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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 |
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. |
[2] |
O. E. Rössler, An equation for continuous chaos, Phys. Lett. A., 57 (1976), 397-398.
doi: 10.1016/0375-9601(76)90101-8. |
[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. |
[4] |
G. R. Chen and J. H. Lü, "Dynamical Analysis, Control and Synchronizations for the Family of Lorenz System," Science Press, Beijing, 2003. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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). |
[11] |
G. R. Chen and T. Ueta, Yet another chaotic attractor, Int. J. Bifurcation and Chaos, 9 (1999), 1465-1466. |
[12] |
J. H. Lü and G. R. Chen, A new chaotic attractor coined, Int. J. Bifurcation and Chaos, 3 (2002), 659-661. |
[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. |
[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. |
[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. |
[16] |
A. Vaně ček and S. Čelikovský, "Control Systems: From Linear Analysis to Synthesis of Chaos," Prentice-Hall, London, 1996. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
J. C. Sprott, Some simple chaotic flows, Phys. Rev. E., 50 (1994), R647-R650.
doi: 10.1103/PhysRevE.50.R647. |
[23] |
C. P. Silva, Sil'nikov theorem-a tutorial, IEEE Trans. Circ. Syst.-I, 40 (1993), 675-682. |
show all references
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. |
[2] |
O. E. Rössler, An equation for continuous chaos, Phys. Lett. A., 57 (1976), 397-398.
doi: 10.1016/0375-9601(76)90101-8. |
[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. |
[4] |
G. R. Chen and J. H. Lü, "Dynamical Analysis, Control and Synchronizations for the Family of Lorenz System," Science Press, Beijing, 2003. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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). |
[11] |
G. R. Chen and T. Ueta, Yet another chaotic attractor, Int. J. Bifurcation and Chaos, 9 (1999), 1465-1466. |
[12] |
J. H. Lü and G. R. Chen, A new chaotic attractor coined, Int. J. Bifurcation and Chaos, 3 (2002), 659-661. |
[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. |
[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. |
[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. |
[16] |
A. Vaně ček and S. Čelikovský, "Control Systems: From Linear Analysis to Synthesis of Chaos," Prentice-Hall, London, 1996. |
[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. |
[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. |
[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. |
[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. |
[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. |
[22] |
J. C. Sprott, Some simple chaotic flows, Phys. Rev. E., 50 (1994), R647-R650.
doi: 10.1103/PhysRevE.50.R647. |
[23] |
C. P. Silva, Sil'nikov theorem-a tutorial, IEEE Trans. Circ. Syst.-I, 40 (1993), 675-682. |
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