen WANG, Yiqun LI, Lin WANG, Zhouping YIN. The quaternion beam model for hard-magnetic flexible cantilevers[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(5): 787-808.
[5]	Jian CHEN, Yawei WANG, Xianfang LI. Antiplane shear crack in a prestressed elastic medium based on the couple stress theory[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(4): 583-602.
[6]	Zhimin CHEN, W. G. PRICE. Secondary steady-state and time-periodic flows from a basic flow with square array of odd number of vortices[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(3): 447-458.
[7]	Peng WANG, Shaopu YANG, Yongqiang LIU, Pengfei LIU, Xing ZHANG, Yiwei ZHAO. Research on hunting stability and bifurcation characteristics of nonlinear stochastic wheelset system[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(3): 431-446.
[8]	Chenxue JIA, Zihao WANG, Donghui ZHANG, Taihua ZHANG, Xianhong MENG. Fracture of films caused by uniaxial tensions: a numerical model[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(12): 2093-2108.
[9]	Juan PENG, Dengke LI, Zaixing HUANG, Guangjian PENG, Peijian CHEN, Shaohua CHEN. Interfacial behavior of a thermoelectric film bonded to a graded substrate[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(11): 1853-1870.
[10]	Lele WANG, Liang WANG, Yueting ZHU, Zhanli LIU, Yongtao SUN, Jie WANG, Hongge HAN, Shuyi XIANG, Huibin SHI, Qian DING. Nonlinear dynamic response and stability analysis of the stapes reconstruction in human middle ear[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(10): 1739-1760.
[11]	A. MOSLEMI, M. R. HOMAEINEZHAD. Effects of viscoelasticity on the stability and bifurcations of nonlinear energy sinks[J]. Applied Mathematics and Mechanics (English Edition), 2023, 44(1): 141-158.
[12]	Duquan ZUO, B. SAFAEI, S. SAHMANI, Guoling MA. Nonlinear free vibrations of porous composite microplates incorporating various microstructural-dependent strain gradient tensors[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(6): 825-844.
[13]	Yaode YIN, Demin ZHAO, Jianlin LIU, Zengyao XU. Nonlinear dynamic analysis of dielectric elastomer membrane with electrostriction[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(6): 793-812.
[14]	Lei LI, Hanbiao LIU, Jianxin HAN, Wenming ZHANG. Nonlinear modal coupling in a T-shaped piezoelectric resonator induced by stiffness hardening effect[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(6): 777-792.
[15]	Pengyu PEI, Ming DAI. Elliptical inclusion in an anisotropic plane: non-uniform interface effects[J]. Applied Mathematics and Mechanics (English Edition), 2022, 43(5): 667-688.