In the paper by Liu et al. (2009, “A Novel Three-Dimensional Autonomous Chaos System,” Chaos Solitons Fractals, 39(4), pp. 1950–1958), the three-dimensional (3D) chaotic system is investigated, and some of its dynamics according to theoretical and numerical analyses only for the parameters (a, e, b, k, c, m) = (1, 1, 2.5, 4, 5, 4) are discussed. In 2013, the same chaotic system by Li et al. (2013, “Analysis of a Novel Three-Dimensional Chaotic System,” Optik, 124(13), pp. 1516–1522) was mainly discussed by numerical simulation. In this article, by some deeper investigations, combining some numerical simulations, we formulate some new results of the system. First, after some problems in the first paper are pointed out, we display that its parameters e, k, and m may be kicked out by some homothetic transformations. Second, some of its rich nonlinear dynamics hiding and not found previously, such as the stability and Hopf bifurcation of its isolated equilibria, the behavior of its nonisolated equilibria, the existence of singular orbits (including singularly degenerate heteroclinic cycle, homoclinic and heteroclinic orbits, etc.), and its dynamics at infinity, etc., are clearly formulated. What's more interesting, we find, this system has two different kinds of nonisolated equilibria Ex and Ez, and new chaotic attractors can be bifurcated out with the disappearance of Ex, but this system has no such properties at Ez. In the meantime, several problems about the existence of singular orbits deserving further investigations are presented. Our results better complement and improve the known ones.
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November 2015
Research-Article
New Results to a Three-Dimensional Chaotic System With Two Different Kinds of Nonisolated Equilibria
Xianyi Li
Xianyi Li
1
College of Mathematical Science,
e-mail: mathxyli@yzu.edu.cn
Yangzhou University
,Yangzhou 225002
, China
e-mail: mathxyli@yzu.edu.cn
1Corresponding author.
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Haijun Wang
Xianyi Li
College of Mathematical Science,
e-mail: mathxyli@yzu.edu.cn
Yangzhou University
,Yangzhou 225002
, China
e-mail: mathxyli@yzu.edu.cn
1Corresponding author.
Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received September 11, 2014; final manuscript received January 23, 2015; published online June 9, 2015. Assoc. Editor: Sotirios Natsiavas.
J. Comput. Nonlinear Dynam. Nov 2015, 10(6): 061021 (14 pages)
Published Online: November 1, 2015
Article history
Received:
September 11, 2014
Revision Received:
January 23, 2015
Online:
June 9, 2015
Citation
Wang, H., and Li, X. (November 1, 2015). "New Results to a Three-Dimensional Chaotic System With Two Different Kinds of Nonisolated Equilibria." ASME. J. Comput. Nonlinear Dynam. November 2015; 10(6): 061021. https://doi.org/10.1115/1.4030028
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