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

2024/06/29 by Katz, Rami, Giordano, Giulia, Margaliot, Michael · 1 citation
#FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2407.00461

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.

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