1887
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
The full text of this article is not currently available.
/content/aip/journal/chaos/23/3/10.1063/1.4813227
1.
1. E. N. Lorenz, “ Deterministic nonperiodic flow,” J. Atmos. Sci. 20, 130141 (1963).
http://dx.doi.org/10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2
2.
2. C. Sparrow, The Lorenz Equations (Springer, New York, 1982).
3.
3. M. Kús, “ Integrals of motion for the Lorenz system,” J. Phys. A 16, L689L691 (1983).
http://dx.doi.org/10.1088/0305-4470/16/18/002
4.
4. J. Pade, A. Rauh, and G. Tsarouhas, “ Analytical investigation of the Hopf bifurcation in the Lorenz model,” Phys. Lett. A 115, 9396 (1986).
http://dx.doi.org/10.1016/0375-9601(86)90031-9
5.
5. W. Tucker, “ The Lorenz attractor exists,” C. R. Acad. Sci., Ser. I: Math. 328, 11971202 (1999).
http://dx.doi.org/10.1016/S0764-4442(99)80439-X
6.
6. I. Stewart, “ The Lorenz attractor exists,” Nature 406, 948949 (2000).
http://dx.doi.org/10.1038/35023206
7.
7. J. Llibre and X. Zhang, “ Invariant algebraic surfaces of the Lorenz system,” J. Math. Phys. 43, 16221645 (2002).
http://dx.doi.org/10.1063/1.1435078
8.
8. P. Swinnerton-Dyer, “ The invariant algebraic surfaces of the Lorenz system,” Math. Proc. Cambridge Philos. Soc. 132, 385393 (2002).
http://dx.doi.org/10.1017/S0305004101005667
9.
9. J. Cao and X. Zhang, “ Dynamics of the Lorenz system having an invariant algebraic surface,” J. Math. Phys. 48, 082702 (2007).
http://dx.doi.org/10.1063/1.2767007
10.
10. M. Messias, “ Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system,” J. Phys. A: Math. Theor. 42, 115101 (2009).
http://dx.doi.org/10.1088/1751-8113/42/11/115101
11.
11. J. Llibre, M. Messias, and P. R. da Silva, “ Global dynamics of the Lorenz system with invariant algebraic surfaces,” Int. J. Bifurcation Chaos 20, 31373155 (2010).
http://dx.doi.org/10.1142/S0218127410027593
12.
12. J. and G. Chen, “ A new chaotic attractor coined,” Int. J. Bifurcation Chaos 12, 659661 (2002).
http://dx.doi.org/10.1142/S0218127402004620
13.
13. C. Liu, T. Liu, L. Liu, and K. Liu, “ A new chaotic attractor,” Chaos, Solitons Fractals 22, 10311038 (2004).
http://dx.doi.org/10.1016/j.chaos.2004.02.060
14.
14. B. Jiang, X. Han and Q. Bi, “Hopf bifurcation analysis in the T-system,” Nonlinear Anal. Real World Appl. 11, 522527 (2010).
http://dx.doi.org/10.1016/j.nonrwa.2009.01.007
15.
15. Z. Wei, “ Dynamical behaviors of a chaotic system with no equilibria,” Phys. Lett. A 376, 102108 (2011).
http://dx.doi.org/10.1016/j.physleta.2011.10.040
16.
16. G. Chen and T. Ueta, “ Yet another chaotic attractor,” Int. J. Bifurcation Chaos 9, 14651466 (1999).
http://dx.doi.org/10.1142/S0218127499001024
17.
17. T. Ueta and G. R. Chen, “ Bifurcation analysis of Chen's equation,” Int. J. Bifurcation Chaos 10, 19171931 (2000).
http://dx.doi.org/10.1142/S0218127400001183
18.
18. J. H. , T. S. Zhou, G. R. Chen, and S. C. Zhang, “ Local bifurcation of the Chen system,” Int. J. Bifurcation Chaos 12, 22572270 (2002).
http://dx.doi.org/10.1142/S0218127402005819
19.
19. C. Li and G. Chen, “ A note on Hopf bifurcation in Chen's system,” Int. J. Bifurcation Chaos 13, 16091615 (2003).
http://dx.doi.org/10.1142/S0218127403007394
20.
20. T. S. Zhou, Y. Tang, and G. R. Chen, “ Complex dynamical behaviors of the chaotic Chen's system,” Int. J. Bifurcation Chaos 13, 25612574 (2003).
http://dx.doi.org/10.1142/S0218127403008089
21.
21. T. C. Li, G. R. Chen, and Y. Tang, “ On stability and bifurcation of Chen's system,” Chaos, Solitons Fractals 19, 12691282 (2004).
http://dx.doi.org/10.1016/S0960-0779(03)00334-5
22.
22. T. Zhou, Y. Tang, and G. Chen, “ Chen's attractor exists,” Int. J. Bifurcation Chaos 14, 31673177 (2004).
http://dx.doi.org/10.1142/S0218127404011296
23.
23. Y. Chang and G. Chen, “ Complex dynamics in Chen's system,” Chaos, Solitons Fractals 27, 7586 (2006).
http://dx.doi.org/10.1016/j.chaos.2004.12.011
24.
24. T. Li, G. Chen, and G. Chen, “ On homoclinic and heteroclinic orbits of Chen's system,” Int. J. Bifurcation Chaos 16, 30353041 (2006).
http://dx.doi.org/10.1142/S021812740601663X
25.
25. T. Lu and X. Zhang, “ Darboux polynomials and algebraic integrability of the Chen system,” Int. J. Bifurcation Chaos 17, 27392748 (2007).
http://dx.doi.org/10.1142/S0218127407018725
26.
26. W. X. Qin and G. R. Chen, “ On the boundedness of the solutions of the Chen system,” J. Math. Anal. Appl. 329, 445451 (2007).
http://dx.doi.org/10.1016/j.jmaa.2006.06.091
27.
27. J. Cao, C. Chen, and X. Zhang, “ The Chen system having an algebraic surface,” Int. J. Bifurcation Chaos 18, 37533758 (2008).
http://dx.doi.org/10.1142/S0218127408022706
28.
28. Z. Hou, N. Kang, X. Kong, G. Chen, and G. Yan, “ On the nonequivalence of Lorenz system and Chen system,” Int. J. Bifurcation and Chaos 20, 557560 (2010).
http://dx.doi.org/10.1142/S0218127410025612
29.
29. R. Barboza and G. Chen, “ On the global boundedness of the Chen system,” Int. J. Bifurcation Chaos 21, 33733385 (2011).
http://dx.doi.org/10.1142/S021812741103060X
30.
30. X. Deng and A. Chen, “ Invariant algebraic surfaces of the Chen system,” Int. J. Bifurcation Chaos 21, 16451651 (2011).
http://dx.doi.org/10.1142/S0218127411029331
31.
31. J. Llibre, M. Messias, and P. R. da Silva, “ Global dynamics in the Poincaré ball of the Chen system having invariant algebraic surfaces,” Int. J. Bifurcation Chaos 22, 1250154 (2012).
http://dx.doi.org/10.1142/S0218127412501544
32.
32. X. Wang and G. Chen, “ A Gallery of Lorenz-like and Chen-like attractors,” Int. J. Bifurcation Chaos 23, 1330011 (2013).
http://dx.doi.org/10.1142/S0218127413300115
33.
33. J. , G. Chen, D. Cheng, and S. Celikovský, “ Bridge the gap between the Lorenz system and the Chen system,” Int. J. Bifurcation Chaos 12, 29172926 (2002).
http://dx.doi.org/10.1142/S021812740200631X
34.
34. D. Li, J. Lu, X. Wu, and G. Chen, “ Estimating the bounds for the Lorenz family of chaotic systems,” Chaos, Solitons Fractals 23, 529534 (2005).
http://dx.doi.org/10.1016/j.chaos.2004.05.021
35.
35. D. Li, J. Lu, X. Wu, and G. Chen, “ Estimating the ultimate bound and positively invariant set for the Lorenz system and a unified chaotic system,” J. Math. Anal. Appl. 323, 844853 (2006).
http://dx.doi.org/10.1016/j.jmaa.2005.11.008
36.
36. X. Chen and C. Liu, “ Passive control on a unified chaotic system,” Nonlinear Anal.: Real World Appl. 11, 683687 (2010).
http://dx.doi.org/10.1016/j.nonrwa.2009.01.014
37.
37. Y. Pan, M. J. Er, and T. Sun, “ Composite adaptive fuzzy control for synchronizing generalized Lorenz systems,” Chaos 22, 023144 (2012).
http://dx.doi.org/10.1063/1.4721901
38.
38. C. Lainscsek, “ A class of Lorenz-like systems,” Chaos 22, 013126 (2012).
http://dx.doi.org/10.1063/1.3689438
39.
39. L. Runzi and W. Yinglan, “ Finite-time stochastic combination synchronization of three different chaotic systems and its application in secure communication,” Chaos 22, 023109 (2012).
http://dx.doi.org/10.1063/1.3702864
40.
40. C. F. Chuang, Y. J. Sun, and W. J. Wang, “ A novel synchronization scheme with a simple linear control and guaranteed convergence time for generalized Lorenz chaotic systems,” Chaos 22, 043108 (2012).
http://dx.doi.org/10.1063/1.4761818
41.
41. L. Runzi, W. Yinglan, and D. Shucheng, “ Combination synchronization of three classic chaotic systems using active backstepping design,” Chaos 21, 043114 (2011).
http://dx.doi.org/10.1063/1.3655366
42.
42. L. Chen, Y. Chai, and R. Wu, “ Linear matrix inequality criteria for robust synchronization of uncertain fractional-order chaotic systems,” Chaos 21, 043107 (2011).
http://dx.doi.org/10.1063/1.3650237
43.
43. C. J. Kim and D. Chwa, “ Synchronization of the bidirectionally coupled unified chaotic system via sum of squares method,” Chaos 21, 013104 (2011).
http://dx.doi.org/10.1063/1.3553183
44.
44. X. Wu, W. X. Zheng, and J. Zhou, “ Generalized outer synchronization between complex dynamical networks,” Chaos 19, 013109 (2009).
http://dx.doi.org/10.1063/1.3072787
45.
45. F. Sun, H. Peng, Q. Luo, L. Li, and Y. Yang, “ Parameter identification and projective synchronization between different chaotic systems,” Chaos 19, 023109 (2009).
http://dx.doi.org/10.1063/1.3127599
46.
46. Y. Tang, Z. Wang, and J. Fang, “ Pinning control of fractional-order weighted complex networks,” Chaos 19, 013112 (2009).
http://dx.doi.org/10.1063/1.3068350
47.
47. L. Wang, “ Yet another 3D quadratic autonomous system generating three-wing and four-wing chaotic attractors,” Chaos 19, 013107 (2009).
http://dx.doi.org/10.1063/1.3070648
48.
48. L. Zhou and F. Chen, “ Sil'nikov chaos of the Liu system,” Chaos 18, 013113 (2008).
http://dx.doi.org/10.1063/1.2839909
49.
49. F. Liu, Q. Song, and J. Cao, “ Improvements and applications of entrainment control for nonlinear dynamical systems,” Chaos 18, 043120 (2008).
http://dx.doi.org/10.1063/1.3029670
50.
50. W. Xia and J. Cao, “ Adaptive synchronization of a switching system and its applications to secure communications,” Chaos 18, 023128 (2008).
http://dx.doi.org/10.1063/1.2937017
51.
51. W. Yu, J. Cao, K. W. Wong, and J. , “ New communication schemes based on adaptive synchronization,” Chaos 17, 033114 (2007).
http://dx.doi.org/10.1063/1.2767407
52.
52. Z. Chen, W. Lin, and J. Zhou, “ Complete and generalized synchronization in a class of noise perturbed chaotic systems,” Chaos 17, 023106 (2007).
http://dx.doi.org/10.1063/1.2718491
53.
53. C. Letellier, G. F. V. Amaral, and L. A. Aguirre, “ Insights into the algebraic structure of Lorenz-like systems using feedback circuit analysis and piecewise affine models,” Chaos 17, 023104 (2007).
http://dx.doi.org/10.1063/1.2645725
54.
54. X. J. Wu, “ A new chaotic communication scheme based on adaptive synchronization,” Chaos 16, 043118 (2006).
http://dx.doi.org/10.1063/1.2401058
55.
55. S. Yu, J. , W. K. S. Tang, and G. Chen, “ A general multiscroll Lorenz system family and its realization via digital signal processors,” Chaos 16, 033126 (2006).
http://dx.doi.org/10.1063/1.2336739
56.
56. A. Algaba, F. Fernández-Sánchez, M. Merino, and A. J. Rodríguez-Luis, “ Comment on ‘Sil'nikov chaos of the Liu system’ [Chaos 18, 013113 (2008)],” Chaos 21, 048101 (2011).
http://dx.doi.org/10.1063/1.3657921
57.
57. A. Algaba, F. Fernández-Sánchez, M. Merino, and A. J. Rodríguez-Luis, “ Comment on ‘Heteroclinic orbits in Chen circuit with time delay’ [Commun. Nonlinear Sci. Numer. Simulat. 15, 3058-3066 (2010)],” Commun. Nonlinear Sci. Numer. Simulat. 17, 27082710 (2012).
http://dx.doi.org/10.1016/j.cnsns.2011.10.011
58.
58. A. Algaba, F. Fernández-Sánchez, M. Merino, and A. J. Rodríguez-Luis, “ Comment on ‘Existence of heteroclinic orbits of the Shil'nikov type in a 3D quadratic autonomous chaotic system’ [J. Math. Anal. Appl. 315, 106-119 (2006)],” J. Math. Anal. Appl. 392, 99101 (2012).
http://dx.doi.org/10.1016/j.jmaa.2012.01.040
59.
59. A. Algaba, F. Fernández-Sánchez, M. Merino, and A. J. Rodríguez-Luis, “ Comment on ‘Silnikov-type orbits of Lorenz-family systems’ [Physica A 375, 438-446 (2007)],” Physica A (published online).
http://dx.doi.org/10.1016/j.physa.2013.05.030
60.
60. T. Zhou, G. Chen, and S. Celikovský, “ Si'lnikov chaos in the generalized Lorenz canonical form of dynamical systems,” Nonlinear Dyn. 39, 319334 (2005).
http://dx.doi.org/10.1007/s11071-005-4195-8
61.
61. Z. Zheng and G. Chen, “ Existence of heteroclinic orbits of the Shil'nikov type in a 3D quadratic autonomous chaotic system,” J. Math. Anal. Appl. 315, 106119 (2006).
http://dx.doi.org/10.1016/j.jmaa.2005.09.087
62.
62. J. Wang, M. Zhao, Y. Zhang, and X. Xiong, “ Šilnikov-type orbits of Lorenz-family systems,” Physica A 375, 438446 (2007).
http://dx.doi.org/10.1016/j.physa.2006.10.007
63.
63. A. Algaba, F. Fernández-Sánchez, M. Merino, and A. J. Rodríguez-Luis, “ Comments on ‘Analysis and application of a novel three-dimensional energy-saving and emission-reduction dynamic evolution system’ [Energy 40, 291-299 (2012)],” Energy 47, 630633 (2012).
http://dx.doi.org/10.1016/j.energy.2012.07.033
64.
64. A. Algaba, F. Fernández-Sánchez, M. Merino, and A. J. Rodríguez-Luis, “ Lü system is a particular case of Lorenz system,” Phys. Lett. A (submitted).
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/3/10.1063/1.4813227
Loading
/content/aip/journal/chaos/23/3/10.1063/1.4813227
Loading

Data & Media loading...

Loading

Article metrics loading...

/content/aip/journal/chaos/23/3/10.1063/1.4813227
2013-07-10
2015-03-29

Abstract

In this paper, we show, by means of a linear scaling in time and coordinates, that the Chen system, given by is, generically ( ), a special case of the Lorenz system. First, we infer that it is enough to consider two parameters to study its dynamics. Furthermore, we prove that there exists a homothetic transformation between the Chen and the Lorenz systems and, accordingly, all the dynamical behavior exhibited by the Chen system is present in the Lorenz system (since the former is a special case of the second). We illustrate our results relating Hopf bifurcations, periodic orbits, invariant surfaces, and chaotic attractors of both systems. Since there has been a large literature that has ignored this equivalence, the aim of this paper is to review and clarify this field. Unfortunately, a lot of the previous papers on the Chen system are unnecessary or incorrect.

Loading

Full text loading...

/deliver/fulltext/aip/journal/chaos/23/3/1.4813227.html;jsessionid=yei4aa2gwn4d.x-aip-live-06?itemId=/content/aip/journal/chaos/23/3/10.1063/1.4813227&mimeType=html&fmt=ahah&containerItemId=content/aip/journal/chaos
true
true
This is a required field
Please enter a valid email address
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Chen's attractor exists if Lorenz repulsor exists: The Chen system is a special case of the Lorenz system
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/3/10.1063/1.4813227
10.1063/1.4813227
SEARCH_EXPAND_ITEM