Oscillations in I/O monotone systems under negative feedback

IEEE Transactions on Automatic Control

Volumn 53, Issue SPECIAL ISSUE, 2008, Pages 166-176

  Angeli, David ^a,b   Sontag, Eduardo D ^c  

^a UNIVERSITY OF FLORENCE   (Italy)

^b IMPERIAL COLLEGE LONDON   (United Kingdom)

^c RUTGERS UNIVERSITY   (United States)

Author keywords

Circadian rhythms; Monotone systems; Negative feedback; Periodic behaviors

Indexed keywords

BIOLOGICAL MODELS; CIRCADIAN OSCILLATIONS; CIRCADIAN RHYTHMS; CLASSICAL MODEL; GLOBAL ASYMPTOTIC STABILITY; INPUT/OUTPUT; MODEL REDUCTION; MONOTONE SYSTEMS; MONOTONICITY; NEGATIVE FEEDBACK; OSCILLATORY BEHAVIORS; PERIODIC BEHAVIOR; PERIODIC SOLUTION; SMALL GAIN THEOREM; STABILITY PROPERTIES; SUFFICIENT CONDITIONS; CIRCADIAN OSCILLATION; MONOTONE SYSTEM;

ASYMPTOTIC STABILITY; FEEDBACK; UNCERTAINTY ANALYSIS; BIOLOGICAL SYSTEMS; MOLECULAR BIOLOGY; PERTURBATION TECHNIQUES; ROBUSTNESS (CONTROL SYSTEMS);

SYSTEM STABILITY; FEEDBACK CONTROL;

EID: 39849105873     PISSN: 00189286     EISSN: None     Source Type: Journal    
DOI: 10.1109/TAC.2007.911320     Document Type: Article

References (38)

1. D. Angeli and E. D. Sontag, "Monotone control systems," IEEE Trans. Autom. Control, vol. 48, no. 10, pp. 1684-1698, Oct. 2003.
2. D. Angeli and E. D. Sontag, "Multi-stability in monotone input/output systems," Syst. Control Lett., vol.51, pp. 185-202, 2004.
3. D. Angeli and E. D. Sontag, "Interconnections of monotone systems with steady-state characteristics," in Optimal Control, Stabilization and Nonsmooth Analysis. Berlin, Germany: Springer, 2004, vol.301, Lecture Notes in Control and Inform. Sci., pp. 135-154.
4. D. Angeli, P. de Leenheer, and E. D. Sontag, "A small-gain theorem for almost global convergence of monotone systems," Syst. Control Lett., vol. 52, no. 5, pp. 407-414, 2004.
5. L. Chisci and P. Falugi, "Asymptotic tracking for constrained monotone systems," IEEE Trans. Autom. Control, vol. 51, no. 5, pp. 873-879, May 2006.
6. E. N. Dancer, "Some remarks on a boundedness assumption for monotone dynamical systems," in Proc. AMS, 1998, vol.126, pp. 801-807.
7. P. de Leenheer, D. Angeli, and E. D. Sontag, "On predator-prey systems and small-gain theorems," Math. Biosci. Eng., vol. 2, no. 1, pp. 25-42, 2005.
8. D. V. Efimov and A. L. Fradkov, "Oscillatority conditions for nonlinear systems with delays," J. Appl. Math., vol.2007, 2007, 72561.
9. G. Enciso and E. D. Sontag, "Monotone systems under positive feedback: Multistability and a reduction theorem," Syst. Control Lett., vol. 54, no. 2, pp. 159-168, 2005.
10. G. Enciso, "A dichotomy for a class of cyclic delay systems," Math. Biosci., vol.208, pp. 63-75, 2007.
11. G. Enciso, H. L. Smith, and E. D. Sontag, "Non-monotone systems decomposable into monotone systems with negative feedback," J. Diff. Equations, vol. 224, no. 224, pp. 205-227, 2006.
12. G. Enciso and E. D. Sontag, "Global attractivity, I/O monotone smallgain theorems, and biological delay systems," Discrete Contin. Dynam. Syst., vol.14, pp. 549-578, 2006.
13. G. Enciso and E. D. Sontag, "On the stability of a model of testosterone dynamics," J. Math. Biol., vol.49, pp. 627-634, 2004.
14. T. Gedeon and E. D. Sontag, "Oscillations in multi-stable monotone systems with slowly varying feedback," J. Diff. Equations, vol.239, pp. 273-295, 2007.
15. A. Goldbeter, "A model for circadian oscillations in the Drosophila period protein (PER)," Proc. Royal Soc. London B., vol.261, pp. 319-324, 1995.
16. A. Goldbeter, Biochemical Oscillations and Cellular Rhythms. Cambridge, U.K.: Cambridge Univ., 1996.
17. F. Grognard, Y. Chitour, and G. Bastin, "Equilibria and stability analysis of a branched metabolic network with feedback inhibition," AIMS J. Netw. Heterogeneous Media, vol.1, pp. 219-239, 2006.
18. D. Gromov and J. Raisch, "Detecting and enforcing monotonicity for hybrid control systems synthesis," in Proc. 2nd IFAC Conf. Anal. Design of Hybrid Syst., Alghero, Italy, Jun. 7-9, 2006, pp. 395-402.
19. J. K. Hale and S. M. V. Lunel, Introduction to Functional Differential Equations. New York: Springer, 1993.
20. M. Hirsch, "Differential equations and convergence almost everywhere in strongly monotone flows," Contemp. Math., vol.17, pp. 267-285, 1983.
21. M. Hirsch, "Systems of differential equations that are competitive or cooperative ii: Convergence almost everywhere," SIAM J. Math. Anal., vol.16, pp. 423-439, 1985.
22. M. Hirsch and H. L. Smith, "Monotone dynamical systems," in Handbook of Differential Equations, Ordinary Differential Equations, 2nd ed. Amsterdam, The Netherlands: Elsevier, 2005.
23. A. F. Ivanov and B. Lani-Wayda, "Periodic solutions for three-dimensional non-monotone cyclic systems with time delays," Discrete Contin. Dynam.Syst., vol.A 11, pp. 667-692, 2004.
24. J. F. Jiang, "On the global stability of cooperative systems," Bull. London Math. Soc., vol.6, pp. 455-458, 1994.
25. J.-C. Leloup and A. Goldbeter, "A model for circadian rhythms in Drosophila incorporating the formation of a complex between the PER and TIM proteins," J. Biol. Rhythms, vol.13, pp. 70-87, 1998.
26. J.-C. Leloup and A. Goldbeter, "Chaos and birhythmicity in a model for circadian oscillations of the PER and TIM proteins in Drosophila," J. Theor. Biol., vol.198, pp. 445-459, 1999.
27. M. Malisoff and P. De Leenheer, "A small-gain theorem for monotone systems with multivalued input-state characteristics," IEEE Trans. Automat. Control, vol. 51, no. 2, pp. 287-292, Feb. 2006.
28. J. Mallet-Paret and G. R. Sell, "The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay," J. Diff. Equations, vol.125, pp. 441-489, 1996.
29. J. Mallet-Paret and H. Smith, "The Poincare-Bendixon theorem for monotone cyclic feedback systems," J. Dynam. Diff. Equations, vol.2, pp. 367-421, 1990.
30. J. Mallet-Paret and R. Nussbaum, "Global continuation and asymptotic behaviour for periodic solutions of a differential-delay equation," Ann. Mat. Pura Appl., vol.145, pp. 33-128, 1986.
31. H. Schneider and M. Vidyasagar, "Cross-positive matrices," SIAM J. Numer. Anal., vol.7, pp. 508-519, 1970.
32. J. Smillie, "Competitive and cooperative tridiagonal systems of differential equations," SIAM J. Math. Anal., vol.15, pp. 530-534, 1984.
33. H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. Ann Arbor, MI: Amer. Math. Soc., 1995, vol.41, Mathematical Surveys and Monographs.
34. H. L. Smith, "Periodic tridiagonal competitive and cooperative systems of differential equations," SIAM J. Math. Anal., vol.22, pp. 1102-1109, 1991.
35. E. D. Sontag, "Molecular systems biology and control," Eur. J. Control, vol. 11, no. 4-5, pp. 396-435, 2005.
36. P. Volkmann, "Gewohnliche differentialungleichungen mit quasimonoton wachsenden funktionen in topologischen vektorraumen," Math. Z., vol.127, pp. 157-164, 1972.
37. S. Walcher, "On cooperative systems with respect to arbitrary orderings," J. Math. Anal. Appl., vol.263, pp. 543-554, 2001.
38. W. Walter, Differential and Integral Inequalities. Berlin, Germany: Springer-Verlag, 1970.