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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**J. F. Selgrade,*Mathematical analysis of a cellular control process with positive feedback*, SIAM J. Appl. Math.**36**, 219-229 (1979)**[2]**J. F. Selgrade,*Asymptotic behavior of solutions to single loop positive feedback systems*, J. Differential Equations**38**, 80-103 (1980)**[3]**J. F. Selgrade,*A Hopf bifurcation in single-loop positive feedback systems*, Quart. Appl. Math.**40**, 347-351 (1982)**[4]**M. W. Hirsch,*Systems of differential equations that are competitive or cooperative*. II:*Convergence almost everywhere*, SIAM J. Math. Anal.**16**, 423-439 (1985)**[5]**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)**[6]**J. S. Griffith,*Mathematics of cellular control processes*. II:*Positive feedback to one gene*, J. Theoret. Biol.**20**, 209-216 (1968)**[7]**J. J. Tyson and H. G. Othmer,*The dynamics of feedback control circuits in biochemical pathways*, Progr. Theoret. Biol.**5**, 1-62 (1978)**[8]**J. P. LaSalle,*Stability theory for ordinary differential equations*, J. Differential Equations**4**, 57-65 (1968)**[9]**Jiang Ji-Fa,*A Liapunov function for*3-*dimensional feedback systems*, Proc. Amer. Math. Soc.**114**, 1009-1013 (1992)**[10]**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)**[11]**Jiang Ji-Fa,*On the asymptotic behavior of a class of nonlinear differential equations*, Nonlinear Anal. TMA**14**, 453-467 (1990)**[12]**Jiang Ji-Fa,*A note on a global stability theorem of M. W. Hirsch*, Proc. Amer. Math. Soc.**112**, 803-806 (1991)

Retrieve articles in *Quarterly of Applied Mathematics*
with MSC:
34C11,
92C40

Retrieve articles in all journals with MSC: 34C11, 92C40

Additional Information

DOI:
https://doi.org/10.1090/qam/1306039

Article copyright:
© Copyright 1994
American Mathematical Society