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MULTIPLE BIFURCATION ANALYSIS IN A NEURAL NETWORK MODEL WITH DELAYS

YUAN YUAN

Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John's NL A1C 5S7, Canada

Research supported by the National Sciences and Engineering Research Council of Canada.

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and

JUNJIE WEI

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, P. R. China

Research supported by the National Natural Science Foundation of China.

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https://doi.org/10.1142/S0218127406016537Cited by:17 (Source: Crossref)

Abstract

A synchronized neural network model with delays is considered. The bifurcations arising from the zero root of the corresponding characteristic equation have been studied by employing the center manifold theorem, normal form method and bifurcation theory. It is shown that the system may exhibit transcritical/pitchfork bifurcation, or Bogdanov–Takens bifurcation. Some numerical simulation examples are given to justify the theoretical results.

Keywords:

References

Campbell, S. A., Ncube, I. & Wu, I. [2005a] "Multistability and stable asynchronous periodic oscillations in a multiple-delayed neural system," preprint . Google Scholar
S. A. Campbell, Y. Yuan and S. Bungay, Nonlinearity 18, 2827 (2005), DOI: 10.1088/0951-7715/18/6/022. Crossref, Web of Science, Google Scholar
J. Carrr , Applications of Centre Manifold Theory ( Springer-Verlag , NY , 1981 ) . Crossref, Google Scholar
Y. Chen and J. Wu, Diff. Integr. Eqs. 14, 1181 (2001). Web of Science, Google Scholar
J. Dieudonn , Foundations of Modern Analysis ( Academic Press , NY , 1969 ) . Google Scholar
T. Faria and L. T. Magalhães, J. Diff. Eqs. 122, 201 (1995), DOI: 10.1006/jdeq.1995.1145. Crossref, Web of Science, Google Scholar
T. Faria, J. Diff. Eqs. 168, 129 (2000), DOI: 10.1006/jdeq.2000.3881. Crossref, Web of Science, Google Scholar
T. Faria, J. Dyn. Contin. Discr. Impuls. Syst., Ser. A 10, 357 (2003). Google Scholar
F. Giannkopoulos and A. Zapp, Physica D 159, 215 (2001), DOI: 10.1016/S0167-2789(01)00337-2. Crossref, Web of Science, Google Scholar
J. Guckenheimer and P. Holmes , Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields ( Springer-Verlag , 1997 ) . Google Scholar
S. Guo and L. Huang, Physica D 183, 19 (2003), DOI: 10.1016/S0167-2789(03)00159-3. Crossref, Web of Science, Google Scholar
P. Hirschberg and E. Knobloch, Quart. Appl. Math. XLIX, 281 (1991). Google Scholar
C. M. Marcus and R. M. Westervelt, Neural Information Processing Systems, Denver, CO, 1987, ed. D. S. Anderson (AIP, 1988) pp. 524–533. Google Scholar
I. Ncube, S. A. Campbell and J. Wu, Dynamical Systems and Their Applications to Biology, Fields Institute Communications 36, eds. S. Ruan, G. S. K. Wolkowicz and J. Wu (AMS, 2003) pp. 179–193. Google Scholar
B. F. Redmond, V. G. LeBlanc and A. Longtin, Physica D 166, 131 (2002), DOI: 10.1016/S0167-2789(02)00423-2. Crossref, Web of Science, Google Scholar
S. Ruan and J. Wei, J. Dyn. Contin. Discr. Impuls. Syst., Ser A 10, 863 (2003). Google Scholar
L. P. Shayer and S. A. Campbell, SIAM J. Appl. Math. 61, 673 (2000), DOI: 10.1137/S0036139998344015. Crossref, Web of Science, Google Scholar
J. Wei and M. Y. Li, Physica D 198, 106 (2004), DOI: 10.1016/j.physd.2004.08.023. Crossref, Web of Science, Google Scholar
J. Wei and Y. Yuan, J. Math. Anal. Appl. 312, 205 (2005), DOI: 10.1016/j.jmaa.2005.03.049. Crossref, Web of Science, Google Scholar
J. Wu, T. Faria and Y. S. Huang, Math. Comp. Modeling 30, 117 (1999), DOI: 10.1016/S0895-7177(99)00120-X. Crossref, Web of Science, Google Scholar
D. Xiao and S. Ruan, J. Diff. Eqs. 176, 494 (2001), DOI: 10.1006/jdeq.2000.3982. Crossref, Web of Science, Google Scholar
Y. Yuan and S. A. Campbell, J. Dyn. Diff. Eqs. 16, 709 (2004), DOI: 10.1007/s10884-004-6114-y. Crossref, Google Scholar