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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 weak sign variation to its interior. Strongly $2$-cooperative systems enjoy a strong Poincare-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.