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Global Hopf bifurcation in networks with fast feedback cycles. (English) Zbl 1454.34060

Summary: 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.

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
37G15 Bifurcations of limit cycles and periodic orbits in dynamical systems
05C38 Paths and cycles
92C40 Biochemistry, molecular biology
92C42 Systems biology, networks
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