Asymptotic Solutions and Circuit Implementations of a Rayleigh Oscillator Including Cubic Fractional Damping Terms

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• 作者：Min Xiao ; Guoping Jiang ; Jinde Cao
• 关键词：Caputo derivative ; Fractional ; order Rayleigh oscillator ; Asymptotic solution ; Two ; scale expansion method ; Circuit implementation
• 刊名：Circuits, Systems, and Signal Processing
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：35
• 期：6
• 页码：2041-2053
• 全文大小：1,720 KB
• 参考文献：1.W.M. Ahmad, J.C. Sprott, Chaos in fractional-order autonomous nonlinear systems. Chaos Solitons Fract. 16(2), 339–351 (2003)CrossRef MATH
  2.A. Alexopoulos, G.V. Weinberg, Fractional-order formulation of power-law and exponential distributions. Phys. Lett. A 378(34), 2478–2481 (2014)MathSciNet CrossRef MATH
  3.A.A.M. Arafa, S.Z. Rida, M. Khalil, The effect of anti-viral drug treatment of human immunodeficiency virus type 1 (HIV-1) described by a fractional order model. Appl. Math. Model. 37(4), 2189–2196 (2013)MathSciNet CrossRef
  4.R.S. Barbosa, J.A.T. Machado, B.M. Vinagre, A.J. Calderon, Analysis of the van der Pol oscillator containing derivatives of fractional order. J. Vib. Control 13(9–10), 1291–1301 (2007)CrossRef MATH
  5.S. Chatterjee, S. Dey, Nonlinear dynamics of two harmonic oscillators coupled by Rayleigh type self-exciting force. Nonlinear Dyn. 72(1–2), 113–128 (2013)MathSciNet CrossRef
  6.D.Y. Chen, C.F. Liu, C. Wu, Y. Liu, X. Ma, Y. You, A new fractional-order chaotic system and its synchronization with circuit simulation. Circuits Syst. Signal Process. 31(5), 1599–1613 (2012)MathSciNet CrossRef
  7.G.P. Chen, Y. Yang, Robust finite-time stability of fractional order linear time-varying impulsive systems. Circuits Syst. Signal Process. 34(4), 1325–1341 (2015)MathSciNet CrossRef
  8.A.C. de Pina, M.S. Dutra, L.S.C. Raptopoulos, Modeling of a bipedal robot using mutually coupled Rayleigh oscillators. Biol. Cybern. 92(1), 1–7 (2005)MathSciNet CrossRef MATH
  9.K. Diethelm, N.J. Ford, A.D. Freed, A predictor–corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn. 29(1–4), 3–12 (2002)MathSciNet CrossRef MATH
  10.T.J. Freeborn, A survey of fractional-order circuit models for biology and biomedicine. IEEE J. Emerg. Selected Topics Circuits Syst. 3(3), 416–424 (2013)CrossRef
  11.S. Ghosh, D.S. Ray, Chemical oscillator as a generalized Rayleigh oscillator. J. Chem. Phys. 139(16), 164112 (2013)CrossRef
  12.Z.J. Guo, A.Y.T. Leung, H.X. Yang, Oscillatory region and asymptotic solution of fractional van der Pol oscillator via residue harmonic balance technique. Appl. Math. Model. 35(8), 3918–3925 (2011)MathSciNet CrossRef MATH
  13.H. Hasegawa, Jarzynski equality in van der Pol and Rayleigh oscillators. Phys. Rev. E 84(6), 061112 (2011)CrossRef
  14.R. Hilfer, Applications of Fractional Calculus in Physics (World Scientific, Singapore, 2000)CrossRef MATH
  15.N. Inaba, S. Mori, Folded torus breakdown in the forced Rayleigh oscillator with a diode pair. IEEE Trans. Circuits Syst. I. Fundam. Theory Appl 39(5), 402–411 (1992)CrossRef MATH
  16.H.Y. Jia, Z.Q. Chen, G.Y. Qi, Chaotic characteristics analysis and circuit implementation for a fractional-order system. IEEE Trans. Circuits Syst. I. Regul. Pap 61(3), 845–853 (2014)CrossRef
  17.B.Z. Kaplan, Y. Horen, Switching-mode counterparts of the Rayleigh and Van-der-Pol oscillators. Int. J. Circuits Theor. Appl. 28(1), 31–49 (2000)CrossRef MATH
  18.M. Khan, S.H. Ali, C. Fetecau, H. Qi, Decay of potential vortex for a viscoelastic fluid with fractional. Appl. Math. Model. 33(5), 2526–2533 (2009)MathSciNet CrossRef MATH
  19.I. Kovacic, M. Zukovic, Oscillators with a power-form restoring force and fractional derivative damping: application of averaging. Mech. Res. Commun. 41, 37–43 (2012)CrossRef
  20.C.A.K. Kwuimy, B.R.N. Nbendjo, Active control of horseshoes chaos in a driven Rayleigh oscillator with fractional order deflection. Phys. Lett. A 375(39), 3442–3449 (2011)CrossRef MATH
  21.C. Letellier, L.A. Aguirre, Dynamical analysis of fractional-order Rossler and modified Lorenz systems. Phys. Lett. A 377(28–30), 1707–1719 (2013)MathSciNet CrossRef MATH
  22.A.Y.T. Leung, Z.J. Guo, H.Y. Yang, Fractional derivative and time delay damper characteristics in Duffing–van der Pol oscillators. Commun. Nonlinear Sci. Numer. Simul. 18(10), 2900–2915 (2013)MathSciNet CrossRef MATH
  23.A.Y.T. Leung, H.Y. Yang, P. Zhu, Periodic bifurcation of Duffing–van der Pol oscillators having fractional derivatives and time delay. Commun. Nonlinear Sci. Numer. Simul. 19(4), 1142–1155 (2014)MathSciNet CrossRef
  24.F.R.S. Lord Rayleigh, On maintained vibrations. Philos. Mag. 15, 229–235 (1883)CrossRef MATH
  25.E. Naseri et al., Solving linear fractional-order differential equations via the enhanced homotopy perturbation method. Phys. Scr. T136, 014035 (2009)CrossRef
  26.P.S.V. Nataraj, R. Kalla, Computation of limit cycles for uncertain nonlinear fractional-order systems. Phys. Scr. T136, 014021 (2009)CrossRef
  27.I. N’Doye, H. Voos, M. Darouach, Observer-based approach for fractional-order chaotic synchronization and secure communication. IEEE J. Emerg. Selected Topics Circuits Syst. 3(3), 442–450 (2013)MathSciNet CrossRef
  28.A. Pálfalvi, Efficient solution of a vibration equation involving fractional derivatives. Int. J. Non-Linear Mech. 45(2), 169–175 (2010)CrossRef
  29.I. Petráš, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (Springer, London, 2011)CrossRef MATH
  30.I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications (Academic Press, San Diego, 1999)MATH
  31.A.G. Radwan, K.N. Salama, Fractional-order RC and RL circuits. Circuits Syst. Signal Process. 31(6), 1901–1905 (2012)MathSciNet CrossRef
  32.M. Rostami, M. Haeri, Study of limit cycles and stability analysis of fractional Arneodo oscillator. J. Optimiz. Theory App. 156(1), 68–78 (2013)MathSciNet CrossRef MATH
  33.M. Rostami, M. Haeri, Undamped oscillations in fractional-order Duffing oscillator. Signal Process. 107, 361–367 (2014)CrossRef
  34.M.S. Siewe, C. Tchawoua, S. Rajasekar, Parametric resonance in the Rayleigh–Duffing oscillator with time-delayed feedback. Commun. Nonlinear Sci. Numer. Simul. 17(11), 4485–4493 (2012)MathSciNet CrossRef MATH
  35.M.S. Tavazoei, M. Haeri, M. Siami, S. Bolouki, Maximum number of frequencies in oscillations generated by fractional order LTI systems. IEEE Trans. Signal Process. 58(8), 4003–4012 (2010)MathSciNet CrossRef MATH
  36.H.H. Wang, K.H. Sun, S.B. He, Dynamic analysis and implementation of a digital signal processor of a fractional-order Lorenz–Stenflo system based on the Adomian decomposition method. Phys. Scr. 90(1), 015206 (2015)CrossRef
  37.M. Xiao, W.X. Zheng, J.D. Cao, Approximate expressions of a fractional order Van der Pol oscillator by the residue harmonic balance method. Math. Comput. Simul. 89, 1–12 (2013)MathSciNet CrossRef
  38.M. Xiao, W.X. Zheng, G.P. Jiang, J.D. Cao, Undamped oscillations generated by Hopf bifurcations in fractional-order recurrent neural networks with Caputo derivative. IEEE Trans. Neural Netw. Learn. Syst. 26(12), 3201–3214 (2015)CrossRef
  39.F. Xie, X. Lin, Asymptotic solution of the van der Pol oscillator with small fractional damping. Phys. Scr. T136, 014033 (2009)CrossRef
  40.Z. Xu, C.X. Liu, T. Yang, Controlling fractional-order new chaotic system based on Lyapunov equation. Acta Phys. Sin. 59(3), 1524–1531 (2010)MathSciNet MATH
  41.J.H. Yang, H. Zhu, Bifurcation and resonance induced by fractional-order damping and time delay feedback in a Duffing system. Commun. Nonlinear Sci. Numer. Simul. 18(5), 1316–1326 (2013)MathSciNet CrossRef MATH
  42.P. Zhou, K. Huang, A new 4-D non-equilibrium fractional-order chaotic system and its circuit implementation. Commun. Nonlinear Sci. Numer. Simul. 19(6), 2005–2011 (2014)MathSciNet CrossRef
  43.M. Zolfaghari et al., Application of the enhanced homotopy perturbation method to solve the fractional-order Bagley–Torvik differential equation. Phys. Scr. T136, 014032 (2009)CrossRef
  
• 作者单位：Min Xiao (1)
  Guoping Jiang (1)
  Jinde Cao (2)
  
  1. College of Automation, Nanjing University of Posts and Telecommunications, Nanjing, 210003, China
  2. Research Center for Complex Systems and Network Sciences, and Department of Mathematics, Southeast University, Nanjing, 210096, China
  
• 刊物类别：Engineering
• 刊物主题：Electronic and Computer Engineering
  
• 出版者：Birkh盲user Boston
• ISSN：1531-5878

文摘

This paper proposes a fractional-order Rayleigh oscillator model, which involves a cubic damping term described by fractional derivatives. The presence of such fractional damping term makes the analysis more difficult. A two-scale expansion method is employed for asymptotic solutions of the fractional-order Rayleigh oscillator. Then, an example is provided to compare the asymptotic solutions with the numerical solutions. The numerical results demonstrate the validity and applicability of the proposed method to solve fractional differential equations with high order fractional terms. Furthermore, an electronic circuit is designed to realize the fractional-order Rayleigh oscillator.