A Liapunov function for three-dimensional feedback systems

Jiang, Ji Fa

doi:10.1090/S0002-9939-1992-1092922-1

A Liapunov function for three-dimensional feedback systems
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Proc. Amer. Math. Soc. 114 (1992), 1009-1013 Request permission

Abstract:

For a three-dimensional model of a positive feedback loop, Selgrade [11, 12] proved that every positive-time trajectory in the nonnegative orthant converges. Hirsch [6] gave another proof of this result under slightly different assumptions. This paper provides a new proof of Selgrade’s result that is much shorter and presents a generalization that can be applied to positive and negative feedback loops and other systems.

References

U. an der Heiden, Existence of periodic solutions of a nerve equation, Biol. Cybernet. 21 (1976), no. 1, 37–39. MR 477293, DOI 10.1007/BF00326671

Oscillatory behavior in enzymatic control processes

Mathematics of cellular control processes

Negative feedback to one gene

20

S. P. Hastings, On the uniqueness and global asymptotic stability of periodic solutions for a third order system, Rocky Mountain J. Math. 7 (1977), no. 3, 513–538. MR 470338, DOI 10.1216/RMJ-1977-7-3-513
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 10.1137/0516030
Morris W. Hirsch, Systems of differential equations that are competitive or cooperative. V. Convergence in $3$-dimensional systems, J. Differential Equations 80 (1989), no. 1, 94–106. MR 1003252, DOI 10.1016/0022-0396(89)90097-1
Ji Fa Jiang, On the asymptotic behavior of a class of nonlinear differential equation, Nonlinear Anal. 14 (1990), no. 5, 453–467. MR 1041509, DOI 10.1016/0362-546X(90)90086-V
Ji Fa Jiang, The asymptotic behavior of a class of second-order differential equations with applications to electrical circuit equations, J. Math. Anal. Appl. 149 (1990), no. 1, 26–37. MR 1054791, DOI 10.1016/0022-247X(90)90283-L
J. P. LaSalle, Stability theory for ordinary differential equations, J. Differential Equations 4 (1968), 57–65. MR 222402, DOI 10.1016/0022-0396(68)90048-X
H. G. Othmer, The qualitative dynamics of a class of biochemical control circuits, J. Math. Biol. 3 (1976), no. 1, 53–78. MR 406568, DOI 10.1007/BF00307858
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 10.1016/0022-0396(80)90026-1
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 10.1137/0136019
J. J. Tyson, On the existence of oscillatory solutions in negative feedback cellular control processes, J. Math. Biol. 1 (1974/75), no. 4, 311–315. MR 390383, DOI 10.1007/BF00279849