Primary resonance of fractional-order van der Pol oscillator

In this paper the primary resonance of van der Pol (VDP) oscillator with fractional-order derivative is studied analytically and numerically. At first the approximately analytical solution is obtained by the averaging method, and it is found that the fractional-order derivative could affect the dynamical properties of VDP oscillator, which is characterized by the equivalent damping coefficient and the equivalent stiffness coefficient. Moreover, the amplitude–frequency equation for steady-state solution is established, and the corresponding stability condition is also presented based on Lyapunov theory. Then, the comparisons of several different amplitude–frequency curves by the approximately analytical solution and the numerical integration are fulfilled, and the results certify the correctness and satisfactory precision of the approximately analytical solution. At last, the effects of the two fractional parameters, i.e., the fractional coefficient and the fractional order, on the amplitude–frequency curves are investigated for some typical excitation amplitudes, which are different from the traditional integer-order VDP oscillator.

[1]  Hsien-Keng Chen,et al.  Chaotic dynamics of the fractionally damped Duffing equation , 2007 .

[2]  Zhongjin Guo,et al.  The residue harmonic balance for fractional order van der Pol like oscillators , 2012 .

[3]  M. Shitikova,et al.  Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results , 2010 .

[4]  Jerzy T. Sawicki,et al.  Nonlinear Vibrations of Fractionally Damped Systems , 1998 .

[5]  Marina V. Shitikova,et al.  Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems , 1997 .

[6]  Zaihua Wang,et al.  Stability of a linear oscillator with damping force of the fractional-order derivative , 2010 .

[7]  J. A. Tenreiro Machado,et al.  Complex-order forced van der Pol oscillator , 2012 .

[8]  Shaopu Yang,et al.  Primary resonance of Duffing oscillator with two kinds of fractional-order derivatives , 2012 .

[9]  Ivo Petras,et al.  Fractional-Order Nonlinear Systems , 2011 .

[10]  Li Gen-guo,et al.  Dynamical stability of viscoelastic column with fractional derivative constitutive relation , 2001 .

[11]  G. Litak,et al.  Vibration of the Duffing oscillator: Effect of fractional damping , 2006, nlin/0601033.

[12]  Junyi Cao,et al.  Nonlinear Dynamics of Duffing System With Fractional Order Damping , 2009 .

[13]  S. Das,et al.  Functional Fractional Calculus for System Identification and Controls , 2007 .

[14]  Pankaj Wahi,et al.  Averaging Oscillations with Small Fractional Damping and Delayed Terms , 2004 .

[15]  F. Mainardi,et al.  Recent history of fractional calculus , 2011 .

[16]  Teodor M. Atanackovic,et al.  On a numerical scheme for solving differential equations of fractional order , 2008 .

[17]  Haitao Qi,et al.  Unsteady flow of viscoelastic fluid with fractional Maxwell model in a channel , 2007 .

[18]  Juhn-Horng Chen,et al.  Chaotic dynamics of the fractionally damped van der Pol equation , 2008 .

[19]  Luigi Fortuna,et al.  Fractional Order Systems: Modeling and Control Applications , 2010 .

[20]  H. Srivastava,et al.  Theory and Applications of Fractional Differential Equations , 2006 .

[21]  I. Podlubny.  Fractional differential equations , 1998 .

[22]  Yang Shaopu,et al.  Dynamical analysis of linear single degree-of-freedom oscillator with fractional-order derivative , 2012 .

[23]  Shaopu Yang,et al.  Primary resonance of Duffing oscillator with fractional-order derivative , 2012 .

[24]  W. Zhu,et al.  The First Passage Failure of SDOF Strongly Nonlinear Stochastic System with Fractional Derivative Damping , 2009 .

[25]  Xiaoling Jin,et al.  Response and stability of a SDOF strongly nonlinear stochastic system with light damping modeled by a fractional derivative , 2009 .

[26]  I. Petráš.  Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation , 2011 .

[27]  Z. Wang,et al.  Asymptotical Behavior of the Solution of a SDOF Linear Fractionally Damped Vibration System , 2011 .

[28]  J. A. Tenreiro Machado,et al.  Fractional Dynamics: A Statistical Perspective , 2007 .

[29]  YangQuan Chen,et al.  Fractional-order Systems and Controls , 2010 .

[30]  YangQuan Chen,et al.  Fractional-order systems and control : fundamentals and applications , 2010 .

[31]  Feng Xie,et al.  Asymptotic solution of the van der Pol oscillator with small fractional damping , 2009 .

[32]  Hongtao Lu,et al.  Synchronization of a new fractional-order hyperchaotic system , 2009 .

[33]  鈴木 増雄.  A. H. Nayfeh and D. T. Mook: Nonlinear Oscillations, John Wiley, New York and Chichester, 1979, xiv+704ページ, 23.5×16.5cm, 10,150円. , 1980 .

[34]  Junguo Lu.  Chaotic dynamics of the fractional-order Lü system and its synchronization , 2006 .

[35]  Carla M. A. Pinto,et al.  Complex order van der Pol oscillator , 2011 .

[36]  Tridip Sardar,et al.  The analytical approximate solution of the multi-term fractionally damped Van der Pol equation , 2009 .

[37]  Shaopu Yang,et al.  Recent advances in dynamics and control of hysteretic nonlinear systems , 2009 .

[38]  Milad Siami,et al.  More Details on Analysis of Fractional-order Van der Pol Oscillator , 2009 .