Chaos excited chaos synchronizations of integral and fractional order generalized van der Pol systems

Chaos, Solitons and Fractals

Volumn 36, Issue 3, 2008, Pages 592-604

  Ge, Zheng Ming ^a   Hsu, Mao Yuan ^a  

^a NATIONAL CHIAO TUNG UNIVERSITY   (Taiwan)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER SIMULATION; INTEGRAL EQUATIONS; NUMERICAL METHODS; SYNCHRONIZATION;

CHAOS SYNCHRONIZATIONS; CHAOTIC STATES; FRACTIONAL ORDER SYSTEMS; STATE ERROR PLOTS; VAN DER POL SYSTEMS;

CHAOTIC SYSTEMS;

EID: 37349088581     PISSN: 09600779     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.chaos.2006.06.093     Document Type: Article

References (62)

1. Bagley R.L., and Calico R.A. Fractional order state equations for the control of viscoelastically damped structures. J Guid Contr Dyn 14 (1991) 304-311
2. Li C., and Chen G. Chaos in the fractional order Chen system and its control. Chaos, Solitons & Fractals 22 (2004) 549-554
3. Ahmad W.M., and Harb A.M. On nonlinear control design for autonomous chaotic systems of integer and fractional orders. Chaos, Solitons & Fractals 18 (2003) 693-701
4. Ahmad W.M., Reyad E.-K., and Yousef A.-A. Stabilization of generalized fractional order chaotic systems using state feedback control. Chaos, Solitons & Fractals 22 (2004) 141-150
5. Ahmad W.M. Hyperchaos in fractional order nonlinear systems. Chaos, Solitons & Fractals 26 (2005) 1459-1465
6. Nimmo S., and Evans A.K. The effects of continuously varying the fractional differential order of chaotic nonlinear systems. Chaos, Solitons & Fractals 10 (1999) 1111-1118
7. Ahmad W.M., and Sprott J.C. Chaos in a fractional-order autonomous nonlinear systems. Chaos, Solitons & Fractals 16 (2003) 339-351
8. Barbosa Ramiro S, Tenreiro Machado JA, Ferreira Isabel M, Tar József K. Dynamics of the fractional-order van der Pol oscillator. In: Proc of the 2nd IEEE international conference on computational cybernetics (ICCC'04), 2004. p. 373-78, ISBN 3-902463-01-5.
9. Li C., and Peng G. Chaos in Chen's system with a fractional order. Chaos, Solitons & Fractals 22 (2004) 443-450
10. Charef A., Sun H.H., Tsao Y.Y., and Onaral B. Fractal system as represented by singularity function. IEEE Trans Automat Contr 37 (1992) 1465-1470
11. Hartley T.T., Lorenzo C.F., and Qammer H.K. Chaos in a fractional order Chua's system. IEEE Trans CAS-I 42 (1995) 485-490
12. Glass L. Theory of heart (1990), Springer, New York-Heidelberg-Berlin
13. Mahmoud G.M., and Farghaly A.A.M. Chaos control of chaotic limit cycles of real and complex van der Pol oscillators. Chaos, Solitons & Fractals 21 (2004) 915-924
14. Chen H.-K., and Ge Z.-M. Bifurcation and chaos of a two-degree-of freedom dissipative gyroscope. Chaos, Solitons & Fractals 24 (2005) 125-136
15. van der Pol B. Forced oscillations in a circuit with nonlinear resistance (receptance with reactive triode). Lond, Edinburgh, Dublin Philos Mag 3 (1927) 65-80
16. Addo-Asah W., Akpati H.C., and Mickens R.E. Investigation of a generalized van der Pol oscillator differential equation. J Sound Vibrat 179 (1995) 733-735
17. Waluya S.B., and van Horssen W.T. On the periodic solutions of a generalized nonlinear van der Pol oscillator. J Sound Vibrat 268 (2003) 209-215
18. Mickens R.E. Analysis of nonlinear oscillators having non-polynomial elastic terms. J Sound Vibrat 255 (2002) 789-792
19. Ge Z.-M., and Chen Y.-S. Synchronization of unidirectional coupled chaotic systems via partial stability. Chaos, Solitons & Fractals 21 (2004) 101-111
20. Ge Z.-M., and Leu W.-Y. Anti-control of chaos of two-degrees-of-freedom louderspeaker system and chaos synchronization of different order systems. Chaos, Solitons & Fractals 20 (2004) 503-521
21. Ge Z.-M., and Leu W.-Y. Chaos synchronization and parameter identification for louderspeaker system. Chaos, Solitons & Fractals 21 (2004) 1231-1247
22. Ge Z.-M., and Chen C.-C. Phase synchronization of coupled chaotic multiple time scale systems. Chaos, Solitons & Fractals 20 (2004) 639-647
23. Ge Z.-M., and Chang C.-M. Chaos synchronization and parameter identification of single time scale brushless DC motors. Chaos, Solitons & Fractals 20 (2004) 883-903
24. Ge Z.-M., Cheng J.-W., and Chen Y.-S. Chaos anticontrol and synchronization of three time scales brushless DC motor system. Chaos, Solitons & Fractals 22 (2004) 1165-1182
25. Ge Z.-M., and Cheng J.-W. Chaos synchronization and parameter identification of three time scales brushless DC motor system. Chaos, Solitons & Fractals 24 (2005) 597-616
26. Ge Z.-M., and Lee C.-I. Control, anticontrol and synchronization of chaos for an autonomous rotational machine system with time-delay. Chaos, Solitons & Fractals 23 (2005) 1855-1864
27. Ge Z.-M., and Chen Y.-S. Adaptive synchronization of unidirectional and mutual coupled chaotic systems. Chaos, Solitons & Fractals 26 (2005) 881-888
28. Jiang G., Zheng W., and Chen G. Global chaos synchronization with channel time-delay. Chaos, Solitons & Fractals 20 (2004) 267-275
29. Chen G., and Liu S. On generalized synchronization of spatial chaos. Chaos, Solitons & Fractals 15 (2003) 311-318
30. Li Z., and Xu D. A secure communication scheme using projective chaos synchronization. Chaos, Solitons & Fractals 22 (2004) 477-481
31. Jiang G.-P., Tang W.K.-S., and Chen G. A simple global synchronization criterion for coupled chaotic systems. Chaos, Solitons & Fractals 15 (2003) 925-935
32. Ge Zheng-Ming, Yang Cheng-Hsiung. Generalized synchronization of quantum-CNN chaotic oscillator with different order systems. Chaos, Solitons & Fractals, accepted for publication.
33. Ge Z.-M., Chang C.-M., and Chen Y.-S. Anti-control of chaos of single time scale brushless DC motor and chaos synchronization of different order systems. Chaos, Solitons & Fractals 27 (2006) 1298-1315
34. Bi Q. Dynamical analysis of two coupled parametrically excited van der Pol oscillators. Int J Nonlinear Mech 39 (2004) 33-54
35. dos Santos A.M., Lopes S.R., and Viana R.L. Rhythm synchronization and chaotic modulation of coupled van der Pol oscillators in a model for the heartbeat. Physica A 338 (2004) 335-355
36. Pecora L.M., and Carroll T.L. Synchronization in chaotic systems. Phys Rev Lett 64 (1990) 821-824
37. Carroll T.L., Heagy J.F., and Pecora L.M. Transforming signals with chaotic synchronization. Phys Rev E 54 (1996) 4676-4680
38. Kocarev L., and Parlitz U. Generalized synchronization, predictability, and equivalence of unidirectionally coupled dynamical systems. Phys Rev Lett 76 (1996) 1816-1819
39. Rosenblum M.G., Pikovsky A.S., and Kurths J. Phase synchronization of chaotic oscillators. Phys Rev Lett 76 (1996) 1804-1807
40. Yang S.S., and Duan C.K. Generalized synchronization in chaotic systems. Chaos, Solitons & Fractals 9 (1998) 1703-1707
41. Chen G., and Liu S.T. On generalized synchronization of spatial chaos. Chaos, Solitons & Fractals 15 (2003) 311-318
42. Kim C.M., Rim S., Kye W.H., Ryu J.W., and Park Y.J. Anti-synchronization of chaotic oscillators. Phys Lett A 320 (2003) 39-46
43. Yang S.P., Niu H.Y., Tian G., et al. Synchronizing chaos by driving parameter. Acta Phys Sin 50 (2001) 619-623
44. Dai D., and Ma X.K. Chaos synchronization by using intermittent parametric adaptive control method. Phys Lett A 288 (2001) 23-28
45. Chen H.K. Synchronization of two different chaotic systems: a new system and each of the dynamical systems Lorenz, Chen and Lü. Chaos, Solitons & Fractals 25 (2005) 1049-1056
46. Chen H.K., and Lin T.N. Synchronization of chaotic symmetric Gyros by one-way coupling conditions. ImechE Part C: J Mech Eng Sci 217 (2003) 331-340
47. Chen H.K. Chaos and chaos synchronization of a symmetric gyro with linear-plus-cubic damping. J Sound Vibrat 255 (2002) 719-740
48. Ge Z.M., Yu T.C., and Chen Y.S. Chaos synchronization of a horizontal platform system. J Sound Vibrat (2003) 731-749
49. Ge Z.M., and Lin T.N. Chaos, chaos control and synchronization of electro-mechanical gyrostat system. J Sound Vibrat 259 3 (2003)
50. Ge Z.M., Lin C.C., and Chen Y.S. Chaos, chaos control and synchronization of vibrometer system. J Mech Eng Sci 218 (2004) 1001-1020
51. Chen H.K., Lin T.N., and Chen J.H. The stability of chaos synchronization of the Japanese attractors and its application. Jpn J Appl Phys 42 12 (2003) 7603-7610
52. Ge Z.M., and Shiue. Non-linear dynamics and control of chaos for tachometer. J Sound Vibrat 253 4 (2002)
53. Ge Z.M., and Lee C.I. Non-linear dynamics and control of chaos for a rotational machine with a hexagonal centrifugal governor with a spring. J Sound Vibrat 262 (2003) 845-864
54. Ge Z.M., Hsiao C.M., and Chen Y.S. Non-linear dynamics and chaos control for a time delay duffing system. Int J Nonlinear Sci Numer 6 2 (2005) 187-199
55. Ge Z.M., Tzen P.C., and Lee S.C. Parametric analysis and fractal-like basins of attraction by modified interpolates cell mapping. J Sound Vibrat 253 3 (2002)
56. Ge Z.M., and Lee S.C. Parameter used and accuracies obtain in MICM global analyses. J Sound Vibrat 272 (2004) 1079-1085
57. Ge Z.M., and Lee J.K. Chaos synchronization and parameter identification for gyroscope system. Appl Math Comput 63 (2004) 667-682
58. Chen H.K. Global chaos synchronization of new chaotic systems via nonlinear control. Chaos, Solitons & Fractals 4 (2005) 1245-1251
59. Chen H.K., and Lee C.I. Anti-control of chaos in rigid body motion. Chaos, Solitons & Fractals 21 (2004) 957-965
60. Ge Z.M., and Wu H.W. Chaos synchronization and chaos anticontrol of a suspended track with moving loads. J Sound Vibrat 270 (2004) 685-712
61. Ge Z.M., Yu C.Y., and Chen Y.S. Chaos synchronization and chaos anticontrol of a rotational supported simple pendulum. JSME Int J Ser C 47 1 (2004) 233-241
62. Ge Z.M., and Lee C.I. Anticontrol and synchronization of chaos for an autonomous rotational machine system with a hexagonal centrifugal governor. Chaos, Solitons & Fractals 282 (2005) 635-648