[1]OLDHAM K B, SPANIER J.The fractional calculus: theory and applications of differentiation and integration to arbitrary order[M].New York: Academic Press, 1974.
[2]ROSS B.Brief history and exposition of the fundamental theory of fractional calculus[M].New York: Spring-Verlag, 1975.
[3]ROSS B.The development of fractional calculus 1695-1900[J].Historia Mathernatica, 1977,4(1): 75-89.
[4]PETRAS I.Fractional-order nonlinear system[M].Beijing: Higher Education Press, 2011.
[5]CAPONETTO R, EBRARY I.Fractional order systems : modeling and control applications[M].Singapore: World Scientific, 2010.
[6]MACHADO J T, KIRYAKOVA V, MAINARDI F.Recent history of fractional calculus[J].Communications in Nonlinear Science and Numerical Simulation, 2011,16(3): 1140-1153.
[7]LI C P, ZENG F H.Numerical methods for fractional calculus[M].New York: CRC Press, 2015.
[8]SUN H G, ZHANG Y, BALEANU D, et al.A new collection of real world applications of fractional calculus in science and engineering[J].Communications in Nonlinear Science & Numerical Simulation, 2018,64: 213-231.
[9]NAYFEH A H, MOOK A D.Nonlinear oscillations[M].New York: John Wiley & Sons, 1979.
[10]李占龙, 孙大刚, 宋勇,等.基于分数阶导数的黏弹性悬架减振模型及其数值方法[J].振动与冲击, 2016,35(16): 123-129.
LI Zhanlong, SUN Dagang, SONG Yong, et al.A fractional calculus-based vibration suppression model and its numerical solution for viscoelastic suspension[J].Journal of Vibration and Shock, 2016,35(16): 123-129.
[11]SHEN Y J, WANG L, YANG S P, et al.Nonlinear dynamical analysis and parameters optimization of four semi-active on-off dynamic vibration absorbers[J].Journal of Vibration and Control, 2013,19(1): 143-160.
[12]KOVACIC I, BRENNAN M J.The duffing equation: nonlinear oscillators and their behaviour[M].New York: John Wiley & Sons, 2011.
[13]SHEN Y J, YANG S P, LIU X D.Nonlinear dynamics of a spur gear pair with time-varying stiffness and backlash based on incremental harmonic balance method[J].International Journal of Mechanical Sciences, 2006,48(11): 1256-1263.
[14]LI X H, HOU J Y, SHEN Y J.Slow-fast effect and generation mechanism of brusselator based on coordinate transformation[J].Open Physics, 2016,14(1): 261-268.
[15]PODLUBNY I.Fractional differential equations[M].London: Academic Press, 1999.
[16]DIETHELM K, FORD N J, FREED A D.A predictor-corrector approach for the numerical solution of fractional differential equations[J].Nonlinear Dynamics, 2002,29(1/2/3/4): 3-22.
[17]FIRDOUS A, SHAH R, ABASS L D.Numerical solution of fractional differential equations using haar wavelet operational matrix method[J].International Journal of Applied and Computational Mathematics, 2017,3(3): 2423-2445.
[18]JAVIDI M, AHMAD B.Numerical solution of fractional partial differential equations by numerical Laplace inversion technique[J].Advances in Difference Equations, 2013,375(1): 1-18.
[19]TRIGEASSOU J C, MAAMRI N, SABATIER J, et al.A Lyapunov approach to the stability of fractional differential equations[J].Signal Processing, 2011,91(3): 437-445.
[20]QI Y F, PENG Y H.Stability analysis of fractional nonlinear dynamic systems with order lying in(1,2)[J].Chinese Quarterly Journal of Mathematics, 2019,34(2): 188-195.
[21]SHEN Y J, YANG S P, XING H J, et al.Primary resonance of duffing oscillator with fractional-order derivative[J].Communications in Nonlinear Science and Numerical Simulation, 2012,17(7): 3092-3100.
[22]SHEN Y J, WEI P, YANG S P.Primary resonance of fractional-order van der Pol oscillator[J].Nonlinear Dynamics, 2014,77(4): 1629-1642.
[23]申永军, 杨绍普, 邢海军.含分数阶微分的线性单自由度振子的动力学分析[J].物理学报, 2012,61(11): 158-163.
SHEN Yongjun, YANG Shaopu, XING Haijun.Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative[J].Acta Physica Sinica, 2012,61(11): 158-163.
[24]姜源,申永军,温少芳,等.分数阶达芬振子的超谐与亚谐联合共振[J].力学学报, 2017,49(5): 1008-1019.
JIANG Yuan, SHEN Yongjun, WEN Shaofang, et al.Super-harmonic and sub-harmonic simultaneous resonances of fractional-order Duffing oscillator [J].Chinese Journal of Theoretical and Applied Mechanics, 2017,49(5): 1008-1019.
[25]顾晓辉,杨绍普,申永军,等.分数阶Duffing振子的组合共振[J].振动工程学报,2017,30(1): 28-32.
GU Xiaohui, YANG Shaopu, SHEN Yongjun, et al.Combined resonance of fractional Duffing oscillator[J].Journal of Vibration Engineering, 2017,30(1): 28-32.
[26]TABEJIEU L M A, NBENDJO B R N, WOAFO P.On the dynamics of Rayleigh beams resting on fractional-order viscoelastic Pasternak foundations subjected to moving loads[J].Chaos Solitons & Fractals, 2016,93: 39-47.
[27]SHEN Y J, WEN S F, YANG S P, et al.Analytical threshold for chaos in a Duffing oscillator with delayed feedbacks[J].International Journal of Non-Linear Mechanics, 2018,98: 1-8.
[28]SUN Z K, XU W, YANG X L, et al.Inducing or suppressing chaos in a double-well Duffing oscillator by time delay feedback[J].Chaos, Solitons and Fractals, 2006,27(3): 705-714.
[29]WEN S F, CHEN J F, GUO S Q.Heteroclinic bifurcation behaviors of a Duffing oscillator with delayed feedback[J].Shock and Vibration, 2018,2018: 1-12.
[30]张思进,王紧业,文桂林.二自由度碰振准哈密顿系统亚谐轨道分析[J].振动与冲击, 2018,37(2): 102-107.
ZHANG Sijin, WANG Jinye, WEN Guilin.Subharmonic orbits analysis for a 2DOF vibro-impact quasi-Hamiltonian system[J].Journal of Vibration and Shock, 2018,37(2): 102-107.
[31]ABTAHI S M.Melnikov-based analysis for chaotic dynamics of spin-orbit motion of a gyrostat satellite[J].Proceedings of the Institution of Mechanical Engineers, 2019,233(4): 931-941.
[32]CHEN L C, HU F, ZHU W Q.Stochastic dynamics and fractional optimal control of quasi integrable Hamiltonian systems with fractional derivative damping[J].Fractional Calculus and Applied Analysis, 2013,16(1): 189-225.
[33]GUCKENHEIMER J, HOLMES P.Nonlinear oscillations, dynamical system and bifurcations of vector fields[M].New York: Springer-Verlag, 1983.
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