Essential unstable motifs for specific nonlinear dynamics

Determine which unstable motifs in reaction networks—formalized as minimal unstable cores such as autocatalytic cores and unstable-negative feedback cores—are essential for realizing specific types of nonlinear dynamical behavior (e.g., periodic oscillations or multistationarity) under monotone kinetics. Provide a structural characterization that identifies the motifs whose presence is necessary to generate each targeted dynamical phenomenon.

Background

The paper develops a stoichiometric framework for oscillations using parameter-rich kinetics and introduces oscillatory cores (Class I and II) as minimal subnetworks guaranteeing the potential for periodic orbits. Prior work identified unstable cores, including autocatalytic cores, as sufficient for instability, but a general mapping from structural motifs to specific nonlinear behaviors (such as oscillations versus multistationarity) is not yet established.

By highlighting positive- and negative-feedback-based oscillatory cores and the principle of length, the paper advances sufficient conditions for oscillations. The open question asks for a necessary characterization: which minimal unstable motifs are structurally required to realize each class of nonlinear dynamics across reaction networks.