The Jacobian nonresonance conjecture for oscillation in chemical reaction networks

Let $(\mathcal{R},\mathcal{K})$ be a fully open chemical reaction network with kinetics, where $\mathcal{K}$ is any scaling-invariant subset of positive general kinetics. For example, $\mathcal{K}$ may be given by mass-action kinetics or physical power-law kinetics. Let $\Gamma$ be the stoichiometric matrix of $\mathcal{R}$, so that $\mathcal{R}$ gives rise to the family of ordinary differential equations on $\mathbb{R}^n_{\gg 0}$

$$\dot{x}=\Gamma v(x),\qquad v\in\mathcal{K}.$$

Jacobian nonresonance conjecture. If, for all $x\in\mathbb{R}^n_{\gg 0}$ and all $v\in\mathcal{K}$, the Jacobian matrix $\Gamma Dv(x)$ has no purely imaginary eigenvalues, then $\mathcal{R}$ does not admit a positive periodic orbit.

The conjecture proposes a Jacobian-based obstruction to positive periodic behavior in fully open chemical reaction networks, uniformly over the specified class of kinetics. It is intended to guide searches for stable oscillations by excluding networks whose Jacobians satisfy the stated spectral condition; the supplied source does not establish the conjecture.