1. M. Margaliot and E.D. Sontag. Revisiting totally positive differential systems: A tutorial and new results. Automatica, 101:1-14, 2019. [PDF] Keyword(s): tridiagonal systems, cooperative systems, monotone systems. Abstract:

   A matrix is totally nonnegative (resp., totally positive) if all its minors are nonnegative (resp., positive). This paper draws connections between B. Schwarz's 1970 work on TN and TP matrices to Smillie's 1984 and Smith's 1991 work on stability of nonlinear tridiagonal cooperative systems, simplifying proofs in the later paper and suggesting new research questions.




2. L. Wang, P. de Leenheer, and E.D. Sontag. Conditions for global stability of monotone tridiagonal systems with negative feedback. Systems and Control Letters, 59:138-130, 2010. [PDF] Keyword(s): systems biology, monotone systems, tridiagonal systems, global stability. Abstract:

   This paper studies monotone tridiagonal systems with negative feedback. These systems possess the Poincar{\'e}-Bendixson property, which implies that, if orbits are bounded, if there is a unique steady state and this unique equilibrium is asymptotically stable, and if one can rule out periodic orbits, then the steady state is globally asymptotically stable. Different approaches are discussed to rule out period orbits. One is based on direct linearization, while the other uses the theory of second additive compound matrices. Among the examples that will illustrate our main theoretical results is the classical Goldbeter model of circadian rhythms.




3. D. Angeli and E.D. Sontag. Oscillations in I/O monotone systems. IEEE Transactions on Circuits and Systems, Special Issue on Systems Biology, 55:166-176, 2008. Note: Preprint version in arXiv q-bio.QM/0701018, 14 Jan 2007. [PDF] Keyword(s): monotone systems, hopf bifurcations, circadian rhythms, tridiagonal systems, nonlinear dynamics, systems biology, reaction networks, oscillations, periodic behavior, delay-differential systems. Abstract:

   In this note, we show how certain properties of Goldbeter's 1995 model for circadian oscillations can be proved mathematically, using techniques from the recently developed theory of monotone systems with inputs and outputs. The theory establishes global asymptotic stability, and in particular no oscillations, if the rate of transcription is somewhat smaller than that assumed by Goldbeter, based on the application of a tight small gain condition. This stability persists even under arbitrary delays in the feedback loop. On the other hand, when the condition is violated a Poincare'-Bendixson result allows to conclude existence of oscillations, for sufficiently high delays.

1. M. Margaliot and E.D. Sontag. Analysis of nonlinear tridiagonal cooperative systems using totally positive linear differential systems. In Proc. 2018 IEEE Conf. Decision and Control, pages 3104-3109, 2018. [PDF] Keyword(s): tridiagonal systems, cooperative systems, monotone systems. Abstract:

   This is a conference version of "Revisiting totally positive differential systems: A tutorial and new results".




2. L. Wang, P. de Leenheer, and E.D. Sontag. Global stability for monotone tridiagonal systems with negative feedback. In Proc. IEEE Conf. Decision and Control, Cancun, Dec. 2008, pages 4091-4096, 2008. Keyword(s): systems biology, monotone systems, tridiagonal systems, global stability. Abstract:

   Conference version of paper "Conditions for global stability of monotone tridiagonal systems with negative feedback"




3. D. Angeli and E.D. Sontag. An analysis of a circadian model using the small-gain approach to monotone systems. In Proc. IEEE Conf. Decision and Control, Paradise Island, Bahamas, Dec. 2004, IEEE Publications, pages 575-578, 2004. [PDF] Keyword(s): circadian rhythms, tridiagonal systems, nonlinear dynamics, systems biology, reaction networks, oscillations, periodic behavior, monotone systems, delay-differential systems. Abstract:

   We show how certain properties of Goldbeter's original 1995 model for circadian oscillations can be proved mathematically. We establish global asymptotic stability, and in particular no oscillations, if the rate of transcription is somewhat smaller than that assumed by Goldbeter, but, on the other hand, this stability persists even under arbitrary delays in the feedback loop. We are mainly interested in illustrating certain mathematical techniques, including the use of theorems concerning tridiagonal cooperative systems and the recently developed theory of monotone systems with inputs and outputs.

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