A Liapunov function for four-dimensional positive feedback systems
Author:
Ji Fa Jiang
Journal:
Quart. Appl. Math. 52 (1994), 601-614
MSC:
Primary 34C11; Secondary 92C40
DOI:
https://doi.org/10.1090/qam/1306039
MathSciNet review:
MR1306039
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Abstract: In this paper, we use elementary symmetric polynomials in parameters to construct a Liapunov function for four-dimensional positive feedback systems. By applying it, we give a sufficient condition for every positive-time trajectory to converge to an equilibrium.
J. F. Selgrade, Mathematical analysis of a cellular control process with positive feedback, SIAM J. Appl. Math. 36, 219–229 (1979)
J. F. Selgrade, Asymptotic behavior of solutions to single loop positive feedback systems, J. Differential Equations 38, 80–103 (1980)
J. F. Selgrade, A Hopf bifurcation in single-loop positive feedback systems, Quart. Appl. Math. 40, 347–351 (1982)
M. W. Hirsch, Systems of differential equations that are competitive or cooperative. II: Convergence almost everywhere, SIAM J. Math. Anal. 16, 423–439 (1985)
M. W. Hirsch, Systems of differential equations that are competitive or cooperative. V: Convergence in 3-dimensional systems, J. Differential Equations 80, 94–106 (1989)
J. S. Griffith, Mathematics of cellular control processes. II: Positive feedback to one gene, J. Theoret. Biol. 20, 209–216 (1968)
J. J. Tyson and H. G. Othmer, The dynamics of feedback control circuits in biochemical pathways, Progr. Theoret. Biol. 5, 1–62 (1978)
J. P. LaSalle, Stability theory for ordinary differential equations, J. Differential Equations 4, 57–65 (1968)
Jiang Ji-Fa, A Liapunov function for 3-dimensional feedback systems, Proc. Amer. Math. Soc. 114, 1009–1013 (1992)
Jiang Ji-Fa, The asymptotic behavior of a class of second order differential equations with applications to electrical circuit equations, J. Math. Anal. Appl. 149, 26–37 (1990)
Jiang Ji-Fa, On the asymptotic behavior of a class of nonlinear differential equations, Nonlinear Anal. TMA 14, 453–467 (1990)
Jiang Ji-Fa, A note on a global stability theorem of M. W. Hirsch, Proc. Amer. Math. Soc. 112, 803–806 (1991)
J. F. Selgrade, Mathematical analysis of a cellular control process with positive feedback, SIAM J. Appl. Math. 36, 219–229 (1979)
J. F. Selgrade, Asymptotic behavior of solutions to single loop positive feedback systems, J. Differential Equations 38, 80–103 (1980)
J. F. Selgrade, A Hopf bifurcation in single-loop positive feedback systems, Quart. Appl. Math. 40, 347–351 (1982)
M. W. Hirsch, Systems of differential equations that are competitive or cooperative. II: Convergence almost everywhere, SIAM J. Math. Anal. 16, 423–439 (1985)
M. W. Hirsch, Systems of differential equations that are competitive or cooperative. V: Convergence in 3-dimensional systems, J. Differential Equations 80, 94–106 (1989)
J. S. Griffith, Mathematics of cellular control processes. II: Positive feedback to one gene, J. Theoret. Biol. 20, 209–216 (1968)
J. J. Tyson and H. G. Othmer, The dynamics of feedback control circuits in biochemical pathways, Progr. Theoret. Biol. 5, 1–62 (1978)
J. P. LaSalle, Stability theory for ordinary differential equations, J. Differential Equations 4, 57–65 (1968)
Jiang Ji-Fa, A Liapunov function for 3-dimensional feedback systems, Proc. Amer. Math. Soc. 114, 1009–1013 (1992)
Jiang Ji-Fa, The asymptotic behavior of a class of second order differential equations with applications to electrical circuit equations, J. Math. Anal. Appl. 149, 26–37 (1990)
Jiang Ji-Fa, On the asymptotic behavior of a class of nonlinear differential equations, Nonlinear Anal. TMA 14, 453–467 (1990)
Jiang Ji-Fa, A note on a global stability theorem of M. W. Hirsch, Proc. Amer. Math. Soc. 112, 803–806 (1991)
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Article copyright:
© Copyright 1994
American Mathematical Society
