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Lü, Jinhu; Chen, Guanrong; Cheng, Daizhan

A new chaotic system and beyond: the generalized Lorenz-like system. (English) Zbl 1129.37323

Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, No. 5, 1507-1537 (2004).
Summary: This article introduces a new chaotic system of three-dimensional quadratic autonomous ordinary differential equations, which can display (i) two 1-scroll chaotic attractors simultaneously, with only three equilibria, and (ii) two 2-scroll chaotic attractors simultaneously, with five equilibria. Several issues such as some basic dynamical behaviors, routes to chaos, bifurcations, periodic windows, and the compound structure of the new chaotic system are then investigated, either analytically or numerically. Of particular interest is the fact that this chaotic system can generate a complex 4-scroll chaotic attractor or confine two attractors to a 2-scroll chaotic attractor under the control of a simple constant input. Furthermore, the concept of generalized Lorenz system is extended to a new class of generalized Lorenz-like systems in a canonical form. Finally, the important problems of classification and normal form of three-dimensional quadratic autonomous chaotic systems are formulated and discussed.

Cited in 2 Reviews
Cited in 126 Documents

MSC:

37D45 Strange attractors, chaotic dynamics of systems with hyperbolic behavior
34C20 Transformation and reduction of ordinary differential equations and systems, normal forms
34C28 Complex behavior and chaotic systems of ordinary differential equations
37C70 Attractors and repellers of smooth dynamical systems and their topological structure

Keywords:

Multi-scroll chaotic attractor; chaotification; three-dimensional quadratic autonomous system; Lorenz-like system; normal form
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\textit{J. Lü} et al., Int. J. Bifurcation Chaos Appl. Sci. Eng. 14, No. 5, 1507--1537 (2004; Zbl 1129.37323)
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References:

[1] DOI: 10.1016/0167-2789(85)90093-4 · Zbl 0595.58030
[2] DOI: 10.1142/S0218127402005467 · Zbl 1043.37023
[3] DOI: 10.1109/81.251821 · Zbl 0850.93355
[4] DOI: 10.1142/3033
[5] DOI: 10.1142/S0218127498001236 · Zbl 0941.93522
[6] Chen G., Controlling Chaos and Bifurcations in Engineering Systems (1999)
[7] DOI: 10.1142/S0218127499001024 · Zbl 0962.37013
[8] Chen G., Dynamical Analysis, Control and Synchronization of the Generalized Lorenz Systems Family (2003)
[9] DOI: 10.1017/CBO9780511665639
[10] DOI: 10.1142/S0218127495001228 · Zbl 0886.58069
[11] DOI: 10.1109/81.915386 · Zbl 0998.94048
[12] DOI: 10.1109/81.995671 · Zbl 1368.37040
[13] DOI: 10.1103/PhysRevE.65.046205 · Zbl 1244.34067
[14] DOI: 10.1007/BF03168594 · Zbl 0991.76082
[15] DOI: 10.1016/0005-1098(92)90177-H · Zbl 0765.93030
[16] Hao B. L., Chaos (1984) · Zbl 0559.58012
[17] DOI: 10.1103/PhysRevE.51.759
[18] Jack H., Nonlinearity 12 pp 617–
[19] DOI: 10.1016/0167-2789(85)90176-9 · Zbl 0579.76051
[20] DOI: 10.1016/S0021-8928(01)00004-1
[21] DOI: 10.1016/S0375-9601(99)00450-8 · Zbl 0935.37008
[22] Liu W. B., Int. J. Bifurcation and Chaos 12 pp 261–
[23] DOI: 10.1142/S0218127497001837 · Zbl 0906.58025
[24] DOI: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 · Zbl 1417.37129
[25] DOI: 10.4324/9780203214589
[26] DOI: 10.1142/S0218127402004620 · Zbl 1063.34510
[27] DOI: 10.1142/S021812740200631X · Zbl 1043.37026
[28] DOI: 10.1142/S0218127402004851 · Zbl 1044.37021
[29] DOI: 10.1016/S0960-0779(02)00007-3 · Zbl 1067.37042
[30] DOI: 10.1142/S0218127402005200
[31] Lü J., Chaotic Time Series Analysis and Its Applications (2002)
[32] DOI: 10.1063/1.1478079
[33] DOI: 10.1142/S0218127402004735 · Zbl 1044.37022
[34] DOI: 10.1142/S0218127402005819 · Zbl 1047.34044
[35] DOI: 10.1016/S0960-0779(02)00456-3 · Zbl 1039.37019
[36] Lü J., IEEE Trans. Circuits Syst.-I 50 pp 198–
[37] Magnitskii N. A., Diff. Eqs. 37 pp 1494–
[38] DOI: 10.1103/PhysRevLett.64.1196 · Zbl 0964.37501
[39] DOI: 10.1103/PhysRevA.33.4253
[40] DOI: 10.1016/0375-9601(76)90101-8 · Zbl 1371.37062
[41] DOI: 10.1111/j.1749-6632.1979.tb29482.x
[42] DOI: 10.1017/S0022112092003392 · Zbl 0747.76089
[43] DOI: 10.1063/1.868614 · Zbl 1023.76582
[44] DOI: 10.1016/0167-2789(93)90292-9 · Zbl 0783.58052
[45] Shilnikov A. L., Int. J. Bifurcation and Chaos 5 pp 1123– · Zbl 0899.58044
[46] DOI: 10.1142/S0218127495001253 · Zbl 0922.76197
[47] DOI: 10.1142/9789812798558
[48] Smale S., Mathematics: Frontiers and Perspectives pp 271–
[49] DOI: 10.1103/PhysRevE.50.R647
[50] DOI: 10.1016/S0375-9601(97)00088-1 · Zbl 1043.37504
[51] Sprott J. C., Int. J. Chaos Th. Appl. 5 pp 3–
[52] DOI: 10.1038/35023206
[53] DOI: 10.1109/81.251829 · Zbl 0844.58063
[54] DOI: 10.1109/81.922468 · Zbl 1010.34033
[55] DOI: 10.1142/9789812705303_0009
[56] DOI: 10.1016/S0764-4442(99)80439-X · Zbl 0935.34050
[57] Ueta T., Int. J. Bifurcation and Chaos 10 pp 1917–
[58] Vanĕcek A., Control Systems: From Linear Analysis to Synthesis of Chaos (1996)
[59] DOI: 10.1142/S0218127499000985 · Zbl 0964.93039
[60] Wang X., Int. J. Bifurcation and Chaos 10 pp 549–
[61] DOI: 10.1063/1.1322358 · Zbl 0967.93045
[62] DOI: 10.1007/978-1-4757-4067-7
[63] Yang X. S., Far East J. Dyn. Syst. 4 pp 27–
[64] Yu Y., Chinese Phys. 11 pp 1249–
[65] Yu Y., Chaos Solit. Fract. 15 pp 897–
[66] Zhang F., Nonlinearity 10 pp 1289–
[67] Zhou Z., J. Xianning Teachers College 22 pp 19–
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