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Feed-forward networks, center manifolds, and forcing
1. | Mathematical Biosciences Institute, The Ohio State University, Columbus, OH 43215, United States |
2. | Department of Mathematics, University of Auckland, Auckland 1142, New Zealand |
References:
[1] |
N. N. Bogoliubov and Y. A. Mitropolsky, "Asymptotic Methods in the Theory of Non-linear Oscillations," Translated from the second revised Russian edition, International Monographs on Advanced Mathematics and Physics, Hindustan Publ. Corp., Delhi, Gordon and Breach Science Publishers, New York, 1961. |
[2] |
J. Carr, "Applications of Centre Manifold Theory," Applied Mathematical Sciences, 35, Springer-Verlag, New York-Berlin, 1981. |
[3] |
T. Elmhirst and M. Golubitsky, Nilpotent Hopf bifurcations in coupled cell systems, SIAM J. Appl. Dynam. Sys., 5 (2006), 205-251.
doi: 10.1137/050635559. |
[4] |
J.-M. Gambaudo, Perturbation of a Hopf bifurcation by an external time-periodic forcing, J. Diff. Eqns., 57 (1985), 172-199.
doi: 10.1016/0022-0396(85)90076-2. |
[5] |
M. Golubitsky, M. Nicol and I. Stewart, Some curious phenomena in coupled cell networks, J. Nonlinear Sci., 14 (2004), 207-236.
doi: 10.1007/s00332-003-0593-6. |
[6] |
M. Golubitsky, C. Postlethwaite, L.-J. Shiau and Y. Zhang, The feed-forward chain as a filter amplifier motif, in "Coherent Behavior in Neuronal Networks," (eds. K. Josic, M. Matias, R. Romo and J. Rubin), Springer Ser. Comput. Neurosci., 3, Springer, New York, (2009), 95-120. |
[7] |
M. Golubitsky and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory," Vol. I, Appl. Math. Sci., 51, Springer-Verlag, New York, 1985. |
[8] |
M. Golubitsky, I. N. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory," Vol. II, Appl. Math. Sci., 69, Springer-Verlag, New York, 1988. |
[9] |
N. J. McCullen, T. Mullin and M. Golubitsky, Sensitive signal detection using a feed-forward oscillator network, Phys. Rev. Lett., 98 (2007), 254101.
doi: 10.1103/PhysRevLett.98.254101. |
[10] |
Y. Zhang, "Periodic Forcing of a System Near a Hopf Bifurcation Point," Ph.D Thesis, Department of Mathematics, Ohio State University, 2010. |
[11] |
Y. Zhang and M. Golubitsky, Periodically forced Hopf bifurcation, SIAM J. Appl. Dynam. Sys., to appear. |
show all references
References:
[1] |
N. N. Bogoliubov and Y. A. Mitropolsky, "Asymptotic Methods in the Theory of Non-linear Oscillations," Translated from the second revised Russian edition, International Monographs on Advanced Mathematics and Physics, Hindustan Publ. Corp., Delhi, Gordon and Breach Science Publishers, New York, 1961. |
[2] |
J. Carr, "Applications of Centre Manifold Theory," Applied Mathematical Sciences, 35, Springer-Verlag, New York-Berlin, 1981. |
[3] |
T. Elmhirst and M. Golubitsky, Nilpotent Hopf bifurcations in coupled cell systems, SIAM J. Appl. Dynam. Sys., 5 (2006), 205-251.
doi: 10.1137/050635559. |
[4] |
J.-M. Gambaudo, Perturbation of a Hopf bifurcation by an external time-periodic forcing, J. Diff. Eqns., 57 (1985), 172-199.
doi: 10.1016/0022-0396(85)90076-2. |
[5] |
M. Golubitsky, M. Nicol and I. Stewart, Some curious phenomena in coupled cell networks, J. Nonlinear Sci., 14 (2004), 207-236.
doi: 10.1007/s00332-003-0593-6. |
[6] |
M. Golubitsky, C. Postlethwaite, L.-J. Shiau and Y. Zhang, The feed-forward chain as a filter amplifier motif, in "Coherent Behavior in Neuronal Networks," (eds. K. Josic, M. Matias, R. Romo and J. Rubin), Springer Ser. Comput. Neurosci., 3, Springer, New York, (2009), 95-120. |
[7] |
M. Golubitsky and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory," Vol. I, Appl. Math. Sci., 51, Springer-Verlag, New York, 1985. |
[8] |
M. Golubitsky, I. N. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory," Vol. II, Appl. Math. Sci., 69, Springer-Verlag, New York, 1988. |
[9] |
N. J. McCullen, T. Mullin and M. Golubitsky, Sensitive signal detection using a feed-forward oscillator network, Phys. Rev. Lett., 98 (2007), 254101.
doi: 10.1103/PhysRevLett.98.254101. |
[10] |
Y. Zhang, "Periodic Forcing of a System Near a Hopf Bifurcation Point," Ph.D Thesis, Department of Mathematics, Ohio State University, 2010. |
[11] |
Y. Zhang and M. Golubitsky, Periodically forced Hopf bifurcation, SIAM J. Appl. Dynam. Sys., to appear. |
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