1. M. Margaliot, C. Wu, and E.D.Sontag. Compact attractors of an antithetic integral feedback system have a simple structure. 2025. Note: Submitted.Keyword(s): Poincaré-Bendixson, synthetic biology, nonlinear control. Abstract:

   Since its introduction by Briat, Gupta and Khammash, the antithetic feedback controller design has attracted considerable attention in both theoretical and experimental systems biology. The case in which the plant is a two-dimensional linear system (making the closed-loop system a four-dimensional nonlinear system) has been analyzed in much detail. This system has a unique equilibrium~$e$ but, depending on parameters, it may exhibit periodic orbits. An interesting question is for what parameter values periodic orbits exist. Another open question is whether other dynamical behaviors, such as chaotic attractors, might be possible for some parameter choices. We show that, for any parameter choices, every compact omega-limit set that does not include~$e$ is a periodic solution. We also show that if the Jacobian of the vector field at the equilibrium is unstable then a (non-trivial) periodic orbit exists. The analysis is based on the theory of strongly~$2$-cooperative systems.




2. 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.




3. D. Angeli, M.W. Hirsch, and E.D. Sontag. Attractors in coherent systems of differential equations. J. of Differential Equations, 246:3058-3076, 2009. [PDF] Keyword(s): monotone systems, positive feedback systems. Abstract:

   Attractors of cooperative dynamical systems are particularly simple; for example, a nontrivial periodic orbit cannot be an attractor. This paper provides characterizations of attractors for the wider class of systems defined by the property that all directed feedback loops are positive. Several new results for cooperative systems are obtained in the process.




4. E.D. Sontag and Y. Wang. A cooperative system which does not satisfy the limit set dichotomy. J. of Differential Equations, 224:373-384, 2006. [PDF] Keyword(s): dynamical systems, monotone systems. Abstract:

   The fundamental property of strongly monotone systems, and strongly cooperative systems in particular, is the limit set dichotomy due to Hirsch: if x < y, then either Omega(x) < Omega (y), or Omega(x) = Omega(y) and both sets consist of equilibria. We provide here a counterexample showing that this property need not hold for (non-strongly) cooperative systems.

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. 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.

This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders.

This document was translated from BibT_EX by bibtex2html