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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

 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.
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
##### References:
 [1] E. N. Lorenz, Deterministic non periodic flow, J. Atmos. Sci., 20 (1963), 130-141. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar [2] O. E. Rössler, An equation for continuous chaos, Phys. Lett. A., 57 (1976), 397-398. doi: 10.1016/0375-9601(76)90101-8.  Google Scholar [3] L. O. Chua, M. Komuro and T. Matsumoto, The double scroll family. Part I: Rigorous proof of chaos, IEEE Trans. Circ. Syst., 33 (1986), 1072-1096. doi: 10.1109/TCS.1986.1085869.  Google Scholar [4] G. R. Chen and J. H. Lü, "Dynamical Analysis, Control and Synchronizations for the Family of Lorenz System," Science Press, Beijing, 2003. Google Scholar [5] A. Jorba and J. C. Tatjer, A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 537-567. doi: 10.3934/dcdsb.2008.10.537.  Google Scholar [6] A. Neishtadtand, C. Sim, D. Treschev and A. Vasiliev, Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 621-650. doi: 10.3934/dcdsb.2008.10.621.  Google Scholar [7] H. W. Broer, C. Sim and R. Vitolo, Chaos and quasi-periodicity in diffeomorphisms of the solid torus, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 871-905. doi: 10.3934/dcdsb.2010.14.871.  Google Scholar [8] G. R. Chen and D. Lai, Feedback control of Lyapunov exponents from discrete-time dynamical systems, Int. J. Bifurcation and Chaos, 6 (1996), 1341-1349. doi: 10.1142/S021812749600076X.  Google Scholar [9] G. R. Chen and D. Lai, Feedback anticontrol discrete of chaos, Int. J. Bifurcation and Chaos, 8 (1998), 1585-1590. doi: 10.1142/S0218127498001236.  Google Scholar [10] G. R. Chen, Chaotification via feedback: the discrete case, 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] G. R. Chen and T. Ueta, Yet another chaotic attractor, Int. J. Bifurcation and Chaos, 9 (1999), 1465-1466.  Google Scholar [12] J. H. Lü and G. R. Chen, A new chaotic attractor coined, Int. J. Bifurcation and Chaos, 3 (2002), 659-661.  Google Scholar [13] J. H. Lü, G. R. Chen, D. Chen and S.Celikovsky, Bridge the gap between the Lorenz system and the Chen system, Int. J. Bifurcation and Chaos, 12 (2002), 2917-2926. doi: 10.1142/S021812740200631X.  Google Scholar [14] W. B. Liu and G. R. Chen, A new chaotic system and its generation, Int. J. Bifurcation and Chaos, 13 (2003), 261-267. doi: 10.1142/S0218127403006509.  Google Scholar [15] W. B. Liu and G. R. Chen, Can a three-dimensional smooth autonomous quadratic chaotic system generate single four-scroll attractors, Int. J. Bifurcation and Chaos, 14 (2004), 1395-1403. doi: 10.1142/S0218127404009880.  Google Scholar [16] A. Vaně ček and S. Čelikovský, "Control Systems: From Linear Analysis to Synthesis of Chaos," Prentice-Hall, London, 1996. Google Scholar [17] S. Čelikovský and G. R. Chen, Hyperbolic-type generalized Lorenz system and its canonical form, In "Proceedings of the 15th Triennial World Congress of IFAC" (CD Rom), Barcelona, Spain, 2002. Google Scholar [18] S. Čelikovský and G. R. Chen, On a generalized Lorenz canonical form of chaotic system, Int. J. Bifurcation and Chaos, 12 (2002), 1789-1812. doi: 10.1142/S0218127402005467.  Google Scholar [19] S. Čelikovský and G. Chen, On the generalized Lorenz canonical form, Chaos, Solitons and Fractal, 26 (2005), 1271-1276. doi: 10.1016/j.chaos.2005.02.040.  Google Scholar [20] A. L. Shil'nikov, On bifurcations of the Lorenz attractor in the Shimizu-Morioka model, Physica D, 62 (1993), 338-346. doi: 10.1016/0167-2789(93)90292-9.  Google Scholar [21] Q. G. Yang, G. R. Chen and T. S. Zhou, A unified Lorenz-type system and its canonical form, Int. J. Bifurcation and Chaos, 14 (2006), 2855-2871. doi: 10.1142/S0218127406016501.  Google Scholar [22] J. C. Sprott, Some simple chaotic flows, Phys. Rev. E., 50 (1994), R647-R650. doi: 10.1103/PhysRevE.50.R647.  Google Scholar [23] C. P. Silva, Sil'nikov theorem-a tutorial, IEEE Trans. Circ. Syst.-I, 40 (1993), 675-682. Google Scholar

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##### References:
 [1] E. N. Lorenz, Deterministic non periodic flow, J. Atmos. Sci., 20 (1963), 130-141. doi: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.  Google Scholar [2] O. E. Rössler, An equation for continuous chaos, Phys. Lett. A., 57 (1976), 397-398. doi: 10.1016/0375-9601(76)90101-8.  Google Scholar [3] L. O. Chua, M. Komuro and T. Matsumoto, The double scroll family. Part I: Rigorous proof of chaos, IEEE Trans. Circ. Syst., 33 (1986), 1072-1096. doi: 10.1109/TCS.1986.1085869.  Google Scholar [4] G. R. Chen and J. H. Lü, "Dynamical Analysis, Control and Synchronizations for the Family of Lorenz System," Science Press, Beijing, 2003. Google Scholar [5] A. Jorba and J. C. Tatjer, A mechanism for the fractalization of invariant curves in quasi-periodically forced 1-D maps, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 537-567. doi: 10.3934/dcdsb.2008.10.537.  Google Scholar [6] A. Neishtadtand, C. Sim, D. Treschev and A. Vasiliev, Periodic orbits and stability islands in chaotic seas created by separatrix crossings in slow-fast systems, Discrete Contin. Dyn. Syst. Ser. B, 10 (2008), 621-650. doi: 10.3934/dcdsb.2008.10.621.  Google Scholar [7] H. W. Broer, C. Sim and R. Vitolo, Chaos and quasi-periodicity in diffeomorphisms of the solid torus, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 871-905. doi: 10.3934/dcdsb.2010.14.871.  Google Scholar [8] G. R. Chen and D. Lai, Feedback control of Lyapunov exponents from discrete-time dynamical systems, Int. J. Bifurcation and Chaos, 6 (1996), 1341-1349. doi: 10.1142/S021812749600076X.  Google Scholar [9] G. R. Chen and D. Lai, Feedback anticontrol discrete of chaos, Int. J. Bifurcation and Chaos, 8 (1998), 1585-1590. doi: 10.1142/S0218127498001236.  Google Scholar [10] G. R. Chen, Chaotification via feedback: the discrete case, 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] G. R. Chen and T. Ueta, Yet another chaotic attractor, Int. J. Bifurcation and Chaos, 9 (1999), 1465-1466.  Google Scholar [12] J. H. Lü and G. R. Chen, A new chaotic attractor coined, Int. J. Bifurcation and Chaos, 3 (2002), 659-661.  Google Scholar [13] J. H. Lü, G. R. Chen, D. Chen and S.Celikovsky, Bridge the gap between the Lorenz system and the Chen system, Int. J. Bifurcation and Chaos, 12 (2002), 2917-2926. doi: 10.1142/S021812740200631X.  Google Scholar [14] W. B. Liu and G. R. Chen, A new chaotic system and its generation, Int. J. Bifurcation and Chaos, 13 (2003), 261-267. doi: 10.1142/S0218127403006509.  Google Scholar [15] W. B. Liu and G. R. Chen, Can a three-dimensional smooth autonomous quadratic chaotic system generate single four-scroll attractors, Int. J. Bifurcation and Chaos, 14 (2004), 1395-1403. doi: 10.1142/S0218127404009880.  Google Scholar [16] A. Vaně ček and S. Čelikovský, "Control Systems: From Linear Analysis to Synthesis of Chaos," Prentice-Hall, London, 1996. Google Scholar [17] S. Čelikovský and G. R. Chen, Hyperbolic-type generalized Lorenz system and its canonical form, In "Proceedings of the 15th Triennial World Congress of IFAC" (CD Rom), Barcelona, Spain, 2002. Google Scholar [18] S. Čelikovský and G. R. Chen, On a generalized Lorenz canonical form of chaotic system, Int. J. Bifurcation and Chaos, 12 (2002), 1789-1812. doi: 10.1142/S0218127402005467.  Google Scholar [19] S. Čelikovský and G. Chen, On the generalized Lorenz canonical form, Chaos, Solitons and Fractal, 26 (2005), 1271-1276. doi: 10.1016/j.chaos.2005.02.040.  Google Scholar [20] A. L. Shil'nikov, On bifurcations of the Lorenz attractor in the Shimizu-Morioka model, Physica D, 62 (1993), 338-346. doi: 10.1016/0167-2789(93)90292-9.  Google Scholar [21] Q. G. Yang, G. R. Chen and T. S. Zhou, A unified Lorenz-type system and its canonical form, Int. J. Bifurcation and Chaos, 14 (2006), 2855-2871. doi: 10.1142/S0218127406016501.  Google Scholar [22] J. C. Sprott, Some simple chaotic flows, Phys. Rev. E., 50 (1994), R647-R650. doi: 10.1103/PhysRevE.50.R647.  Google Scholar [23] C. P. Silva, Sil'nikov theorem-a tutorial, IEEE Trans. Circ. Syst.-I, 40 (1993), 675-682. Google Scholar
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