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The Poincaré-Bendixson theorem for monotone cyclic feedback systems. (English) Zbl 0712.34060
The authors consider systems of the form (*) \(\dot x^ i=f^ i(x^ i,x^{i-1})\), \(i=1,2,...,n\), where we agree to interpret \(x^ 0\) as \(x^ n\). They assume that the nonlinearity \(f=(f^ 1,f^ 2,...,f^ n)\) is defined on a nonempty open set \(U\subset R^ n\) with the property that each coordinate projection \(U^ i\subset {\mathbb{R}}^ 2\) of U onto the \((x^ i,x^{i-1})\)-plane is convex and that \(f^ i\in C^ 1(U^ i).They\) assume also for some \(\delta^ i\in \{1,+1\}\), that (**): \(\delta^ i\partial f^ j(x^ i,x^{i-1})/\partial x^{i-1}>0\) for all \((x^ i,x^{i-1})\in U^ i\) and \(i=1,2,...,n\). Such a system, of the form (*) satisfying (**), is called a monotone cyclic feedback system. The main result of this paper is that the Poincaré-Bendixson theorem holds for monotone cyclic feedback systems. The organization of this paper is as follows. In § 1 a principal tool, an integer valued Lyapunov function N is developed. § 2 is concerned with the Floquet theory of linear monotone cyclic feedback systems. § 3 is devoted to the proof of the main result. Finally, in § 4 various applications are treated.
Reviewer: Chungyou He

MSC:
34C25 Periodic solutions to ordinary differential equations
93C15 Control/observation systems governed by ordinary differential equations
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[1] D. J. Allwright, ?A Global Stability Criterion for Simple Control Loops,?J. Math. Biol. 4 (1977), 363-373. · Zbl 0372.93042
[2] U. an der Heiden, ?Delays in Physiological Systems,?J. Math. Biol. 8 (1979), 345-364. · Zbl 0429.92009
[3] U. an der Heiden, ?Existence of Periodic Solutions of a Nerve Equation,?Biol. Cybern. 21 (1976), 37-39. · Zbl 0316.92001
[4] U. an der Heiden and H. O. Walther, ?Existence of Chaos in Control Systems with Delayed Feedback,?J. Diff. Eq. 47 (1983), 273-295. · Zbl 0498.93027
[5] H. T. Banks and J. M. Mahaffy, ?Mathematical Models for Protein Biosynthesis,? Technical Report, Lefschetz Center for Dynamical Systems, Brown University, Providence, R. I., 1979.
[6] H. T. Banks and J. Mahaffy, ?Stability of Cyclic Gene Models for Systems Involving Repression,?J. Theor. Biol. 74 (1978), 323-334. · Zbl 0406.92011
[7] H. T. Banks and J. M. Mahaffy, ?Global Asymptotic Stability of Certain Models for Protein Synthesis and Repression,?Q. App. Math. 36 (1978), 209-220. · Zbl 0406.92011
[8] C. Berding and T. Harbich, ?On the Dynamics of a Simple Biochemical Control Circuit,?Biol. Cybern. 49 (1984), 209-219. · Zbl 0533.92010
[9] P. Brunovsky and B. Fiedler, ?Zero Numbers on Invariant Manifolds in Scalar Reaction Diffusion Equations,?Nonlin. Anal. TMA 10 (1986), 129-194. · Zbl 0594.35056
[10] S. N. Chow, O. Djekmann, and J. Mallet-Paret, ?Stability, Multiplicity, and Global Continuation of Symmetric Periodic Solutions of a Nonlinear Volterra Integral Equation,?Jap. J. Appl. Math. 2 (1985), 433-469. · Zbl 0596.45009
[11] B. Fiedler and J. Mallet-Paret, ?A Poincaré-Bendixson Theorem for Scalar Reaction Diffusion Equations,?Arch. Rat. Mech. Anal. 107 (1989), 325-345. · Zbl 0704.35070
[12] A. Fraser and J. Tiwari, ?Genetical Feedback-Repression. II. Cyclic Genetic Systems,?J. Theor. Biol. 47 (1974), 397-12.
[13] G. Fusco and W. Oliva, ?Jacobi Matrices and Transversality,?Proc. Roy. Soc. Edinburg Sect. A 109 (1988), 231-243. · Zbl 0692.58019
[14] B. C. Goodwin,Temporal Organization in Cells, Academic Press, New York, 1963.
[15] B. C. Goodwin, ?Oscillatory Behavior in Enzymatic Control Processes,?Adv. Enzyme Reg. 3 (1965), 425-439.
[16] J. S. Griffith, ?Mathematics of Cellular Control Processes I, II,?J. Theor. Biol. 20 (1968), 202-208, 209-216.
[17] J. K. Hale,Ordinary Differential Equations, R. E. Krieger, 1980.
[18] S. P. Hastings, ?On the Uniqueness and Global Asymptotic Stability of Periodic Solutions for a Third Order System,?Rocky Mt. J. Math. 7 (1977), 513-538. · Zbl 0386.34039
[19] S. P. Hastings, J. Tyson, and D. Webster, ?Existence of Periodic Solutions for Negative Feedback Cellular Control Systems,?J. Diff. Eq. 25 (1977), 39-64. · Zbl 0361.34038
[20] M. W. Hirsch, ?The Dynamical Systems Approach to Differential Equations,?Bull. A.M.S. 11 (1984), 1-64. · Zbl 0541.34026
[21] M. W. Hirsch, ?Stability and Convergence in Strongly Monotone Dynamical Systems,?J. Reine Angewendte Math. 383 (1988), 1-53. · Zbl 0624.58017
[22] M. W. Hirsch, ?Systems of Differential Equations which are Competitive or Cooperative, I: Limit Sets,?SIAM J. Math. Anal. 13 (1982), 167-179; ?II: Convergence Almost Everywhere,?SIAM J. Math. Anal. 16 (1985) 432-439. · Zbl 0494.34017
[23] G. A. Lenov, ?An Analog of Bendixson’s Criterion for Third Order Equations,?Diff. Eq. 13 (1977), 367-368.
[24] N. MacDonald, ?Bifurcation Theory Applied to a Simple Model of a Biochemical Oscillator,?J. Theor. Biol. 65 (1977), 727-734.
[25] N. MacDonald, ?Time Lag in a Model of a Biochemical Reaction Sequence with End Product Inhibition,? /.Theor. Biol. 67 (1977), 549-556.
[26] M. C. Mackey and L. Glass, ?Oscillations and Chaos in Physiological Control Systems,?Science 197 (1977), 287-289. · Zbl 1383.92036
[27] J. M. Mahaffy, ?Periodic Solutions for Certain Protein Synthesis Models,?JMAA 74 (1980), 72-105.
[28] J. M. Mahaffy, ?Stability of Periodic Solutions for a Model of Genetic Repression with Delays,?J. Math. Biol. 22 (1985), 137-144. · Zbl 0563.92008
[29] J. Mallet-Paret and G. Sell. In preparation.
[30] H. Matano, ?Convergence of Solutions of One-Dimensional Semilinear Parabolic Equations,?J. Math. Kyoto Univ. (1978), 221-227. · Zbl 0387.35008
[31] K. Nickel, ?Gestaltaussagen über Lösungen parabolischer Differentialgleichungen,?J. Reine Angew. Math. 211 (1962), 78-94. · Zbl 0127.31801
[32] R. Reissig, G. Sansone, and R. Conti,Nonlinear Differential Equations of Higher Order, Rome, 1969. · Zbl 0275.34001
[33] J. F. Selgrade, ?Asymptotic Behavior of Solutions to Single Loop Positive Feedback Systems,?J. Diff. Eq. 38 (1980), 80-103. · Zbl 0438.34052
[34] J. Smillie, ?Competitive and Cooperative Tridiagonal Systems of Differential Equations,?SIAM J. Math. Anal. 15 (1984), 530-534. · Zbl 0546.34007
[35] H. L. Smith, ?Periodic Orbits of Competitive and Cooperative Systems,?J. Diff. Eq. 65 (1986), 361-373. · Zbl 0615.34027
[36] H. L. Smith, ?Oscillations and Multiple Steady States in a Cyclic Gene Model with Repression,?J. Math. Biol. 25 (1987), 169-190. · Zbl 0619.92004
[37] H. L. Smith, ?Systems of Ordinary Differential Equations which Generate an Order Preserving Flow. A Survey of Results,?SIAM Rev. 30 (1988), 87-113. · Zbl 0674.34012
[38] R. A. Smith, ?The Poincaré-Bendixson Theorem for Certain Differential Equations of Higher Order,?Proc. Roy. Soc. Edinburgh 83A (1979), 63-79. · Zbl 0408.34042
[39] R. A. Smith, ?Existence of Periodic Orbits of Autonomous Ordinary Differential Equations,?Proc. Roy. Soc. Edinburgh 85A (1980), 153-172. · Zbl 0429.34040
[40] R. A. Smith, ?Orbital Stability for Ordinary Differential Equations,?J. Diff. Eq. 69 (1987), 265-287. · Zbl 0632.34054
[41] R. B. Stein, K. V. Leung, D. Mangeron, and M. Oguztöreli, ?Improved Neuronal Models for Studying Neural Networks,?Kybernetic 15 (1974), 1-9. · Zbl 0297.92002
[42] J. J. Tyson, ?On the Existence of Oscillatory Solutions in Negative Feedback Cellular Control Processes,?J. Math. Biol. 1 (1975), 311-315. · Zbl 0301.34044
[43] J. J. Tyson and H. G. Othmer, ?The Dynamics of Feedback Control Circuits in Biochemical Pathways,? inProg. Theor. Biol. (R. Rosen and F. M. Snell, Eds.), Academic Press, New York, 1978. · Zbl 0448.92010
[44] J. K. Hale, ?Topics in Dynamic Bifurcation Theory,?CBMS 47 (1980), A.M.S. · Zbl 0482.35010
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