- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 417678, 10 pages
Multiple Nonlinear Oscillations in a -Symmetrical Coupled System of Identical Cells with Delays
College of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan 410076, China
Received 31 January 2013; Accepted 8 May 2013
Academic Editor: Zhiming Guo
Copyright © 2013 Haijun Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
- M. Golubitsky, I. Stewart, and D. G. Schaeffer, Singularities and Groups in Bifurcation Theory, Vol. II, Springer, New York, NY, USA, 1988.
- A. P. S. Dias and A. Rodrigues, “Hopf bifurcation with -symmetry,” Nonlinearity, vol. 22, no. 3, pp. 627–666, 2009.
- J. Wu, “Symmetric functional-differential equations and neural networks with memory,” Transactions of the American Mathematical Society, vol. 350, no. 12, pp. 4799–4838, 1998.
- L. Huang and J. Wu, “Nonlinear waves in networks of neurons with delayed feedback: pattern formation and continuation,” SIAM Journal on Mathematical Analysis, vol. 34, no. 4, pp. 836–860, 2003.
- S. Guo and L. Huang, “Hopf bifurcating periodic orbits in a ring of neurons with delays,” Physica D, vol. 183, no. 1-2, pp. 19–44, 2003.
- S. Guo and L. Huang, “Stability of nonlinear waves in a ring of neurons with delays,” Journal of Differential Equations, vol. 236, no. 2, pp. 343–374, 2007.
- S. A. Campbell, Y. Yuan, and S. D. Bungay, “Equivariant Hopf bifurcation in a ring of identical cells with delayed coupling,” Nonlinearity, vol. 18, no. 6, pp. 2827–2846, 2005.
- M. Peng, “Bifurcation and stability analysis of nonlinear waves in symmetric delay differential systems,” Journal of Differential Equations, vol. 232, no. 2, pp. 521–543, 2007.
- C. Zhang, Y. Zhang, and B. Zheng, “A model in a coupled system of simple neural oscillators with delays,” Journal of Computational and Applied Mathematics, vol. 229, no. 1, pp. 264–273, 2009.
- Y. Song, M. O. Tadé, and T. Zhang, “Bifurcation analysis and spatio-temporal patterns of nonlinear oscillations in a delayed neural network with unidirectional coupling,” Nonlinearity, vol. 22, no. 5, pp. 975–1001, 2009.
- Y. Jiang and S. Guo, “Linear stability and Hopf bifurcation in a delayed two-coupled oscillator with excitatory-to-inhibitory connection,” Nonlinear Analysis: Real World Applications, vol. 11, no. 3, pp. 2001–2015, 2010.
- V. Kolmanovskiĭ and A. Myshkis, Applied Theory of Functional-Differential Equations, Kluwer Academic, Dordrecht, The Netherlands, 1992.
- W. Krawcewicz, P. Vivi, and J. Wu, “Computation formulae of an equivariant degree with applications to symmetric bifurcations,” Nonlinear Studies, vol. 4, no. 1, pp. 89–119, 1997.
- W. Krawcewicz and J. Wu, “Theory and applications of Hopf bifurcations in symmetric functional-differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 35, no. 7, pp. 845–870, 1999.
- C. M. Marcus and R. M. Westervelt, “Stability of analog neural networks with delay,” Physical Review A, vol. 39, no. 1, pp. 347–359, 1989.
- J. J. Hopfield, “Neurons with graded response have collective computational properties like twostate neurons,” Proceedings of the National Academy of Sciences of the United States of America, vol. 81, pp. 3088–3092, 1984.
- J. C. Eccles, M. Ito, and J. Szenfagothai, The Cerebellum as Neuronal Machine, Springer, New York, NY, USA, 1967.
- M. W. Hirsch, “Convergent activation dynamics in continuous time networks,” Neural Networks, vol. 2, no. 5, pp. 331–349, 1989.
- E. R. Kandel, J. H. Schwartz, and T. M. Jessell, Principles of Neural Science, McGraw-Hill, New York, NY, USA, 2000.
- G. Dangelmayr, W. Güttinger, and M. Wegelin, “Hopf bifurcation with -symmetry,” Zeitschrift für Angewandte Mathematik und Physik, vol. 44, no. 4, pp. 595–638, 1993.
- M. Wegelin, J. Oppenländer, J. Tomes, W. Güttinger, and G. Dangelmayr, “Synchronized patterns in hierarchical networks of neuronal oscillators with symmetry,” Physica D, vol. 121, no. 1-2, pp. 213–232, 1998.
- O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H.-O. Walther, Delay Equations, Functional-, Complex-, and Nonlinear Analysis, Springer, New York, NY, USA, 1995.
- B. W. Levinger, “A folk theorem in functional differential equations,” Journal of Differential Equations, vol. 4, pp. 612–619, 1968.
- T. Faria and L. T. Magalháes, “Normal forms for retarded functional-differential equations with parameters and applications to Hopf bifurcation,” Journal of Differential Equations, vol. 122, no. 2, pp. 181–200, 1995.
- T. Faria and L. T. Magalháes, “Normal forms for retarded functional-differential equations and applications to Bogdanov-Takens singularity,” Journal of Differential Equations, vol. 122, no. 2, pp. 201–224, 1995.