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.

1. M. Margaliot and E.D. Sontag. Compact attractors of an antithetic integral feedback system have a simple structure. Technical report, bioRxiv 2019/868000v1, 2019. [PDF] Keyword(s): Poincare-Bendixson, k-cooperative dynamical systems, sign-regular matrices, synthetic biology, antithetic feedback. 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 nonlinear four-dimensional system) has been analyzed in much detail. This system has a unique equilibrium but, depending on parameters, it may exhibit periodic orbits. This note shows that, for any parameter choices, every bounded trajectory satisfies a Poincare'-Bendixson property: the dynamics in the omega-limit set of any precompact solution is conjugate to the dynamics in a compact invariant subset of a two-dimensional Lipschitz dynamical system, thus precluding chaotic and other strange attractors.

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