Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

A Hopf bifurcation in single-loop positive-feedback systems


Author: James F. Selgrade
Journal: Quart. Appl. Math. 40 (1982), 347-351
MSC: Primary 58F14; Secondary 34C15, 34C25, 92A09
DOI: https://doi.org/10.1090/qam/678206
MathSciNet review: 678206
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Abstract: This paper gives sufficient conditions for a Hopf bifurcation in a five-dimensional system of ordinary differential equations which provides a model for positive feedback in biochemical control circuits. These conditions only depend on the feedback function and its first and second derivative. The conditions are used to exhibit Hopf bifurcations for the Griffith equations and the Tyson-Othmer equations.


References [Enhancements On Off] (What's this?)

  • [1] W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath, Boston, 1965 MR 0190463
  • [2] J. S. Griffith, Mathematics of cellular control processes, II: Positive feedback to one gene, J. Theor. Biol. 20, 209-216 (1968)
  • [3] M. W. Hirsch, Systems of differential equations that are competitive or cooperative, II. Convergence almost everywhere, to appear MR 783970
  • [4] J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag, New York, 1976 MR 0494309
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  • [6] -, Asymptotic behavior of solutions to single loop positive feedback systems, J. Diff. Eq. 38, 80-103 (1980) MR 592869
  • [7] J. J. Tyson and H. G. Othmer, The dynamics of feedback control circuits in biochemical pathways, Progr. Theor. Biol. 5, 1-62 (1978)

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DOI: https://doi.org/10.1090/qam/678206
Article copyright: © Copyright 1982 American Mathematical Society

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