Generalized projective synchronization of the fractional-order Chen hyperchaotic system

Nonlinear Dynamics

Volumn 57, Issue 1-2, 2009, Pages 25-35

  Wu, Xiangjun ^a   Lu, Yang ^a  

^a HENAN UNIVERSITY   (China)

Author keywords

Chen hyperchaotic system; Fractional order system; Generalized projective synchronization

Indexed keywords

CHEN HYPERCHAOTIC SYSTEM; CONDITIONAL LYAPUNOV EXPONENT; FRACTIONAL CALCULUS; FRACTIONAL ORDER SYSTEM; GENERALIZED PROJECTIVE SYNCHRONIZATION; HYPER-CHAOTIC SYSTEMS; HYPERCHAOS; HYPERCHAOTIC; LYAPUNOV EXPONENT; NUMERICAL SIMULATION; SYNCHRONIZATION SCHEME;

DIFFERENTIAL EQUATIONS; LAPLACE TRANSFORMS; LYAPUNOV METHODS; SYNCHRONIZATION;

CHAOTIC SYSTEMS;

EID: 67449143115     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s11071-008-9416-5     Document Type: Article

References (53)

1. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
2. Hifer, R.: Applications of Fractional Calculus in Physics. World Scientific, New Jersey (2001)
3. R.C. Koeller 1984 Application of fractional calculus to the theory of viscoelasticity J. Appl. Mech. 51 299 307 (Pubitemid 14583151)
4. H.H. Sun A.A. Abdelwahad B. Onaral 1984 Linear approximation of transfer function with a pole of fractional order IEEE Trans. Autom. Control 29 441 444 10.1109/TAC.1984.1103551 (Pubitemid 14588227)
5. M. Ichise Y. Nagayanagi T. Kojima 1971 An analog simulation of noninteger order transfer functions for analysis of electrode process J. Electroanal. Chem. 33 253 265
6. Heaviside, O.: Electromagnetic Theory. Chelsea, New York (1971)
7. A. Oustaloup F. Levron F. Nanot B. Mathieu 2000 Frequency band complex noninteger differentiator: characterization and synthesis IEEE Trans. Circuits Syst.-I 47 25 40
8. Y.Q. Chen K. Moore 2000 Discretization schemes for fractional order differentiators and integrators IEEE Trans. Circuits Syst.-I 49 363 367
9. T.T. Hartley C.F. Lorenzo 2002 Dynamics and control of initialized fractional-order systems Nonlinear Dyn. 29 201 233 10.1023/A:1016534921583
10. C. Hwang J.F. Leu S.Y. Tsay 2002 A note on time-domain simulation of feedback fractional-order systems IEEE Trans. Autom. Control 47 625 631 10.1109/9.995039
11. T.T. Hartley C.F. Lorenzo H.K. Qammer 1995 Chaos in a fractional order Chua's system IEEE Trans. Circuits Syst.-I 42 485 490
12. Arena, P., Caponetto, R., Fortuna, L., Porto, D.: Chaos in a fractional order Duffing system. In: Proceedings of European Conference on Circuit Theory and Design, Budapest, pp. 1259-1262 (1997)
13. I. Grigorenko E. Grigorenko 2003 Chaotic dynamics of the fractional Lorenz system Phys. Rev. Lett. 91 034101 10.1103/PhysRevLett.91.034101
14. W.M. Ahmad J.C. Sprott 2003 Chaos in fractional-order autonomous nonlinear systems Chaos Solitons Fractals 16 339 351 10.1016/S0960-0779(02) 00438-1
15. W.M. Ahmad A.M. Harb 2003 On nonlinear control design for autonomous chaotic systems of integer and fractional orders Chaos Solitons Fractals 18 693 701 10.1016/S0960-0779(02)00644-6
16. C.G. Li G. Chen 2004 Chaos and hyperchaos in the fractional-order Rössler equations Physica A 341 55 61 10.1016/j.physa.2004.04.113
17. C.P. Li G.J. Peng 2004 Chaos in Chen's system with a fractional order Chaos Solitons Fractals 22 443 450 10.1016/j.chaos.2004.02.013
18. C.G. Li G. Chen 2004 Chaos in the fractional order Chen system and its control Chaos Solitons Fractals 22 549 554 10.1016/j.chaos.2004.02.035
19. G.M. Zaslavsky 2002 Chaos, fractional kinetics, and anomalous transport Phys. Rep. 371 461 580 10.1016/S0370-1573(02)00331-9
20. L.M. Pecora T.L. Carroll 1990 Synchronization in chaotic systems Phys. Rev. Lett. 64 821 824 10.1103/PhysRevLett.64.821
21. L. Kocarev U. Parlitz 1995 General approach for chaotic synchronization with application to communication Phys. Rev. Lett. 74 5028 5031 10.1103/PhysRevLett.74.5028
22. Lakshmanan, M., Murali, K.: Chaos in Nonlinear Oscillators: Controlling and Synchronization. World Scientific, Singapore (1996)
23. Chen, G., Dong, X.: From Chaos to Order: Methodologies, Perspectives and Applications. World Scientific, Singapore (1998)
24. X.J. Wu 2006 A new chaotic communication scheme based on adaptive synchronization Chaos 16 043118 10.1063/1.2401058
25. M.G. Rosenblum A.S. Pikovsky J. Kurtz 1996 Phase synchronization of chaotic oscillators Phys. Rev. Lett. 76 1804 1807 10.1103/PhysRevLett.76.1804 (Pubitemid 126638871)
26. A.S. Pikovsky M.G. Rosenlum G. Osipov J. Kurtz 1997 Phase synchronization of chaotic oscillators by external driving Physica D 104 219 238 10.1016/S0167-2789(96)00301-6 (Pubitemid 127701449)
27. M.G. Rosenblum A.S. Pikovsky J. Kurths 1997 From phase to lag synchronization in coupled chaotic oscillators Phys. Rev. Lett. 78 4193 4196 10.1103/PhysRevLett.78.4193 (Pubitemid 127655459)
28. Y. Zhang J. Sun 2004 Chaotic synchronization and anti-synchronization based on suitable separation Phys. Lett. A 330 442 447 10.1016/j.physleta.2004. 08.023
29. R. Mainieri J. Rehacek 1999 Projective synchronization in three-dimensional chaotic system Phys. Rev. Lett. 82 3042 3045 10.1103/PhysRevLett.82.3042 (Pubitemid 129689207)
30. D. Xu Z. Li 2001 Manipulating the scaling factor of projective synchronization in three-dimensional chaotic systems Chaos 11 439 442 10.1063/1.1380370
31. D. Xu C.Y. Chee C.P. Li 2004 A necessary condition of projective synchronization in discrete-time systems of arbitrary dimensions Chaos Solitons Fractals 22 175 180 10.1016/j.chaos.2004.01.012
32. G.H. Li 2007 Modified projective synchronization of chaotic system Chaos Solitons Fractals 32 1786 1790 10.1016/j.chaos.2005.12.009
33. M.L. Hung J.J. Yan T.L. Liao 2008 Generalized projective synchronization of chaotic nonlinear gyros coupled with dead-zone input Chaos Solitons Fractals 35 181 187 10.1016/j.chaos.2006.05.050
34. C.G. Li X.F. Liao J.B. Yu 2003 Synchronization of fractional order chaotic systems Phys. Rev. E Stat. Nonlinear Soft Matter Phys. 68 067203 10.1103/PhysRevE.68.067203
35. J.G. Lu 2005 Chaotic dynamics and synchronization of fractional-order Arneodo's systems Chaos Solitons Fractals 26 1125 1133 10.1016/j.chaos.2005.02. 023
36. C.P. Li W.H. Deng D. Xu 2006 Chaos synchronization of the Chua system with a fractional order Physica A 360 171 185 10.1016/j.physa.2005.06.078
37. C. Li J. Yan 2007 The synchronization of three fractional differential systems Chaos Solitons Fractals 32 751 757 10.1016/j.chaos.2005.11.020
38. M.S. Tavazoei M. Haeri 2008 Synchronization of chaotic fractional-order systems via active sliding mode controller Physica A 387 57 70 10.1016/j.physa.2007.08.039
39. Z.M. Ge C.Y. Ou 2008 Chaos synchronization of fractional order modified Duffing systems with parameters excited by a chaotic signal Chaos Solitons Fractals 35 705 717 10.1016/j.chaos.2006.05.101
40. Y. Yu H.X. Li 2008 The synchronization of fractional-order Rössler hyperchaotic systems Physica A 387 1393 1403 10.1016/j.physa.2007.10.052
41. N.F. Rulkov M.M. Sushchik L.S. Tsimring H.D.I. Abarbanel 1995 Generalized synchronization of chaos in directionally coupled chaotic systems Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 51 980 994 10.1103/PhysRevE.51.980
42. L. Kocarev U. Parlitz 1996 Generalized synchronization predictability, and equivalence of unidirectionally coupled dynamical systems. Phys. Rev. Lett. 76 1816 1819 10.1103/PhysRevLett.76.1816
43. Y.W. Wang Z.H. Guan 2006 Generalized synchronization of continuous chaotic system Chaos Solitons Fractals 27 97 101 10.1016/j.chaos.2004.12.038
44. M. Caputo 1967 Linear models of dissipation whose Q is almost frequency independent Geophys. J. R. Astron. Soc. 13 529 539
45. Samko, S.G., Klibas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach, Amsterdam (1993)
46. G.R. Chen T. Ueta 1999 Yet another chaotic attractor Int. J. Bifurc. Chaos 9 1465 1466 10.1142/S0218127499001024 (Pubitemid 129292251)
47. Y.X. Li W.K.S. Tang G.R. Chen 2005 Generating hyperchaos via state feedback control Int. J. Bifurc. Chaos 15 3367 3375 10.1142/S0218127405013988
48. Z.Y. Yan 2005 Controlling hyperchaos in the new hyperchaotic Chen system Appl. Math. Comput. 168 1239 1250 10.1016/j.amc.2004.10.016
49. K. Diethelm 1997 An algorithm for the numerical solution of differential equations of fractional order Electron Trans. Numer. Anal. 5 1 6 (Pubitemid 38870919)
50. K. Diethelm N.J. Ford 2002 Analysis of fractional differential equations J. Math. Anal. Appl. 265 229 248 10.1006/jmaa.2000.7194
51. K. Diethelm N.J. Ford A.D. Freed 2002 A predictor-corrector approach for the numerical solution of fractional differential equations Nonlinear Dyn. 29 3 22 10.1023/A:1016592219341
52. A. Wolf J.B. Swinney H.L. Swinney J.A. Vastano 1985 Determining Lyapunov exponents from a time series Physica D 16 285 317 10.1016/0167-2789(85)90011-9
53. Muth, E.J.: Transform Methods with Applications to Engineering and Operations Research. Prentice-Hall, Englewood Cliffs (1977)