# American Institute of Mathematical Sciences

August  2012, 32(8): 2913-2935. doi: 10.3934/dcds.2012.32.2913

## 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

Received  June 2011 Revised  August 2011 Published  March 2012

This paper discusses feed-forward chains near points of synchrony-breaking Hopf bifurcation. We show that at synchrony-breaking bifurcations the center manifold inherits a feed-forward structure and use this structure to provide a simplified proof of the theorem of Elmhirst and Golubitsky that there is a branch of periodic solutions in such bifurcations whose amplitudes grow at the rate of $\lambda^{\frac{1}{6}}$. We also use this center manifold structure to provide a method for classifying the bifurcation diagrams of the forced feed-forward chain where the amplitudes of the periodic responses are plotted as a function of the forcing frequency. The bifurcation diagrams depend on the amplitude of the forcing, the deviation of the system from Hopf bifurcation, and the ratio $\gamma$ of the imaginary part of the cubic term in the normal form of Hopf bifurcation to the real part. These calculations generalize the results of Zhang on the forcing of systems near Hopf bifurcations to three-cell feed-forward chains.
Citation: Martin Golubitsky, Claire Postlethwaite. Feed-forward networks, center manifolds, and forcing. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2913-2935. doi: 10.3934/dcds.2012.32.2913
##### 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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