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A HORSESHOE IN A CELLULAR NEURAL NETWORK OF FOUR-DIMENSIONAL AUTONOMOUS ORDINARY DIFFERENTIAL EQUATIONS

Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan 430074, P. R. China

Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, P. R. China

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QINGDU LI

Department of Electronics and Information Engineering, Huazhong University of Science and Technology, Wuhan 430074, P. R. China

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https://doi.org/10.1142/S0218127407018968Cited by:11 (Source: Crossref)

Abstract

We obtain numerically a horseshoe in a Poincaré map derived from a cellular neural network described by four-dimensional autonomous ordinary differential equations. Contrary to the horseshoe numerically found in the Hodgkin–Huxley model, which showed evidence that the Poincaré map derived from the Hodgkin–Huxley model has just one expanding direction on some invariant subset, the horseshoe obtained in this paper proves that the Poincaré map derived from the neural network have two expanding directions on some invariant subset.

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References

L. O. Chua and T. Roska , Cellular Networks and Visual Computing ( Cambridge University Press , 2002 ) . Crossref, Google Scholar
J. Guckenheimer and A. O. Ricardo, Siam J. Appl. Dyn. Syst. 1, 105 (2002), DOI: 10.1137/S1111111101394040. Crossref, Web of Science, Google Scholar
K. Mischaikow and M. Mrozek, Bull. Amer. Math. Soc. 32, 66 (1995), DOI: 10.1090/S0273-0979-1995-00558-6. Crossref, Web of Science, Google Scholar
K. Mischaikow and M. Mrozek, Math. Comput. 67, 1023 (1998). Crossref, Web of Science, Google Scholar
K. Mischaikow, M. Mrozek and A. Szymczak, J. Diff. Eqs. 169, 17 (2001), DOI: 10.1006/jdeq.2000.3894. Crossref, Web of Science, Google Scholar
C. Robinson , Dynamical Systems: Stability, Symbolic Dynamics and Chaos ( CRC Press, Inc. , 1995 ) . Google Scholar
H. Yan and X.-S. Yang, J. Math. Chem. 38, 107 (2005). Web of Science, Google Scholar
X.-S. Yang and Q. Li, Int. J. Bifurcation and Chaos 14, 1847 (2004). Link, Web of Science, Google Scholar
X.-S. Yang and Y. Tang, Chaos Solit. Fract. 19, 841 (2004), DOI: 10.1016/S0960-0779(03)00202-9. Crossref, Web of Science, Google Scholar
X.-S. Yang and Q. Li, Int. J. Bifurcation and Chaos 15, 2271 (2005). Link, Web of Science, Google Scholar
X.-S. Yang and Q. Li, Int. J. Bifurcation and Chaos 15, 1823 (2005). Link, Web of Science, Google Scholar
X.-S. Yang and Q. Li, Chaos Solit. Fract. 27, 25 (2006), DOI: 10.1016/j.chaos.2005.04.017. Crossref, Web of Science, Google Scholar