Global Hopf bifurcation in networks with fast feedback cycles

Bernold Fiedler

doi:10.3934/dcdss.2020344

Article Contents

2021, Volume 14, Issue 1: 177-203. Doi: 10.3934/dcdss.2020344

This issue Previous Article The spectrum of delay differential equations with multiple hierarchical large delays Next Article Orthogonality of fluxes in general nonlinear reaction networks

Global Hopf bifurcation in networks with fast feedback cycles

Bernold Fiedler

Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195 Berlin, Germany

Dedicated to Alexander Mielke on the occasion of his sixtieth birthday

Received: March 2019

Revised: January 2020

Early access: May 2020

Published: January 2021

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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.

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Mathematics Subject Classification: 34C15, 37G15, 05C38, 92C40, 92C42.

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