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

Rami Katz, Giulia Giordano, Michael Margaliot

TL;DR

This work investigates 3D strongly $2$-cooperative dynamical systems and proves a simple sufficient condition guaranteeing the existence of an invariant compact set without equilibria, from which all trajectories converge to a periodic orbit. By leveraging the strong Poincaré–Bendixson property, spectral analysis of Jacobians, and a coordinate-cylindrical construction, the authors explicitly characterize the basin of attraction in which convergence to periodic behavior occurs. They validate the theory on two classical biological models: the 3D Goodwin oscillator and the Field–Noyes model for the Belousov–Zhabotinskii reaction, showing that unstable equilibria lead to attracting limit cycles for a wide set of initial conditions. The results extend the understanding of oscillatory behavior in higher-dimensional cooperative-like systems and have potential implications for designing robust biochemical oscillators in systems biology and chemical dynamics.

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.

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

TL;DR

This work investigates 3D strongly -cooperative dynamical systems and proves a simple sufficient condition guaranteeing the existence of an invariant compact set without equilibria, from which all trajectories converge to a periodic orbit. By leveraging the strong Poincaré–Bendixson property, spectral analysis of Jacobians, and a coordinate-cylindrical construction, the authors explicitly characterize the basin of attraction in which convergence to periodic behavior occurs. They validate the theory on two classical biological models: the 3D Goodwin oscillator and the Field–Noyes model for the Belousov–Zhabotinskii reaction, showing that unstable equilibria lead to attracting limit cycles for a wide set of initial conditions. The results extend the understanding of oscillatory behavior in higher-dimensional cooperative-like systems and have potential implications for designing robust biochemical oscillators in systems biology and chemical dynamics.

Abstract

The flow of a -cooperative system maps the set of vectors with up to~ sign variations to itself. Strongly -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 -dimensional strongly -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.
Paper Structure (7 sections, 4 theorems, 37 equations, 3 figures)

This paper contains 7 sections, 4 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and Preliminary Results
  3. Convergence to a Periodic Orbit
  4. Two Case Studies
  5. 3-dimensional Goodwin oscillator model
  6. Field-Noyes model
  7. Discussion

Key Result

Theorem 1

Consider the non-linear time-invariant system Suppose that $f\in C^2$, and let $J(x):=\frac{\partial}{\partial x}f(x)$ denote the Jacobian of the vector field. Suppose that eq:nonlin3 is strongly 2-cooperative, and that its trajectories evolve in the closed box ${\mathcal{B}}=\{x\in{\mathbb R}^3\ | \ \underline{x}\leq x \leq \overline{x}\},$ fo Then ${\mathcal{B}}_{16}:= {\mathcal{B}}_1 \cup\dots

Figures (3)

  • Figure 1: The red line is $\operatorname{span}(\tilde{\zeta})$, and the cubes are $\tilde{{\mathcal{B}}}_7$ and $\tilde{{\mathcal{B}}}_8$, intersecting at the origin. Left: Any $z\in \tilde{{\mathcal{B}}}$ such that $\angle (z ,\tilde{\zeta} )$ is sufficiently close to $0$ or $\pi$ lies in the interior of the two cubes. Right: The invariant set is obtained by cutting out from $\tilde{{\mathcal{B}}}_{16}$ a cylinder around $\operatorname{span}(\tilde{\zeta})$. Thus, the invariant set has a positive distance from the equilibrium at the origin.
  • Figure 2: Solution of the 3D Goodwin system in Example \ref{['exa:3DGOOD_num']}, with initial condition $x(0)=0.10.10.1^\top$, converging to a periodic orbit. The equilibrium point $e$ is denoted by $*$.
  • Figure 3: Solution of the Field-Noyes system in Example \ref{['exa:BZ']}.

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • Lemma 1
  • Corollary 1
  • Example 1
  • Corollary 2
  • ...and 1 more