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Autonomous sustained oscillations are ubiquitous in living and nonlivingsystems. As open systems, far from thermodynamic equilibrium, they defyentropic laws which mandate convergence to stationarity. We present structuralconditions on network cycles which support global Hopf bifurcation, i.e. globalbifurcation of non-stationary time-periodic solutions from stationarysolutions. Specifically, we show how monotone feedback cycles of thelinearization at stationary solutions give rise to global Hopf bifurcation, forsufficiently dominant coefficients along the cycle.  We include four example networks which feature such strong feedback cycles oflength three and larger: Oregonator chemical reaction networks, Lotka-Volterraecological population dynamics, citric acid cycles, and a circadian generegulatory network in mammals. Reaction kinetics in our approach are notlimited to mass action or Michaelis-Menten type.

Global Hopf bifurcation in networks with fast feedback cycles