Convergence to periodic orbits in 3-dimensional strongly 2-cooperative systems

Abstract

The flow of a $k$-cooperative system maps the set of vectors with up to~$(k-1)$ sign variations to itself. Strongly $2$-cooperative systems satisfy a strong \Poincare-Bendixson property: any bounded solution that evolves in a compact set containing no equilibria converges to a periodic orbit. For $3$-dimensional strongly $2$-cooperative nonlinear systems, we provide a simple sufficient condition that guarantees the existence, in the state space, of an invariant compact set that includes no equilibrium points. Thus, any solution emanating from this set converges to a periodic orbit. We characterize explicitly the set of initial conditions from which the trajectory converges to a periodic solution. We demonstrate our theoretical results on two well-known models in biochemistry: a 3D Goodwin oscillator model and the 3D Field-Noyes ordinary-differential-equation (ODE) model for the Belousov-Zhabotinskii reaction.