  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.
