Conditions for global stability of monotone tridiagonal systems with negative feedback

This paper studies monotone tridiagonal systems with negative feedback. These systems possess the Poincaré-Bendixson property, which implies that, if orbits are bounded, if there is a unique steady state and this unique steady state is asymptotically stable, and if one can rule out periodic orbits, then the steady state is globally asymptotically stable. Two different approaches are discussed to rule out period orbits, one based on direct linearization and another one based on the theory of second additive compound matrices. Among the examples that illustrate the theoretical results is the classical Goldbeter model of the circadian rhythm.

[1]  Murat Arcak,et al.  A sufficient condition for additive D-stability and application to reaction-diffusion models , 2009, Syst. Control. Lett..

[2]  E. N. Dancer.  Some remarks on a boundedness assumption for monotone dynamical systems , 1998 .

[3]  L. Sanchez,et al.  Dynamics of the modified Michaelis–Menten system , 2006 .

[4]  Mario di Bernardo,et al.  Global Entrainment of Transcriptional Systems to Periodic Inputs , 2009, PLoS Comput. Biol..

[5]  V. Boichenko,et al.  Dimension theory for ordinary differential equations , 2005 .

[6]  Jan C. Willems,et al.  Lyapunov functions for diagonally dominant systems , 1975, Autom..

[7]  A. Goldbeter,et al.  Biochemical Oscillations And Cellular Rhythms: Contents , 1996 .

[8]  James S. Muldowney,et al.  Dynamics of Differential Equations on Invariant Manifolds , 2000 .

[9]  H. H. Rosenbrock,et al.  Computer Aided Control System Design , 1974, IEEE Transactions on Systems, Man, and Cybernetics.

[10]  L. Sanchez,et al.  Global asymptotic stability of the Goodwin system with repression , 2009 .

[11]  A. Goldbeter.  A model for circadian oscillations in the Drosophila period protein (PER) , 1995, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[12]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[13]  Michael Y. Li,et al.  Global stability for the SEIR model in epidemiology. , 1995, Mathematical biosciences.

[14]  G. Sell,et al.  THE POINCARE-BENDIXSON THEOREM FOR MONOTONE CYCLIC FEEDBACK SYSTEMS WITH DELAY , 1996 .

[15]  Chris Cosner,et al.  Book Review: Monotone dynamical systems: An introduction to the theory of competitive and cooperative systems , 1996 .

[16]  James S. Muldowney,et al.  Compound matrices and ordinary differential equations , 1990 .

[17]  David Angeli,et al.  Monotone control systems , 2003, IEEE Trans. Autom. Control..

[18]  David Angeli,et al.  Oscillations in I/O Monotone Systems Under Negative Feedback , 2007, IEEE Transactions on Automatic Control.

[19]  D. iljak.  Connective stability of competitive equilibrium , 1975 .

[20]  M. Hirsch,et al.  4. Monotone Dynamical Systems , 2005 .

[21]  Jean-Jacques E. Slotine,et al.  On Contraction Analysis for Non-linear Systems , 1998, Autom..