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

TL;DR

This paper establishes conditions under which 3D strongly 2-cooperative systems guarantee convergence of solutions to periodic orbits, extending the Poincare-Bendixson property to certain nonlinear systems.

Contribution

It provides a simple sufficient condition for the existence of invariant sets leading to periodic orbits in 3D strongly 2-cooperative systems, with explicit initial condition characterization.

Findings

Solutions converge to periodic orbits under specified conditions.

Theoretical results are demonstrated on biochemical models.

Explicit characterization of initial conditions for convergence.

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.