Instability of Equilibrium and Convergence to Periodic Orbits in Strongly 2- Cooperative Systems

We consider time-invariant nonlinear n-dimensional strongly 2-cooperative systems, that is, systems that map the set of vectors with up to one weak sign variation to its interior. Strongly 2-cooperative systems enjoy a strong Poincaré-Bendixson property: bounded solutions that maintain a positive distance from the set of equilibria converge to a periodic solution. For strongly 2-cooperative systems whose trajectories evolve in a bounded and invariant set that contains a single unstable equilibrium, we provide a simple criterion for the existence of periodic trajectories. Moreover, we explicitly characterize a positive-measure set of initial conditions which yield solutions that asymptotically converge to a periodic trajectory. We demonstrate our theoretical results using two models from systems biology, the n-dimensional Goodwin oscillator and a 4-dimensional biomolecular oscillator with RNA-mediated regulation, and provide numerical simulations that verify the theoretical results.