In this paper, we generalize all the existing results in the current literature for the upper bound of a general 3-D quadratic continuous-time system. In particular, we find large regions in the bifurcation parameter space of this system where it is bounded.
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References
E. N. Lorenz, “Deterministic nonperiodic flow,” J. Atmos. Sci., 20, 130–141 (1963).
G. Chen and T. Ueta, “Yet another chaotic attractor,” Int. J. Bifur. Chaos, 9, 1465–1466 (1999).
O. E. Rössler, “An equation for continuous chaos,” Phys. Lett. A, 57, 397–398 (1976).
J. H. Lu and G. Chen, “A new chaotic attractor coined,” Int. J. Bifur. Chaos, 12, 659–661 (2002).
J. C. Sprott, “Some simple chaotic flows,” Phys. Rev. E, 50, 647–650 (1994).
G. Chen, Controlling Chaos and Bifurcations in Engineering Systems, CRC Press, Boca Raton, FL (1999).
Q. Yang, G. Chen, and T. Zhou, “A unified Lorenz-type system and its canonical form,” Int. J. Bifur. Chaos, 16, 2855–2871 (2006).
G. Qi, G. Chen, and S. Du, “Analysis of a new chaotic system,” Physica A, 352, 295–308 (2005).
T. Zhou, Y. Tang, and G. Chen, “Complex dynamical behaviors of the chaotic Chen’s system,” Int. J. Bifur. Chaos, 9, 2561–2574 (2003).
T. Zhou, G. Chen, and Q. Yang, “Constructing a new chaotic system based on the Shil’nikov criterion,” Chaos, Solitons Fractals, 19, 985–993 (2004).
T. Zhou and G. Chen, “Classification of chaos in 3-D autonomous quadratic systems. I. Basic framework and methods,” Int. J. Bifur. Chaos, 16, 2459–2479 (2006).
S. Celikovsky and G. Chen, “On a generalized Lorenz canonical form of chaotic systems,” Int. J. Bifur. Chaos, 12, 1789–1812 (2002).
J. H. Lu, G. Chen, D. Cheng, and S. Celikovsky, “Bridge the gap between the Lorenz system and the Chen system,” Int. J. Bifur. Chaos, 12, 2917–2926 (2002).
G. Leonov, A. Bunin, and N. Koksch, “Attractor localization of the Lorenz system,” Z. Angew. Math. Mech., 67, 649–656 (1987).
D. Li, J. A. Lu, X. Wu, and G. Chen, “Estimating the bounds for the Lorenz family of chaotic systems,” Chaos, Solitons Fractals, 23, 529–534 (2005).
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Published in Neliniini Kolyvannya, Vol. 13, No. 4, pp. 515–521, October–December, 2010.
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Elhadj, Z., Sprott, J.C. About the boundedness of 3-D continuous-time quadratic systems. Nonlinear Oscill 13, 550–557 (2011). https://doi.org/10.1007/s11072-011-0130-8
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DOI: https://doi.org/10.1007/s11072-011-0130-8