# American Institute of Mathematical Sciences

January  2021, 14(1): 177-203. doi: 10.3934/dcdss.2020344

## Global Hopf bifurcation in networks with fast feedback cycles

 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 Published  May 2020

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.

Citation: Bernold Fiedler. Global Hopf bifurcation in networks with fast feedback cycles. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344
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