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. |