More Details on Analysis of Fractional-order Van der Pol Oscillator

This paper is devoted to the analysis of fractional order Van der Pol system studied in the literature. Based on the existing theorems on the stability of incommensurate fractional order systems, we determine parametric range for which a fractional order Van der Pol system with a specific order can perform as an undamped oscillator. Numerical simulations are presented to support the given analytical results. These results also illuminate a main difference between oscillations in a fractional order Van der Pol oscillator and its integer order counterpart. We show that contrary to integer order case, trajectories in a fractional Van der Pol oscillator do not converge to a unique cycle.

[1]  R. Mickens Fractional Van Der Pol Equations , 2003 .

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

[3]  Igor Podlubny,et al.  Fractional-order systems and PI/sup /spl lambda//D/sup /spl mu//-controllers , 1999 .

[4]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

[5]  Ahmed S. Elwakil,et al.  Fractional-order Wien-bridge oscillator , 2001 .

[6]  C. F. Lorenzo,et al.  Chaos in a fractional order Chua's system , 1995 .

[7]  Mohammad Saleh Tavazoei,et al.  A necessary condition for double scroll attractor existence in fractional-order systems , 2007 .

[8]  Anissa Zergaïnoh-Mokraoui,et al.  State-space representation for fractional order controllers , 2000, Autom..

[9]  E. Ahmed,et al.  Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models , 2007 .

[10]  Alain Oustaloup,et al.  From fractal robustness to the CRONE control , 1999 .

[11]  D. Matignon Stability results for fractional differential equations with applications to control processing , 1996 .

[12]  M. Haeri,et al.  Unreliability of frequency-domain approximation in recognising chaos in fractional-order systems , 2007 .

[13]  J. A. Tenreiro Machado,et al.  Analysis of the Van der Pol Oscillator Containing Derivatives of Fractional Order , 2007 .

[14]  Ivo Petras,et al.  A note on the fractional-order Chua’s system , 2008 .

[15]  S. Westerlund Dead matter has memory , 1991 .

[16]  Changpin Li,et al.  Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle , 2007 .

[17]  Jinhu Lü,et al.  Stability analysis of linear fractional differential system with multiple time delays , 2007 .

[18]  Chunguang Li,et al.  Chaos and hyperchaos in the fractional-order Rössler equations , 2004 .

[19]  Ronald E. Mickens,et al.  ANALYSIS OF NON-LINEAR OSCILLATORS HAVING NON-POLYNOMIAL ELASTIC TERMS , 2002 .

[20]  Weihua Deng,et al.  Design of multidirectional multiscroll chaotic attractors based on fractional differential systems via switching control. , 2006, Chaos.

[21]  J.A.T. Machado,et al.  Dynamics of the fractional-order Van der Pol oscillator , 2004, Second IEEE International Conference on Computational Cybernetics, 2004. ICCC 2004..

[22]  Elena Grigorenko,et al.  Chaotic dynamics of the fractional Lorenz system. , 2003, Physical review letters.

[23]  R. Hilfer Applications Of Fractional Calculus In Physics , 2000 .

[24]  P. Arena,et al.  Bifurcation and Chaos in Noninteger Order Cellular Neural Networks , 1998 .

[25]  Mohammad Saleh Tavazoei,et al.  Fractional controller to stabilize fixed points of uncertain chaotic systems: Theoretical and experimental study , 2008 .

[26]  I. Podlubny Fractional-order systems and PIλDμ-controllers , 1999, IEEE Trans. Autom. Control..

[27]  Vicente Feliú Batlle,et al.  Fractional order control strategies for power electronic buck converters , 2006, Signal Process..