时滞动力系统的稳定性与分岔:从理论走向应用

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doi:10.6052/1000-0992-12-018

王在华^1,2,,
胡海岩^1,3

1.
机械结构力学及控制国家重点实验室, 南京航空航天大学, 南京210016
2. 解放军理工大学理学院, 南京211101
3. 北京理工大学宇航学院, 北京100081

基金项目:国家自然科学基金重点项目(10532050),国家杰出青年科学基金(10825207),全国优秀博士学位论文作者专项基金(200430),国家自然科学基金重点项目(11032009)资助

详细信息

作者简介:
王在华, 1964 年生, 现任解放军理工大学教授和南京航空航天大学兼职教授. 2000 年毕业于南京航空航天大学一般力学与力学基础(动力学与控制) 专业,获理学博士学位, 学位论文被评为2004 年度全国优秀博士学位论文. 曾任德国科隆大学和斯图加特大学洪堡基金研究员以及科隆大学洪堡基金访问教授. 主要学术兴趣为具记忆效应的动力系统的建模、稳定性与非线性动力学,2008 年获得国家杰出青年科学基金和政府特殊津贴.

通讯作者:
王在华

中图分类号:O177.91
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- 被引次数:0
出版历程
- 收稿日期:2012-02-26
- 修回日期:2012-05-03
- 刊出日期:2013-01-24

Stability and bifurcation of delayed dynamic systems: from theory to application

WANG Zaihua^1,2,,
HU Haiyan^1,3

1.
State Key Laboratory of Mechanics and Control of Mechanical Structures Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
2. PLA University of Science and Technology, Institute of Science, Nanjing 211101, China
3. Beijing Institute of Technology, School of Aerospace Engineering, Beijing 100081, China

Funds:This project was supported by Key Program of the National Natural Science Foundation of China (10532050, 11032009), National Natural Science Foundation for Distinguished Young Scholars (10825207), and Foundation for the Authors of National Excellent Doctoral Dissertation of China (200430).

More Information

Corresponding author:WANG Zaihua

摘要

摘要:
本文综述了近年来时滞动力系统稳定性与分岔方面的研究进展, 重点阐述了作者及其团队在稳定性分析、Hopf分岔计算、利用时滞改善系统稳定性等方面的一些理论和方法研究结果, 介绍了时滞对颤振主动控制系统、不稳定系统镇定、网络系统的影响等方面的研究. 基于研究体会, 对进一步的研究提出了若干展望.
- 时滞/
- 稳定性/
- 分岔/
- 时滞反馈控制/
- 非线性系统/
- 记忆/
- 分数阶导数
Abstract:
This paper surveys recent advances in the study on the stability and bifurcation of time-delay systems. The review focuses mainly on the achievements of authors' team in the fundamental studies of stability, Hopf bifurcation and utilization of time delays to improve the system stability, as well as the delay effects on the stability of some typical systems or structures, including an aircraft wing, an inverted double-pendulum and networks. Finally, the paper addresses some open problems for future investigations.
- delay/
- stability/
- bifurcation/
- delayed feedback control/
- nonlinear systems/
- memory/
- fractional-order derivative

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参考文献 (140)

1 秦元勋, 王联, 刘永清, 等. 带有时滞的动力系统的稳定性. 北京: 科学出版社, 1989

2 Stepan G. Retarded Dynamical Systems: Stability and Characteristic Functions. Essex: Longman Scientific & Technical, 1989

3 Kuang Y. Delay Differential Equation with Application Dynamics. San Diego: Academic Press, 1993

4 Hu H Y, Wang Z H. Dynamics of Controlled Mechanical Systems with Delayed Feedback. Berlin: Springer, 2002

5 Niculescu S I. Delay Effects on Stability: A Robust Control Approach. London: Springer, 2001

6 Erneux T. Applied Delay Differential Equations. Berlin: Springer, 2009

7 胡海岩, 王在华. 非线性时滞动力系统的研究进展. 力学进展, 1999, 29(4): 501-512

8 徐鉴, 裴利军. 时滞系统动力学近期研究进展与展望. 力学进展, 2006, 36(1): 17-30

9 van Wiggeren G D, Roy R. Chaotic communication using time-delayed optical systems. International Journal of Bifurcation and Chaos, 1999, 9(11): 2129-2156

10 Lu H T, He Z Y. Chaotic behavior in first-order autonomous continuous-time systems with delay. IEEE Transactions on Circuits and Systems I : Fundamental Theory and Applications, 1996, 43(8): 700-702

11 Wang X. Period-doublings to chaos in a simple neural network: an analytical proof. Complex Systems, 1991, 5(4):425-441

12 Xu X. Complicated dynamics of a ring neural network with time delays. Journal of Physics A: Mathematical and Theoretical, 2008, 41(3): 035102

13 Haraguchi M, Hu H Y. Stability analysis of a noise control system in a duct by using delay differential equation. Acta Mechanica Sinica, 2009, 25(1): 131-137

14 Wang Q Y, Chen G R, Perc M. Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling. PLoS ONE, 2011, 6(1): e15851, doi:10.1371/journal.pone.0015851

15 Wang Q Y, Perc M, Duan Z S, et al. Delay-induced multiple stochastic resonances on scale-free neuronal networks. Chaos, 2009, 19(2): 023112/7

16 Hale J K, Verduyn Lunel S M. Introduction to Functional Differential Equations. New York: Springer, 1993

17 Insperger T, St é p á n G. Semi-Discretization for TimeDelay Systems: Stability and Engineering Applications. New York: Springer, 2011

18 王在华, 李俊余. 时滞状态正反馈在振动控制中的新特征. 力学学报, 2010, 42(5): 933-942

19 Wang Z H, Xu Q. Vibration control via positive delayed feedback. In: Naprstek J, Horacek J, Okrouhlik M, et al. eds. Vibration Problems ICoVP 2011-The 10th International Conference on Vibration Problems. Dordrecht: Springer, 2011. 385-391

20 胡海岩, 王在华. 论迟滞与时滞. 力学学报, 2010, 42(4):740-746

21 Nayfeh A H, Masoud Z N, Nayfeh N A. A smart sway controller for cranes from theory to laboratory to industry. In: Segla S, Tuma J, Petrikova I, et al. eds. Vibration Problems ICoVP 2011 SupplementThe 10th International Conference on Vibration Problems. Technical University of Liberec, 2011. 14-29

22 Liu Z H, Zhu W Q. Asymptotic Lyapunov stability with probability one of quasi-integrable Hamiltonian systems with delayed feedback control. Automatica, 2008, 44(7):1923-1928

23 Zhu W Q, Liu Z H. Response of quasi-integrable Hamiltonian systems with delayed feedback bang-bang control. Nonlinear Dynamics, 2007, 49(1-2): 31-47

24 Liu X P, Liu Z H, Huan R H, et al. Asymptotic Lyapunov stability with probability one of quasi linear systems subject to multi-time-delayed feedback control and wide-band parameter random excitation. Archive of Applied Mechanics, 2009, 79(11): 1051-1061

25 Zhang H Q, Xu W, Xu Y, et al. Delay induced transitions in an asymmetry bistable system and stochastic resonance. Science China: Physics, Mechanics & Astronomy,2010, 53(4): 745-750

26 Sun Z K, Xu W, Yang X L. Influence of time delay and noise on the chaotic motion of a bistable system. Physics Letters A, 2006, 353(1-2): 21-25

27 Li D X, Xu W, Guo Y F, et al. Transient properties of a bistable system with delay time driven by non-Gaussian and Gaussian noises: mean first-passage time. Communications in Theoretical Physics, 2008, 50(3): 669-673

28 Wang Q Y, Lu Q S. Time delay-enhanced synchronization and regularization in two coupled chaotic neurons. Chinese Physics Letters, 2005, 22(3): 543-546

29 Wang Q Y, Lu Q S, Duan Z S. Adaptive lag synchronization in coupled chaotic systems with unidirectional delay feedback. International Journal of Non-Linear Mechanics,2010, 45(6): 640-646

30 Duan L X, Lu Q S, Wang Q Y. Two-parameter bifurcation analysis of firing activity in the Chay neural model. Neurocomputing, 2008, 72(1-3): 341-351

31 Zhen B, Xu J. Bautin bifurcation analysis for synchronous solution of a coupled FHN neural system with delay. Communications in Nonlinear Science and Numerical Simulation,2010, 15(2): 442-458

32 Ge J H, Xu J. Computation of synchronized periodic solution in a BAM network with two delays. IEEE Transactions on Neural Networks, 2010, 21(3): 439-450

33 Ge J H, Xu J. Synchronization and synchronized periodic solution in a simplified five-neuron BAM neural network with delays. Neurocomputing, 2011, 74(6): 993-999

34 Zhou J, Chen T P. Synchronization in general complex delayed dynamical networks. IEEE Transactions on Circuits and Systems I, 2006, 53(3): 733-744

35 Zhou J, Xiang L, Liu Z R. Synchronization in complex delayed dynamical systems with impulsive effects. Physcica A: Statistical Mechanics and Its Applications,2007, 384(2): 684-692

36 Cai S M, Zhou J, Xiang L, et al. Robust impulsive synchronization of complex delayed dynamical networks. Physics Letters A, 2008, 372(12): 4990-4995

37 Li C P, Sun W G, Kurths J. Synchronization between two coupled complex networks. Physical Review E, 2007,76(3):046204

38 Li C P, Xu C X, Sun W G, et al. Outer synchronization of coupled discrete-time networks. Chaos, 2009, 19(1):013106

39 Cai G P, Huang J Z. Optimal control method for seismically excited building structures with time-delay in control. ASCE Journal of Engineering Mechanics, 2002,128(6): 602-612

40 Cai G P, Lim C W. Optimal tracking control of a flexible hub-beam system with time delay. Multibody System Dynamics, 2006, 16(4): 331-350

41 Liu K, Chen L X, Cai G P. Experimental study of delayed positive feedback control for a flexible beam. Theoretical and Applied Mechanics Letters, 2011, 1: 063003

42 Wang Z H, Hu H Y. Robust stability test for dynamic systems with short delays by using Pad′e approximation. Nonlinear Dynamics, 1999, 18(3): 275-287

43 Hu H Y, Dowell E H, Virgin L N. Stability estimation of high dimensional vibrating systems under state delay feedback control. Journal of Sound and Vibration, 1998,214(3): 497-511

44 Kolmanovskii V, Myshkis A. Introduction to the Theory and Applications of Functional Differential Equations. Dordrecht: Kluwer, 1999

45 Fu M Y, Olbrot A W, Polis M P. Robust stability for time-delay systems: the edge theorem and graphical tests. IEEE Transactions on Automatic Control, 1989, 34(8):813-820

46 Hassard B D. Counting roots of the characteristic equation for linear delay differential equations. Journal of Differential Equations, 1997, 136(2): 222-235

47 Buslowicz M. Stability of linear continuous-time fractional order systems with delays of the retarded type. Bulletin of the Polish Academy of Sciences: Technical Sciences,2008, 56(4): 319-324

48 Wang Z H, Du M L, Shi M. Stability test of fractionaldelay systems via integration. Science China: Physics, Mechanics & Astronomy, 2011, 54(10): 1839-1846

49 Yu Y J, Wang Z H. A graphical test for the interval stability of fractional-delay systems. Computers & Mathematics with Applications, 2011, 62(3): 1501-1509

50 Shi M, Wang Z H. An effective analytical criterion for stability testing of fractional-delay systems. Automatica,2011, 47(9): 2001-2005

51 Breda D, Maset S, Vermiglio R. Computing the characteristic roots for delay differential equations. IMA Journal of Numerical Analysis, 2004, 24(1): 1-19

52 Wang Z H, Hu H Y. Calculation of the rightmost characteristic root of retarded time-delay systems via Lambert Wfunction. Journal of Sound and Vibration, 2008, 318(4-5): 757-767

53 Verheyden K, Luzyanina T, Rose D. Efficient computation of characteristic roots of delay differential equations using LMS methods. Journal of Computational and Applied Mathematics, 2008, 214(1): 29-226

54 Wu Z, Michiels W. Reliably computing all characteristic roots of delay differential equations in a given right half plane using a spectral method. Journal of Computational and Applied Mathematics, 2011, 236(9): 2499-2514

55 李俊余, 王在华. 一类时滞系统Hurwitz 稳定性的简单判据. 动力学与控制学报, 2009, 7(2): 136-142

56 Wang Z H. Numerical stability test of neutral delay differential equations. Mathematical Problems in Engineering, Volume 2008, Article ID 698043, 10 pages, doi:10.1155/2008/698043

57 Marshall J E, Gorecki H, Korytowski A, et al. Time-delay Systems: Stability and Performance Criteria with Applications. New York: Prentice Hall, 1992

58 Li J Y, Zhang L, Wang Z H. Two effective stability criteria for linear time-delay systems with complex coefficients. Journal of System Science and Complexity, 2011, 24(5):835-849

59 Xu X, Hu H Y, Wang H L. Stability switches, Hopf bifurcation and chaos of a neuron model with delay-dependent parameters. Physics Letters A, 2006, 354(1-2): 126-136

60 Li J, Wang Z H. Hopf bifurcation of a nonlinear LasotaWazewska-type population model with maturation delay. Dynamics of Continuous Discrete and Impulsive Systems: Series B-Applications and Algorithms, 2007, 14(5): 611-623

61 Li J Y. Hopf bifurcation of the sunflower equation. Nonlinear Analysis: Real World Applications, 2009, 10(4):2574-2580

62 Zheng Y G, Wang Z H. Stability and Hopf bifurcation of a class of TCP/AQM networks. Nonlinear Analysis: Real World Applications, 2010, 11(3): 1552-1559

63 Bretta E, Kuang Y. Geometric stability switch criteria in delay differential equations with delay-dependent parameters. SIAM Journal on Mathematical Analysis, 2002,33(5): 1144-1165

64 Wang Z H. A very simple criterion for characterizing the crossing direction of time-delay systems with delaydependent parameters. International Journal of Bifurcation and Chaos, 2012, 22(3): 1250048 (7p)

65 Xu X, Hu H Y, Wang H L. Stability, bifurcation and chaos of a delayed oscillator with negative damping and delayed feedback control. Nonlinear Dynamics, 2007, 49(1-2):117-129

66 Xu X, Hu H Y, Wang H L. Dynamics of a two dimensional delayed small-world network under delayed feedback control. International Journal of Bifurcation and Chaos, 2006, 16(11): 3257-3273

67 Xu X, Wang Z H. Effects of small world connection on the dynamics of a delayed ring network. Nonlinear Dynamics,2009, 56(1-2): 127-144

68 Xu X. Local and global Hopf bifurcation in a two-neuron network with multiple delays. International Journal of Bifurcation and Chaos, 2008, 18(4): 1015-1028

69 Mao X C, Hu H Y. Stability and bifurcation analysis of a network of four neurons with time delays. ASME Journal of Computational and Nonlinear Dynamics, 2010, 5(4):041001

70 茅晓晨, 胡海岩. 四神经元时滞网络的稳定性与分岔. 力学季刊, 2009, 30(1): 1-7

71 Zhang L, Wang H L, Hu H Y. Symbolic computation of normal form for Hopf bifurcation in a retarded functional differential equation with unknown parameters. Communications in Nonlinear Science and Numerical Simulation,2012, 17(8): 3328-3344

72 Nayfeh A H. Order reduction of retarded nonlinear systems: the method of multiple scales versus centermanifold reduction. Nonlinear Dynamics, 2008, 51(4):483-500

73 Das S L, Chatterjee A. Multiple scales without center manifold reductions for delay differential equations near Hopf bifurcations. Nonlinear Dynamics, 2002, 30(4): 323-335

74 Rand R, Verdugo A. Hopf bifurcation formula for first order differential equations. Communications in Nonlinear Science and Numerical Simulation, 2007, 12(6): 859-864

75 Zheng Y G, Wang Z H. Stability and Hopf bifurcations of an optoelectronic time-delay feedback system. Nonlinear Dynamics, 2009, 57(1-2): 125-134

76 Hu H Y, Wang Z H. Singular perturbation methods for nonlinear dynamic systems with time delay. Chaos, Solitons and Fractals, 2009, 40(1): 13-27

77 Jin Y F, Hu H Y. Principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation. Nonlinear Dynamics, 2007,50(1-2): 213-227

78 Li J Y, Zhang L,Wang Z H. A simple algorithm for testing the stability of periodic solutions of some nonlinear oscillators with large time delay. Science China: Technological Sciences, 2011, 54(8): 2033-2043

79 Wang Z H, Hu H Y. A modified averaging scheme with application to secondary Hopf bifurcation of a delayed van der Pol oscillator. Acta Mechanica Sinica, 2008, 24(4):449-454

80 Wang Z H, Hu H Y. An energy analysis of the local dynamics of a delayed oscillator near a Hopf bifurcation. Nonlinear Dynamics, 2006, 46(1-2): 149-159

81 Wang Z H, Hu H Y. Pseudo-oscillator analysis of scalar nonlinear time-delay systems near a Hopf bifurcation. International Journal of Bifurcation and Chaos, 2007,17(8): 2805-2814

82 Gao F,Wang H L,Wang Z H. Hopf bifurcation of a nonlinear delayed system of machine tool vibration via pseudooscillator analysis. Nonlinear Analysis, Series B: Real World Applications, 2007, 8(5): 1561-1568

83 Zhang L L, Huang L H, Zhang Z Z. Hopf bifurcation of the maglev time-delay feedback system via pseudo-oscillator analysis. Mathematical and Computer Modeling, 2010,52(5-6): 667-673

84 Li J Y, Wang Z H. Local Hopf bifurcation of complex nonlinear systems with time-delay. International Journal of Bifurcation and Chaos, 2009, 19(3): 1069-1079

85 俞亚娟, 王在华. 伪振子分析法的证明及其在高阶Hopf 分岔中的应用. 动力学与控制学报, 2012, 10(3): 202-208

86 Wang Z H. An iteration method for calculating the periodic solution of time-delay systems after a Hopf bifurcation. Nonlinear Dynamics, 2008, 53(1-2): 1-11

87 郑远广, 黄承代, 王在华. 反馈时滞对van der Pol 振子张弛振荡的影响. 力学学报, 2012, 44(1): 148-157

88 Zheng Y G, Wang Z H. Delayed Hopf bifurcation in timedelayed slow-fast systems. Science China: Technological Sciences, 2010, 53(3): 656-663

89 Zheng Y G, Wang Z H. The impact of delayed feedback on the dynamics of class-B lasers. International Journal of Non-Linear Mechanics, 2010, 45(7): 727-733

90 郑远广, 王在华. 含时滞的快慢耦合系统的动力学研究进展. 力学进展, 2011, 41(4): 415-424

91 Wang H L, Hu H Y,Wang Z H. Global dynamics of a Duffing oscillator with delayed displacement feedback. International Journal of Bifurcation and Chaos, 2004, 14(8):2753-2775

92 Zhang L, Wang H L, Hu H Y. Global view of Hopf bifurcations of a van der Pol oscillator with delayed state feedback. Science China: Technological Sciences, 2010,53(3): 595-607

93 Xu J, Chung K W, Chan C L. An efficient method for studying weak resonant double Hopf bifurcation in nonlinear systems with delayed feedback. SIAM Journal on Applied Dynamical Systems, 2007, 6(1): 29-60

94 Xu J, Chung K W. A perturbation-incremental scheme for studying Hopf bifurcation in delayed differential systems. Science China: Technological Sciences, 2009, 52(3): 698-708

95 Xu J, Chung K W. Double Hopf bifurcation with strong resonances in delayed systems with nonlinearities. Mathematical Problems in Engineering, Volume 2009, Article ID 759363, 16 pages, doi:10.1155/2009/759363

96 Ma S Q, Lu Q S, Feng Z S. Double Hopf bifurcation for van der Pol-Duffing oscillator with parametric delay feedback control. Journal of Mathematical Analysis and Applications,2008, 338 (2): 993-1007

97 Ma S Q, Feng Z S, Lu Q S. Dynamics and double-Hopf bifurcations of the Rose-Hindmarsh model with time delay. International Journal of Bifurcation and Chaos, 2009,19(11): 3733-3751

98 王在华, 胡海岩, 王怀磊. 一个时滞反馈受控机电系统中的暂态混沌. 动力学与控制学报, 2004, 2(4): 24-28

99 Hu H Y. Using delayed state feedback to stabilize periodic motions of an oscillator. Journal of Sound and Vibration,2004, 275(3-5): 1009-1025

100 Wang Z H, Hu H Y. Stabilization of vibration systems via delayed state difference feedback. Journal of Sound and Vibration, 2006, 296(1-2): 117-129

101 Liu B, Hu H Y. Stabilization of linear un-damped systems via position and delayed position feedbacks. Journal of Sound and Vibration, 2008, 312(3): 509-525

102 Olgac N, Holm-Hansen B T. A novel active vibration absorption technique: delayed resonator. Journal of Sound and Vibration, 1994, 176(1): 93-104

103 Olgac N, Elmali H, Hosek M, et al. Active vibration control of distributed systems using delayed resonator with acceleration feedback. ASME Journal of Dynamical Systems, Measurement and Control, 1997, 119(3): 380-389

104 Librescu L, Marzocca P. Advances in the linear/nonlinear control of aeroelastic structural systems. Acta Mechanica,2005, 178(3-4): 147-186

105 Yu M L, Hu H Y. Flutter control based on ultrasonic motor for a two-dimensional airfoil section. Journal of Fluids and Structures, 2012, 28(1): 89-102

106 Hu H Y, Yu M L. Robust flutter control of an airfoil section through an ultrasonic motor. In: Hu H Y, Kreuzer E. eds.Proceedings of the IUTAM Symposium on Dynamics and Control of Nonlinear Systems with Uncertainty, Dodrecht: Springer, 2007. 307-316

107 Zhao Y H. Stability of a two-dimensional airfoil with timedelayed feedback control. Journal of Fluids and Structures,2009, 25(1): 1-25

108 Zhao Y H. Stability of a time-delayed aeroelastic system with a control force. Aerospace Science and Technology,2011, 15(1): 72-77

109 Zhao Y H. Flutter suppression of a high aspect-ratio wing with multiple control surfaces. Journal of Sound and Vibration,2009, 324(3-5): 490-513

110 Cai G P, Huang J Z, Yang S X. An optimal control method for linear systems with time delay. Computers & Structures,2003, 81(15): 1539-1546

111 Haraguchi M, Hu H Y. Using a new discretization approach to design delayed LQG controller. Journal of Sound and Vibration, 2008, 314(3-5): 558-570

112 Huang R, Hu H Y, Zhao Y H. Designing active flutter suppression for high-dimensional aeroelastic systems involving a control delay. Journal of Fluids and Structures,2012, 34: 35-50

113 Liu B, Haraguchi M, Hu H Y. A new reduction-based LQ control for dynamic systems with a slowly time-varying delay. Acta Mechanica Sinica, 2009, 25(4): 529-537

114 Awrejcewicz J, Supel B, Kudra G, et al. Numerical and experimental study of regular and chaotic motion of triple physical pendulum. International Journal of Bifurcation and Chaos, 2008, 18(10): 2883-2915

115 Liu B, Hu H Y. Group delay induced instabilities and Hopf bifurcations of a controlled double pendulum. International Journal of Non-Linear Mechanics, 2010, 45(4):442-452

116 刘博, 胡海岩. 群时延引起的受控小车二级摆失稳及其抑制. 振动工程学报, 2009, 22(5): 443-448

117 Watts D J, Strogatz S H. Collective dynamics of ’small world’ networks. Nature, 1998, 393(6684): 440-442

118 Barab à si A L, Albert R. Emergence of scaling in random networks. Science, 1999, 286(5439): 509-512

119 汪小帆, 李翔, 陈关荣. 复杂网络理论及其应用. 北京: 清华大学出版社, 2006

120 Xu X, Luo J W,Gu Y T. Collective dynamics and control of a 3-D small world network with time delays. International Journal of Bifurcation and Chaos, 2012, 22(11):1250281 (17p)

121 Mao X C. Stability switches, bifurcation, and multistability of coupled networks with time delays. Applied Mathematics and Computation, 2012, 218(11): 6263-6274

122 Mao X C. Stability and Hopf bifurcation analysis of a pair of three-neuron loops with time delays. Nonlinear Dynamics, 2012, 68(1-2): 151-159

123 Mao X C, Hu H Y. Hopf bifurcation analysis of a fourneuron network with multiple time delays. Nonlinear Dynamics,2009, 55(1-2): 95-112

124 Mao X C, Hu H Y. Stability and Hopf bifurcation of a delayed network of four neurons with a short-cut connection. International Journal of Bifurcation and Chaos,2008, 18(10): 3053-3072

125 Sipahi R, Niculescu S I, Abdallah C T, et al. Stability and stabilization of systems with time delays: limitation and opportunities. IEEE Control Systems Magazine, 2011,31(1): 38-65

126 赵艳影, 徐鉴. 时滞非线性动力吸振器的减振机理, 力学学报, 2008, 40(1): 98-106

127 Zheng Y G, Wang Z H. Stability analysis of nonlinear dynamic systems with slowly and periodically varying delay. Communications in Nonlinear Science and Numerical Simulation, 2012, 17(10): 3999-4009

128 Zhao H Y, Xie W. Hopf bifurcation for a small-world network model with parameters delay feedback control. Nonlinear Dynamics, 2011, 63(3): 345-357

129 Demir A, Hasanov A, Namachchivaya N S. Delay equations with fluctuating delay related to the regenerative chatter. International Journal of Non-Linear Mechanics,2006, 41(3): 464-474

130 Michiels W. Stabilization of time-delay systems with a controlled time-varying delay and applications. IEEE Transactions on Automatic Control, 2005, 50(4): 493-504

131 Inamura T, Sata T. Stability analysis of cutting under varying spindle speed. Journal of the Faculty of Engineering. The University of Tokyo-B, 1975, 23(1): 13-29

132 Sexton J S, Stone B J. The stability of machining with continuously varying spindle speed. CIRP Annals, 1978,27(1): 321-326

133 王在华, 胡海岩. 含分数阶导数阻尼的线性振动系统的稳定性. 中国科学G 辑, 2009, 39(10): 1495-1502

134 Coronado A, Trindade M A, Sampaio R. Frequencydependent viscoelastic models for passive vibration isolation systems. Shock and Vibration, 2002, 9(1): 253-264

135 Eldred L B, Baker W P, Palazotto A N. Kelvin-Voigt vs fractional derivative model as constitutive relations for viscoelastic materials. AIAA Journal, 1995, 33(3): 547-550

136 Du M L,Wang Z H. Initialized fractional differential equations with Riemann-Liouville fractional-order derivative. European Physics Journal: Special Topics, 2011, 193(1):49-60

137 Wang Z H, Du M L. Asymptotical behavior of the solution of a SDOF linear fractionally damped vibration system.Shock and Vibration, 2011, 18(1-2): 257-268

138 Deng W. Short memory principle and a predictorcorrector approach for fractional differential equations. Journal of Computational and Applied Mathematics,2007, 206(1):174-188

139 Monje C A, Chen Y Q, Vinagre B M, et al. Fractionalorder Systems and Controls: Fundamentals and Applications. London: Springer, 2010

140 Tavazoei M S, Haeri M. Limitations of frequency domain approximation for detecting chaos in fractional order systems. Nonlinear Analysis: Theory, Method & Applications,2008, 69(4): 1299-1320

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