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September  2011, 16(2): 475-488. doi: 10.3934/dcdsb.2011.16.475

Converting a general 3-D autonomous quadratic system to an extended Lorenz-type system

Cuncai Hua 1, , Guanrong Chen 2, , Qunhong Li 3, and  Juhong Ge 4,

1. 

School of Mathematics, Yunnan Normal University, Kunming 650092, China
2. 

Department of Electronic Engineering, City University of Hong Kong, Hong Kong, China
3. 

College of Mathematics and Information Science, Guangxi University, Nanning 530004, China
4. 

School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China

Received  November 2009 Revised  December 2010 Published  June 2011

A problem of reducing a general three-dimensional (3-D) autonomous quadratic system to a Lorenz-type system is studied. Firstly, under some necessary conditions for preserving the basic qualitative properties of the Lorenz system, the general 3-D autonomous quadratic system is converted to an extended Lorenz-type system (ELTS) which contains a large class of existing chaotic dynamical systems. Secondly, some different canonical forms of the ELTS are obtained with the aid of various nonsingular linear transformations and normalization techniques. Thirdly, the conjugate systems of the ELTS are defined and discussed. Finally, a sufficient condition for the nonexistence of chaos in such ELTS is derived.
Keywords: Three-dimensional autonomous quadratic system, bifurcation analysis., Lorenz-type system.
Mathematics Subject Classification: Primary: 34A34, 34A05; Secondary: 37G3.
Citation: Cuncai Hua, Guanrong Chen, Qunhong Li, Juhong Ge. Converting a general 3-D autonomous quadratic system to an extended Lorenz-type system. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 475-488. doi: 10.3934/dcdsb.2011.16.475
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show all references

References:
[1]

J. Atmos. Sci., 20 (1963), 130-141. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar

[2]

Phys. Lett. A., 57 (1976), 397-398. doi: 10.1016/0375-9601(76)90101-8.  Google Scholar

[3]

IEEE Trans. Circ. Syst., 33 (1986), 1072-1096. doi: 10.1109/TCS.1986.1085869.  Google Scholar

[4]

Science Press, Beijing, 2003. Google Scholar

[5]

Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 537-567. doi: 10.3934/dcdsb.2008.10.537.  Google Scholar

[6]

Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 621-650. doi: 10.3934/dcdsb.2008.10.621.  Google Scholar

[7]

Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 871-905. doi: 10.3934/dcdsb.2010.14.871.  Google Scholar

[8]

Int. J. Bifurcation and Chaos, 6 (1996), 1341-1349. doi: 10.1142/S021812749600076X.  Google Scholar

[9]

Int. J. Bifurcation and Chaos, 8 (1998), 1585-1590. doi: 10.1142/S0218127498001236.  Google Scholar

[10]

in "Chaos and Bifurcation Control: Theorey and Applications, Part I: Chaos Control" (eds. G. Chen, X. Yu and D. Hill), Springer-Verlag, Heidelberg, (1971). Google Scholar

[11]

Int. J. Bifurcation and Chaos, 9 (1999), 1465-1466.  Google Scholar

[12]

Int. J. Bifurcation and Chaos, 3 (2002), 659-661.  Google Scholar

[13]

Int. J. Bifurcation and Chaos, 12 (2002), 2917-2926. doi: 10.1142/S021812740200631X.  Google Scholar

[14]

Int. J. Bifurcation and Chaos, 13 (2003), 261-267. doi: 10.1142/S0218127403006509.  Google Scholar

[15]

Int. J. Bifurcation and Chaos, 14 (2004), 1395-1403. doi: 10.1142/S0218127404009880.  Google Scholar

[16]

Prentice-Hall, London, 1996. Google Scholar

[17]

In "Proceedings of the 15th Triennial World Congress of IFAC" (CD Rom), Barcelona, Spain, 2002. Google Scholar

[18]

Int. J. Bifurcation and Chaos, 12 (2002), 1789-1812. doi: 10.1142/S0218127402005467.  Google Scholar

[19]

Chaos, Solitons and Fractal, 26 (2005), 1271-1276. doi: 10.1016/j.chaos.2005.02.040.  Google Scholar

[20]

Physica D, 62 (1993), 338-346. doi: 10.1016/0167-2789(93)90292-9.  Google Scholar

[21]

Int. J. Bifurcation and Chaos, 14 (2006), 2855-2871. doi: 10.1142/S0218127406016501.  Google Scholar

[22]

Phys. Rev. E., 50 (1994), R647-R650. doi: 10.1103/PhysRevE.50.R647.  Google Scholar

[23]

IEEE Trans. Circ. Syst.-I, 40 (1993), 675-682. Google Scholar

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