More details on analysis of fractional-order Van der Pol oscillator

JVC/Journal of Vibration and Control

Volumn 15, Issue 6, 2009, Pages 803-819

  Tavazoei, Mohammad S ^a   Haeri, Mohammad ^a   Attari, Mina ^a   Bolouki, Sadegh ^a   Siami, Milad ^a  

^a SHARIF UNIVERSITY OF TECHNOLOGY   (Iran)

Author keywords

Fractional order system; Fractional order Van der Pol oscillator; Limit set; Oscillation; Stability

Indexed keywords

ANALYTICAL RESULTS; FRACTIONAL ORDER; FRACTIONAL ORDER SYSTEM; FRACTIONAL-ORDER SYSTEMS; FRACTIONAL-ORDER VAN DER POL OSCILLATOR; INTEGER ORDER; INTEGER-ORDER COUNTERPARTS; LIMIT SET; NUMERICAL SIMULATION; OSCILLATION; UNDAMPED OSCILLATORS; VAN DER POL OSCILLATOR; VAN DER POL SYSTEMS;

AUTOMOBILE EXHIBITIONS; CIRCUIT OSCILLATIONS; LYAPUNOV METHODS; OSCILLATORS (MECHANICAL);

SYSTEM STABILITY;

EID: 66249115324     PISSN: 10775463     EISSN: 17412986     Source Type: Journal    
DOI: 10.1177/1077546308096101     Document Type: Article

References (27)

1. Ahlfors, L.V., 1966, Complex Analysis, 2 nd Edition, McGraw-Hill, New York.
2. Ahmad, W., El-Khazali, R., and El-Wakil, A., 2001, " Fractional-order Wien-bridge oscillator, " Electronic Letters 37, 1110-1112.
3. Ahmed, E., El-Sayed, A.M.A., and El-Saka, H.A.A., 2007, " Equilibrium points, stability and numerical solutions of fractional order predator-prey and rabies models, " Journal Mathematical Analysis and Applications 325, 542-553.
4. Arena, P., Caponetto, R., Fortuna, L., and Porto, D., 1998, " Bifurcation and chaos in noninteger order cellular neural networks, " International Journal of Bifurcation and Chaos 8 (7). 1527-1539. (Pubitemid 128338351)
5. Barbosa, R.S., Machado, J. AT., Ferreira, M.I., and Tar, K.J., 2004, " Dynamics of the fractional order Van der Pol oscillator, " in Proceedings of the 2nd IEEE International Conference on Computational Cybernetics (ICCC'04), Vienna University of Technology, Austria, 30 Aug. - 1 Sept. pp. 373-378.
6. Barbosa, R.S., Machado, J. AT., Vingare, B.M., and Calderon A.J., 2007, " Analysis of the Van der Pol oscillator containing derivatives of fractional order, " Journal of Vibration and Control 13 (9-10). 1291-1301. (Pubitemid 47482814)
7. Calderon, A.J., Vinagre, B.M., and Feliu, V., 2006, " Fractional order control strategies for power electronic buck converters, " Signal Processing 86, 2803-2819.
8. Deng, W. and Lu, J., 2006, " Design of multidirectional multiscroll chaotic attractors based on fractional differential systems via switching control, " Chaos 16, 043120.
9. Deng, W., Li, C., and L?1/4, J., 2007, " Stability analysis of linear fractional differential system with multiple time delays, " Nonlinear Dynamics 48, 409-416. (Pubitemid 46407957)
10. Diethelm, K., Ford, N.J., and Freed, A.D., 2002, " A predictor-corrector approach for the numerical solution of fractional differential equations, " Nonlinear Dynamics 29, 3-22.
11. Grigorenko, I. and Grigorenko, E., 2003, " Chaotic dynamics of the fractional Lorenz system, " Physical Review Letters 91, 034101.
12. Hartley, T.T., Lorenzo, C.F., and Qammer, H.K., 1995, " Chaos in a fractional order Chua's system, " IEEE Transactions on Circuits and Systems I 42, 485-490.
13. Hilfer, R. (Editor), 2001, Applications of Fractional Calculus in Physics, World Scientific, New Jersey.
14. Li, C. and Chen, G., 2004, " Chaos and hyperchaos in the fractional order R?¶ssler equations, " Physica A: Statistical Mechanics and its Applications 341, 55-61.
15. Matignon, D., 1996, " Stability results for fractional differential equations with applications to control processing, " in Proceedings of IMACS, IEEE-SMC Computational Engineering in Systems and Application Multi-Conference, July 9-12, Lille, France, Vol. 2, pp. 963-968.
16. Mickens, R.E., 2002 a, " Analysis of nonlinear oscillators having non-polynomial elastic terms ", Journal of Sound and Vibration 255 (4). 789-792.
17. Mickens, R.E., 2002 b, " Fractional Van der Pol equations ", Journal of Sound and Vibration 259 (2). 457-460.
18. Oustaloup, A., Sabatier, J., and Lanusse, P., 1999, " From fractal robustness to CRONE control, " Fractional Calculus and Applied Analysis 2 (1). 1-30.
19. Petras, I., 2008, " A note on the fractional-order Chua's system, " Chaos, Solitons & Fractals 38, 140-147.
20. Podlubny, I., 1999 a, Fractional differential equations, Academic Press, San Diego.
21. Podlubny, I., 1999 b, " Fractional order systems and PI?»D?1/4-controllers, " IEEE Transactions on Automatic Control 44 (1). 208-214.
22. Raynaud, H.F. and Inoh, A.Z., 2000, " State-space representation for fractional order controllers, " Automatica 36, 1017-1021.
23. Tavazoei, M.S. and Haeri, M., 2007 a, " A necessary condition for double scroll attractor existence in fractional order systems, " Physics Letters A 367 (1-2). 102-113.
24. Tavazoei, M.S. and Haeri, M., 2007 b, " Unreliability of frequency domain approximation in recognising chaos in fractional order systems, " IET Signal Processing 1 (4). 171-181.
25. Tavazoei, M.S., Haeri, M., and Jafari, S., 2008, " Fractional controller to stabilize fixed points of uncertain chaotic systems: theoretical and experimental study, " in Proceedings of the I MECH E Part I Journal of Systems & Control Engineering Vol. 222 (3). pp. 175-184.
26. Wang, Y. and Li, C., 2007, " Does the fractional Brusselator with efficient dimension less than 1 have a limit cycle?, " Physics Letters A 363, 414-419.
27. Westerlund, S., " Dead matter has memory! ", 1991, Physica Scripta 43, 174-179.