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

Abstract

This paper offers an overview of some important issues concerning the robustness of chaos in dynamical systems and their applications to the real world.

References

1. Andrecut M Ali M K Inter. J. Mod. Phys.B 2001 15(2)177. doi: 10.1142/S0217979201003715
2. Andrecut M Ali M K Mod. Phys. Lett. B 2001 15(12–13)391. doi: 10.1142/S0217984901001793
3. Andrecut M Ali M K Europhys. Lett. 2001 54300. doi: 10.1209/epl/i2001‐00241‐3
4. Andrecut M Ali M K Phys. Rev. E 2001 64025203(R)
5. Gabriel P Inter. J. Bifur. Chaos 2004 14(7)2431. doi: 10.1142/S0218127404010722
6. Jafarizadeh M A Behnia S J. Nonlinear. Math. Phys. 2002 9(1)26. doi: 10.2991/jnmp.2002.9.1.4
7. Singer D SIAM Journal on Applied Mathematics 1978 35(2)260. doi: 10.1137/0135020
8. Barreto E Hunt B Grebogi C Yorke J A Phys.Rev. Lett. 1997 784561. doi: 10.1103/PhysRevLett.78.4561
9. Lozi R Colloque C5, Supplément au n⁰ 8 1978 399
10. Hénon M Commun. Math. Phys. 1976 5069. doi: 10.1007/BF01608556
11. Robert B Robert C Int. J. Control 2002 75(16–17)1356. doi: 10.1080/0020717021000023771
12. Banerjee S Grebogi C Phy.Rev.E 1999 59(4)4052. doi: 10.1103/PhysRevE.59.4052
13. Banergee S York J A Grebogi C Phys. Rev. Lett. 1998 80(14)3049. doi: 10.1103/PhysRevLett.80.3049
14. Hassouneh M A Habed E Banerjee S Feedback Controlof Border Collision Bifurcationsin Two-dimensional Discrete-time SystemsISR TechnicalResearch Report 2002
15. Ottino J M The Kinematics of Mixing: Stretching, Chaos, and TransportCambridgeCambridgeUniversity Press 1989
16. Ottino J M Muzzion F J Tjahjadi M Franjione J G Jana S C Kusch H A Science 1992 257754. doi: 10.1126/science.257.5071.754
17. Lorenz E N J. Atmos. Sci. 1963 20130. doi: 10.1175/1520‐0469(1963)020<0130:DNF>2.0.CO;2
18. Tucker W C. R. Acad. Sci. Paris Ser. I: Math 1999 328(12)1197
19. Stewart I Nature 2000 406948. doi: 10.1038/35023206
20. Franceschini V Giberti C Zheng Z Nonlinearity 1993 6251. doi: 10.1088/0951‐7715/6/2/006
21. Majumdar M Mitra T Economic Theory 1994 4(5)677. doi: 10.1007/BF01212024
22. Morales C A Pacifico M J Pujalls E R Annals of Mathematics 2004 160(2)375
23. Archibald T Centaurus 1988 31(2)141. doi: 10.1111/j.1600‐0498.1988.tb00684.x
24. Widrow B Lehr M A 30 Years of AdaptiveNeural Networks: Perceptron, Madaline, and Backpropagation, Proc.IEEE 1990 78(9)1415
25. Galias Z Zgliczynski P Physica D 1998 115165. doi: 10.1016/S0167‐2789(97)00233‐9
26. Guckenheimer J A Strange,Strange AttractorIn: The Hopf Bifurcationand Its ApplicationsMarsden and McCrackenNew YorkSpringer-Verlag 1976
27. Guckenheimer J Williams R F Publ. Math. IHES 1979 50307
28. Hasting S P Troy W C Bull. Amer. Math. Soc 1992 27298. doi: 10.1090/S0273‐0979‐1992‐00327‐0
29. Mischaikow K Mrozek M Bull. Amer. Math. Soc. 1995 3266. doi: 10.1090/S0273‐0979‐1995‐00558‐6
30. Mukul M Mitra T Ricerche Economiche 1994 48225. doi: 10.1016/0035‐5054(94)90026‐4
31. Rand D Math. Proc. Camb. Phil. Soc. 1978 83451
32. Robinson C Nonlinearity 1989 2495. doi: 10.1088/0951‐7715/2/4/001
33. Rychlik M Ergod. Th. & Dynam. Sys. 1989 10793
34. Smale S Math. Intelligencer 1998 207
35. Sparrow C The Lorenz Equations: Bifurcations, Chaos, and Strange AttractorsNew YorkSpringer-Verlag 1982
36. Tucker W Foundations of Computational Mathematics 2002 2(1)53
37. Williams R F Publ. Math. IHES 1979 50321
38. Araujo V Pacifico M J Pujals E R Viana M preprint, arXiv:math.DS/0511352
39. Anishchenko V S Strelkova G I Attractors of DynamicalSystems, Control of Oscillations and Chaos, 1997. Proc. 1st InternationalConference 27–29 Aug. 1997 Vol. 3498. doi: 10.1109/COC.1997.626654
40. Banerjee SVergheseG CNonlinear Phenomena in Power Electronics:Attractors, Bifurcations, Chaos, and Nonlinear Control, USANew YorkIEEEPress 2001
41. Sun F-Y Liu S-T Lv Z-W Chinese Physics 2007 16(12)3616. doi: 10.1088/1009‐1963/16/12/011
42. Vikas U Kumar R Chaos, Solitons &Fractals 2004 21(5)1195. doi: 10.1016/j.chaos.2003.12.065
43. Potapov A Ali M K Phy. Lett. A 2000 277(6)310. doi: 10.1016/S0375‐9601(00)00726‐X
44. Mira C Inter. J. Bifur. Chaos 1997 7(9)1911. doi: 10.1142/S0218127497001497
45. Chua L O Komuro M Matsumoto T IEEE Transaction in Circuit and System 1986 CAS-331073
46. Kuznetsov S Seleznev E Journal of Experimentaland Theoretical Physics 2006 102(2)355. doi: 10.1134/S1063776106020166
47. Kowalczyk P Nonlinearity 2005 18485. doi: 10.1088/0951‐7715/18/2/002
48. Dogaru R Murgan A T Ortmann S lesner M G Searchingfor Robust Chaos in Discrete Time Neural Networks Using Weight SpaceExploration, Proc. ICNN'96, Washington D. C. 2–6 June, 1996
49. Henk B Konstantinos E Easwar S Nonlinearity 2008 21(1)13. doi: 10.1088/0951‐7715/21/1/002
50. Kei E Takahiro I Akio T IEICE Trans. Fundamentals of Electronics, Communicationsand Computer Sciences 1998 E81-A(6)1223
51. Bhalla U S Lyengar R Chaos 2001 11221. doi: 10.1063/1.1350440
52. Plykin R V Russ. Math. Surv. 2002 57(6)1163. doi: 10.1070/RM2002v057n06ABEH000574
53. Shastry M C Nagaraj N Vaidya P G The β-ExponentialMap: A Generalization of the Logistic Map, and Its Applications inGenerating Pseudo-random Numberspreprint
54. Nithin N V Prabhakar G Bhat. Kishor G Joint EntropyCoding and Encryption Using Robust Chaos preprint
55. Kantz H Grebogi C Prasad A Lai Y-C Sinde E Phys. Rev. E 2002 65026209. doi: 10.1103/PhysRevE.65.026209
56. Anishenko V S Vadivasova T E Strelkova G I Kopeikin A S DiscreteDynamics in Nature and Society 1998 2249. doi: 10.1155/S1026022698000223
57. Klinshpont N E Sataev E A Plykin R V Journal of Physics: Conference Series 2005 2396. doi: 10.1088/1742‐6596/23/1/011
58. Yulmetyev R Emelyanova N Demin S Gafarov F Hänggi P Yulmetyeva D Fluctuations and Noise in Stochastic Spread of RespiratoryInfection Epidemics in Social Networks, Unsolved Problems of Noiseand Fluctuations UPoN, 3rd International Conference 2002 408
59. Dan T Prog. Theor. Phys., Suppl. 2006 61119
60. Ruxton G D Rohani P Theoretical PopulationBiology 1998 53(3)175. doi: 10.1006/tpbi.1998.1312
61. Drutarovsky M Galajda P A Radioengineering 2007 16(3)120
62. Vano J A Wildenberg J C Anderson M B Noel J K Sprott J C Nonlinearity 2006 19(10)2391. doi: 10.1088/0951‐7715/19/10/006
63. Yau H T Yan J J Chaos, Solitons and Fractals 2004 19(4)891. doi: 10.1016/S0960‐0779(03)00255‐8
64. Adrangi B Chatrath A Applied Financial Economics 2003 13(4)245. doi: 10.1080/09603100110115660
65. Tapan M Gerhard S Japanese Economic Review 1999 50(4)470. doi: 10.1111/1468‐5876.00133
66. Ashwin P Rucklidge A M Physica D 1998 122(1)134. doi: 10.1016/S0167‐2789(98)00174‐2
67. Cornfeld I P Fomin S V Sinai Ya G Ergodic TheoryBerlinSpringer-Verlag 1982
68. Elhadj Z Sprott J C Inter. J. Bifur. Chaos 2008 18(4)to appear
69. Elhadj Z Sprott J C A Unified Piecewise SmoothChaotic Mapping that Contains the Hénon and the Lozi Systemssubmitted
70. Alvarez-Llamoza O Cosenza M G Ponce G A Chaos, Solitons & Fractals 2008 36(1)150. doi: 10.1016/j.chaos.2006.06.017
71. Priel A Kanter I Europhys. Lett. 2000 51(2)230. doi: 10.1209/epl/i2000‐00521‐4
72. Afraimovich V S Bykov V V Shil'nikov L P Trans. Moscow Math. Soc. 1982 44153
73. Hassard B Zhang J Hastings S P Troy W C Appl.Math. Lett. 1994 779. doi: 10.1016/0893‐9659(94)90058‐2
74. Mischaikow K Mrozek M Bull. Amer. Math. Soc. 1995 32(1)66. doi: 10.1090/S0273‐0979‐1995‐00558‐6
75. Mischaikow K Mrozek M Math. Comp. 1998 67(223)1023. doi: 10.1090/S0025‐5718‐98‐00945‐4
76. Chen Y Liu Y Up-embeddability of aGraph by Order and Girth, Graphs and Combinatorics 2007 23(5)521
77. Banerjee S Kastha D Das S Vivek G Grebogi C Robust Chaos-the Theoretical Formulationand Experimental Evidence, ISCAS 1999 5293
78. Vijayaraghavan V Leung Henry L A Robust Chaos Radarfor Collision Detection and Vehicular Ranging in Intelligent TransportationSystems, ITSC 2004 548
79. Shanmugam S Leung H A Robust Chaotic SpreadSpectrum Inter-vehicle Communication Scheme for ITS, IEEE InternationalConference on Intelligent Transportation Systems, Shanghai, China Oct. 2003 1540
80. Dafilis M P Liley D T J Cadusch P J Chaos 2001 11(3)474. doi: 10.1063/1.1394193