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 & Continuous Dynamical Systems - A, 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, (1961).

[2]

J. Carr, "Applications of Centre Manifold Theory,", Applied Mathematical Sciences, 35 (1981).

[3]

T. Elmhirst and M. Golubitsky, Nilpotent Hopf bifurcations in coupled cell systems,, SIAM J. Appl. Dynam. Sys., 5 (2006), 205. 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. 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. 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, 3 (2009), 95.

[7]

M. Golubitsky and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory,", Vol. I, 51 (1985).

[8]

M. Golubitsky, I. N. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory,", Vol. II, 69 (1988).

[9]

N. J. McCullen, T. Mullin and M. Golubitsky, Sensitive signal detection using a feed-forward oscillator network,, Phys. Rev. Lett., 98 (2007). doi: 10.1103/PhysRevLett.98.254101.

[10]

Y. Zhang, "Periodic Forcing of a System Near a Hopf Bifurcation Point,", Ph.D Thesis, (2010).

[11]

Y. Zhang and M. Golubitsky, Periodically forced Hopf bifurcation,, SIAM J. Appl. Dynam. Sys., ().

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, (1961).

[2]

J. Carr, "Applications of Centre Manifold Theory,", Applied Mathematical Sciences, 35 (1981).

[3]

T. Elmhirst and M. Golubitsky, Nilpotent Hopf bifurcations in coupled cell systems,, SIAM J. Appl. Dynam. Sys., 5 (2006), 205. 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. 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. 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, 3 (2009), 95.

[7]

M. Golubitsky and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory,", Vol. I, 51 (1985).

[8]

M. Golubitsky, I. N. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory,", Vol. II, 69 (1988).

[9]

N. J. McCullen, T. Mullin and M. Golubitsky, Sensitive signal detection using a feed-forward oscillator network,, Phys. Rev. Lett., 98 (2007). doi: 10.1103/PhysRevLett.98.254101.

[10]

Y. Zhang, "Periodic Forcing of a System Near a Hopf Bifurcation Point,", Ph.D Thesis, (2010).

[11]

Y. Zhang and M. Golubitsky, Periodically forced Hopf bifurcation,, SIAM J. Appl. Dynam. Sys., ().

[1]

Akinori Awazu. Input-dependent wave propagations in asymmetric cellular automata: Possible behaviors of feed-forward loop in biological reaction network. Mathematical Biosciences & Engineering, 2008, 5 (3) : 419-427. doi: 10.3934/mbe.2008.5.419

[2]

Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71

[3]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997

[4]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045

[5]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805

[6]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152

[7]

Luis Barreira, Claudia Valls. Regularity of center manifolds under nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 55-76. doi: 10.3934/dcds.2011.30.55

[8]

Luis Barreira, Claudia Valls. Reversibility and equivariance in center manifolds of nonautonomous dynamics. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 677-699. doi: 10.3934/dcds.2007.18.677

[9]

Luis Barreira, Claudia Valls. Center manifolds for nonuniform trichotomies and arbitrary growth rates. Communications on Pure & Applied Analysis, 2010, 9 (3) : 643-654. doi: 10.3934/cpaa.2010.9.643

[10]

Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098

[11]

Isaac A. García, Claudia Valls. The three-dimensional center problem for the zero-Hopf singularity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2027-2046. doi: 10.3934/dcds.2016.36.2027

[12]

R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147

[13]

Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197

[14]

Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247

[15]

Redouane Qesmi, Hans-Otto Walther. Center-stable manifolds for differential equations with state-dependent delays. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 1009-1033. doi: 10.3934/dcds.2009.23.1009

[16]

Jun Shen, Kening Lu, Bixiang Wang. Convergence and center manifolds for differential equations driven by colored noise. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4797-4840. doi: 10.3934/dcds.2019196

[17]

Runxia Wang, Haihong Liu, Fang Yan, Xiaohui Wang. Hopf-pitchfork bifurcation analysis in a coupled FHN neurons model with delay. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 523-542. doi: 10.3934/dcdss.2017026

[18]

Fabien Crauste. Global Asymptotic Stability and Hopf Bifurcation for a Blood Cell Production Model. Mathematical Biosciences & Engineering, 2006, 3 (2) : 325-346. doi: 10.3934/mbe.2006.3.325

[19]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735

[20]

Hui Miao, Zhidong Teng, Chengjun Kang. Stability and Hopf bifurcation of an HIV infection model with saturation incidence and two delays. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2365-2387. doi: 10.3934/dcdsb.2017121

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]