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.
- W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath and Co., Boston, Mass., 1965. MR 0190463
J. S. Griffith, Mathematics of cellular control processes, II: Positive feedback to one gene, J. Theor. Biol. 20, 209–216 (1968)
- Morris W. Hirsch, Systems of differential equations that are competitive or cooperative. II. Convergence almost everywhere, SIAM J. Math. Anal. 16 (1985), no. 3, 423–439. MR 783970, DOI https://doi.org/10.1137/0516030
- J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag, New York, 1976. With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale; Applied Mathematical Sciences, Vol. 19. MR 0494309
- James F. Selgrade, Mathematical analysis of a cellular control process with positive feedback, SIAM J. Appl. Math. 36 (1979), no. 2, 219–229. MR 524498, DOI https://doi.org/10.1137/0136019
- James F. Selgrade, Asymptotic behavior of solutions to single loop positive feedback systems, J. Differential Equations 38 (1980), no. 1, 80–103. MR 592869, DOI https://doi.org/10.1016/0022-0396%2880%2990026-1
J. J. Tyson and H. G. Othmer, The dynamics of feedback control circuits in biochemical pathways, Progr. Theor. Biol. 5, 1–62 (1978)
W. A. Coppel, Stability and asymptotic behavior of differential equations, D. C. Heath, Boston, 1965
J. S. Griffith, Mathematics of cellular control processes, II: Positive feedback to one gene, J. Theor. Biol. 20, 209–216 (1968)
M. W. Hirsch, Systems of differential equations that are competitive or cooperative, II. Convergence almost everywhere, to appear
J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Springer-Verlag, New York, 1976
J. Selgrade, Mathematical analysis of a cellular control process with positive feedback, SIAM J. Appl. Math. 36, 219–229 (1979)
---, Asymptotic behavior of solutions to single loop positive feedback systems, J. Diff. Eq. 38, 80–103 (1980)
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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Article copyright:
© Copyright 1982
American Mathematical Society
