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CHEN'S ATTRACTOR EXISTS

TIANSHOU ZHOU

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China

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YUN TANG

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China

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, and

GUANRONG CHEN

Department of Electronic Engineering, City University of Hong Kong, P. R. China

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https://doi.org/10.1142/S0218127404011296Cited by:123 (Source: Crossref)

Abstract

By applying the undetermined coefficient method, this paper finds homoclinic and heteroclinic orbits in the Chen system. It analytically demonstrates that the Chen system has one heteroclinic orbit of Ši'lnikov type that connects two nontrivial singular points. The Ši'lnikov criterion guarantees that the Chen system has Smale horseshoes and the horseshoe chaos. In addition, there also exists one homoclinic orbit joined to the origin. The uniform convergence of the series expansions of these two types of orbits are proved in this paper. It is shown that the heteroclinic and homoclinic orbits together determine the geometric structure of Chen's attractor.

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References

S. Čelikovský and A. Vaněček, Kybernetika 30, 403 (1994). Web of Science, Google Scholar
S. Čelikovský and G. Chen, Int. J. Bifurcation and Chaos 12, 1789 (2002). Link, Web of Science, Google Scholar
G. Chen and T. Ueta, Int. J. Bifurcation and Chaos 9, 1465 (1999). Link, Web of Science, Google Scholar
J. Kennedy, S. Kocak and J. A. Yorke, Amer. Math. Monthly 108, 411 (2001). Crossref, Web of Science, Google Scholar
T. Y. Li and J. A. Yorke, Amer. Math. Monthly 82, 985 (1975). Crossref, Web of Science, Google Scholar
E. N. Lorenz, J. Atmos. Sci. 20, 130 (1963). Crossref, Web of Science, Google Scholar
J. Lü and G. Chen, Int. J. Bifurcation and Chaos 12, 659 (2002). Link, Web of Science, Google Scholar
L. P. Shi'lnikov, Sov. Math. Docklady 6, 163 (1965). Google Scholar
L. P. Shi'lnikov, Math. U.S.S.R.–Shornik 10, 91 (1970). Crossref, Web of Science, Google Scholar
C. P. Silva, IEEE Trans. Circuits Syst.-I 40, 675 (1993). Crossref, Web of Science, Google Scholar
C. Sparrow , The Lorenz Equations: Bifurcation, Chaos, and Strange Attractor ( Springer-Verlag , NY , 1982 ) . Crossref, Google Scholar
W. Tucker, C. R. Acad. Paris Ser. I Math. 328, 1197 (1999). Crossref, Web of Science, Google Scholar
T. Ueta and G. Chen, Int. J. Bifurcation and Chaos 10, 1917 (2000). Link, Web of Science, Google Scholar
A. Vanĕcek and S. Celikovský , Control Systems: From Linear Analysis to Synthesis of Chaos ( Prentice-Hall , London , 1996 ) . Google Scholar
S. Wiggins , Global Bifurcation and Chaos ( Springer-Verlag , NY , 1988 ) . Crossref, Google Scholar
T. S. Zhou, G. Chen and Y. Tang, Int. J. Bifurcation and Chaos 13, 2561 (2003). Link, Web of Science, Google Scholar