# Global Hopf bifurcation in networks with fast feedback cycles

## TL;DR

This paper identifies structural conditions in networks with feedback cycles that lead to global Hopf bifurcation, resulting in sustained oscillations across various biological and chemical systems.

## Contribution

It establishes how monotone feedback cycles in the linearization cause global bifurcation, extending understanding of oscillation emergence in complex networks.

## Key findings

- Feedback cycles of length three or more support oscillations.
- Examples include chemical, ecological, metabolic, and genetic networks.
- Reaction kinetics are not limited to standard models.

## Abstract

Autonomous sustained oscillations are ubiquitous in living and nonliving systems. As open systems, far from thermodynamic equilibrium, they defy entropic laws which mandate convergence to stationarity. We present structural conditions on network cycles which support global Hopf bifurcation, i.e. global bifurcation of non-stationary time-periodic solutions from stationary solutions. Specifically, we show how monotone feedback cycles of the linearization at stationary solutions give rise to global Hopf bifurcation, for sufficiently dominant coefficients along the cycle. We include four example networks which feature such strong feedback cycles of length three and larger: Oregonator chemical reaction networks, Lotka-Volterra ecological population dynamics, citric acid cycles, and a circadian gene regulatory network in mammals. Reaction kinetics in our approach are not limited to mass action or Michaelis-Menten type.