On the robustness of chaos in dynamical systems: Theories and applications

Frontiers of Physics in China

Volumn 3, Issue 2, 2008, Pages 195-204

  Elhadj, Zeraoulia ^a   Sprott, J C ^b  

^a UNIVERSITY OF TEBESSA   (Algeria)

^b University of Wisconsin Madison   (United States)

Author keywords

Methods; Real applications; Robust chaos; Theories

Indexed keywords

EID: 44149085182     PISSN: 16733487     EISSN: 16733606     Source Type: Journal    
DOI: 10.1007/s11467-008-0017-z     Document Type: Article

References (80)

1. M. Andrecut and M. K. Ali, Inter. J. Mod. Phys. B, 2001, 15(2): 177
2. M. Andrecut and M. K. Ali, Mod. Phys. Lett. B, 2001, 15(12-13): 391
3. M. Andrecut and M. K. Ali, Europhys. Lett., 2001, 54: 300
4. M. Andrecut and M. K. Ali, Phys. Rev. E, 2001, 64: 025203(R)
5. P. Gabriel, Inter. J. Bifur. Chaos, 2004, 14(7): 2431
6. M. A. Jafarizadeh and S. Behnia, J. Nonlinear. Math. Phys., 2002, 9(1): 26
7. D. Singer, SIAM Journal on Applied Mathematics, 1978, 35(2): 260
8. E. Barreto, B. Hunt, C. Grebogi, and J. A. Yorke, Phys. Rev. Lett., 1997, 78: 4561
9. 0 8, 1978, 39: 9
10. M. Hénon, Commun. Math. Phys., 1976, 50: 69
11. B. Robert and C. Robert, Int. J. Control, 2002, 75(16-17): 1356
12. S. Banerjee and C. Grebogi, Phy. Rev. E, 1999, 59(4): 4052
13. S. Banergee, J. A. York, and C. Grebogi, Phys. Rev. Lett., 1998, 80(14): 3049
14. M. A. Hassouneh, E. Habed, and S. Banerjee, Feedback Control of Border Collision Bifurcations in Two-dimensional Discrete-time Systems, ISR Technical Research Report, 2002
15. J. M. Ottino, The Kinematics of Mixing: Stretching, Chaos, and Transport, Cambridge: Cambridge University Press, 1989
16. J. M. Ottino, F. J. Muzzion, M. Tjahjadi, J. G. Franjione, S. C. Jana, and H. A. Kusch, Science, 1992, 257: 754
17. E. N. Lorenz, J. Atmos. Sci., 1963, 20: 130
18. W. Tucker, C. R. Acad. Sci. Paris Ser. I: Math, 1999, 328(12): 1197
19. I. Stewart, Nature, 2000, 406: 948
20. V. Franceschini, C. Giberti, and Z. Zheng, Nonlinearity, 1993, 6: 251
21. M. Majumdar and T. Mitra, Economic Theory, 1994, 4(5): 677
22. C. A. Morales, M. J. Pacifico, and E. R. Pujalls, Annals of Mathematics, 2004, 160(2): 375
23. T. Archibald, Centaurus, 1988, 31(2): 141
24. B. Widrow and M. A. Lehr, 30 Years of Adaptive Neural Networks: Perceptron, Madaline, and Backpropagation, Proc. IEEE, 1990, 78 (9): 1415
25. Z. Galias and P. Zgliczynski, Physica D, 1998, 115: 165
26. J. Guckenheimer, A Strange, Strange Attractor, In: The Hopf Bifurcation and Its Applications, edited by Marsden and McCracken, New York: Springer-Verlag, 1976
27. J. Guckenheimer and R. F. Williams, Publ. Math. IHES, 1979, 50: 307
28. S. P. Hasting and W. C. Troy, Bull. Amer. Math. Soc, 1992, 27: 298
29. K. Mischaikow and M. Mrozek, Bull. Amer. Math. Soc., 1995, 32: 66
30. M. Mukul and T. Mitra, Ricerche Economiche, 1994, 48: 225
31. D. Rand, Math. Proc. Camb. Phil. Soc., 1978, 83: 451
32. C. Robinson, Nonlinearity, 1989, 2: 495
33. M. Rychlik, Ergod. Th. & Dynam. Sys., 1989, 10: 793
34. S. Smale, Math. Intelligencer, 1998, 20: 7
35. C. Sparrow, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, New York: Springer-Verlag, 1982
36. W. Tucker, Foundations of Computational Mathematics, 2002, 2(1): 53
37. R. F. Williams, Publ. Math. IHES, 1979, 50: 321
38. V. Araujo, M. J. Pacifico, E. R. Pujals, and M. Viana, preprint, arXiv: math.DS/0511352
39. V. S. Anishchenko and G. I. Strelkova, Attractors of Dynamical Systems, Control of Oscillations and Chaos, 1997. Proc. 1st International Conference, 27-29 Aug. 1997, Vol. 3: 498
40. S. Banerjee and G. C. Verghese (eds.), Nonlinear Phenomena in Power Electronics: Attractors, Bifurcations, Chaos, and Nonlinear Control, USA, New York: IEEE Press, 2001
41. F.-Y. Sun, S.-T. Liu, and Z.-W. Lv, Chinese Physics, 2007, 16(12): 3616
42. U. Vikas and R. Kumar, Chaos, Solitons & Fractals, 2004, 21(5): 1195
43. A. Potapov and M. K. Ali, Phy. Lett. A, 2000, 277(6): 310
44. C. Mira, Inter. J. Bifur. Chaos, 1997, 7(9): 1911
45. L. O. Chua, M. Komuro, and T. Matsumoto, IEEE Transaction in Circuit and System, 1986, CAS-33: 1073
46. S. Kuznetsov and E. Seleznev, Journal of Experimental and Theoretical Physics, 2006, 102(2): 355
47. P. Kowalczyk, Nonlinearity, 2005, 18: 485
48. R. Dogaru, A. T. Murgan, S. Ortmann, M. Glesner, Searching for Robust Chaos in Discrete Time Neural Networks Using Weight Space Exploration, Proc. ICNN'96, Washington D. C., 2-6 June, 1996
49. B. Henk, E. Konstantinos, and S. Easwar, Nonlinearity, 2008, 21(1): 13
50. E. Kei, I. Takahiro, and T. Akio, IEICE Trans. Fundamentals of Electronics, Communications and Computer Sciences, 1998, E81-A (6): 1223
51. U. S. Bhalla and R. Lyengar, Chaos, 2001, 11: 221
52. R. V. Plykin, Russ. Math. Surv., 2002, 57(6): 1163
53. M. C. Shastry, N. Nagaraj, and P. G. Vaidya, The β-Exponential Map: A Generalization of the Logistic Map, and Its Applications in Generating Pseudo-random Numbers, preprint
54. N. V. Nithin, G. Prabhakar, Bhat, and G. Kishor, Joint Entropy Coding and Encryption Using Robust Chaos, preprint
55. H. Kantz, C. Grebogi, A. Prasad, Y.-C. Lai, and E. Sinde, Phys. Rev. E, 2002, 65: 026209
56. V. S. Anishenko, T. E. Vadivasova, G. I. Strelkova, and A. S. Kopeikin, Discrete Dynamics in Nature and Society, 1998, 2: 249
57. N. E. Klinshpont, E. A. Sataev, and R. V. Plykin, Journal of Physics: Conference Series, 2005, 23: 96
58. R. Yulmetyev, N. Emelyanova, S. Demin, F. Gafarov, P. Hänggi, and D. Yulmetyeva, Fluctuations and Noise in Stochastic Spread of Respiratory Infection Epidemics in Social Networks, Unsolved Problems of Noise and Fluctuations UPoN, 3rd International Conference, 2002: 408
59. T. Dan, Prog. Theor. Phys., Suppl., 2006, 61: 119
60. G. D. Ruxton and P. Rohani, Theoretical Population Biology, 1998, 53(3): 175
61. M. Drutarovsky and P. A. Galajda, Radioengineering, 2007, 16(3): 120
62. J. A. Vano, J. C. Wildenberg, M. B. Anderson, J. K. Noel, and J. C. Sprott, Nonlinearity, 2006, 19(10): 2391
63. H. T. Yau and J. J. Yan, Chaos, Solitons and Fractals, 2004, 19(4): 891
64. B. Adrangi and A. Chatrath, Applied Financial Economics, 2003, 13(4): 245
65. M. Tapan and S. Gerhard, Japanese Economic Review, 1999, 50(4): 470
66. P. Ashwin and A. M. Rucklidge, Physica D, 1998, 122(1): 134
67. I. P. Cornfeld, S. V. Fomin, and Ya G. Sinai, Ergodic Theory, Berlin: Springer-Verlag, 1982
68. Z. Elhadj and J. C. Sprott., Inter. J. Bifur. Chaos, 2008, 18(4), to appear
69. Z. Elhadj and J. C. Sprott., A Unified Piecewise Smooth Chaotic Mapping that Contains the Hénon and the Lozi Systems, submitted
70. O. Alvarez-Llamoza, M. G. Cosenza, and G. A. Ponce, Chaos, Solitons & Fractals, 2008, 36(1): 150
71. A. Priel and I. Kanter, Europhys. Lett., 2000, 51(2): 230
72. V. S. Afraimovich, V. V. Bykov, and L. P. Shilńikov, Trans. Moscow Math. Soc., 1982, 44: 153
73. B. Hassard, J. Zhang, S. P. Hastings, and W. C. Troy, Appl. Math. Lett., 1994, 7: 79
74. K. Mischaikow and M. Mrozek, Bull. Amer. Math. Soc., 1995, 32(1): 66
75. K. Mischaikow and M. Mrozek, Math. Comp., 1998, 67 (223): 1023
76. Y. Chen and Y. Liu, Up-embeddability of a Graph by Order and Girth, Graphs and Combinatorics, 2007, 23(5): 521
77. S. Banerjee, D. Kastha, S. Das, G. Vivek, and C. Grebogi, Robust Chaos-the Theoretical Formulation and Experimental Evidence, ISCAS, 1999, 5: 293
78. V. Vijayaraghavan and L. Leung Henry, A Robust Chaos Radar for Collision Detection and Vehicular Ranging in Intelligent Transportation Systems, ITSC, 2004: 548
79. S. Shanmugam and H. Leung, A Robust Chaotic Spread Spectrum Inter-vehicle Communication Scheme for ITS, IEEE International Conference on Intelligent Transportation Systems, Shanghai, China, Oct. 2003: 1540
80. M. P. Dafilis, D. T. J. Liley, and P. J. Cadusch, Chaos, 2001, 11(3): 474