# American Institute of Mathematical Sciences

• Previous Article
Existence and multiplicity of positive solutions for a class of quasilinear Schrödinger equations in $\mathbb R^N$$^\diamondsuit$
• DCDS-S Home
• This Issue
• Next Article
Stability and bifurcation analysis in a delay-induced predator-prey model with Michaelis-Menten type predator harvesting

## 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, doi: 10.3934/dcdss.2020344
##### References:

show all references

##### References:
 [1] Casian Pantea, Heinz Koeppl, Gheorghe Craciun. Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2153-2170. doi: 10.3934/dcdsb.2012.17.2153 [2] Julien Coatléven, Claudio Altafini. A kinetic mechanism inducing oscillations in simple chemical reactions networks. Mathematical Biosciences & Engineering, 2010, 7 (2) : 301-312. doi: 10.3934/mbe.2010.7.301 [3] Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523 [4] Klemens Fellner, Wolfang Prager, Bao Q. Tang. The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinetic & Related Models, 2017, 10 (4) : 1055-1087. doi: 10.3934/krm.2017042 [5] Ginestra Bianconi, Riccardo Zecchina. Viable flux distribution in metabolic networks. Networks & Heterogeneous Media, 2008, 3 (2) : 361-369. doi: 10.3934/nhm.2008.3.361 [6] Yong Zhao, Qishao Lu. Periodic oscillations in a class of fuzzy neural networks under impulsive control. Conference Publications, 2011, 2011 (Special) : 1457-1466. doi: 10.3934/proc.2011.2011.1457 [7] Maya Mincheva, Gheorghe Craciun. Graph-theoretic conditions for zero-eigenvalue Turing instability in general chemical reaction networks. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1207-1226. doi: 10.3934/mbe.2013.10.1207 [8] Maria Conceição A. Leite, Yunjiao Wang. Multistability, oscillations and bifurcations in feedback loops. Mathematical Biosciences & Engineering, 2010, 7 (1) : 83-97. doi: 10.3934/mbe.2010.7.83 [9] Yacine Chitour, Frédéric Grognard, Georges Bastin. Equilibria and stability analysis of a branched metabolic network with feedback inhibition. Networks & Heterogeneous Media, 2006, 1 (1) : 219-239. doi: 10.3934/nhm.2006.1.219 [10] Freddy Dumortier, Robert Roussarie. Bifurcation of relaxation oscillations in dimension two. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 631-674. doi: 10.3934/dcds.2007.19.631 [11] Hirotada Honda. On a model of target detection in molecular communication networks. Networks & Heterogeneous Media, 2019, 14 (4) : 633-657. doi: 10.3934/nhm.2019025 [12] Joo Sang Lee, Takashi Nishikawa, Adilson E. Motter. Why optimal states recruit fewer reactions in metabolic networks. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2937-2950. doi: 10.3934/dcds.2012.32.2937 [13] Nathaniel J. Merrill, Zheming An, Sean T. McQuade, Federica Garin, Karim Azer, Ruth E. Abrams, Benedetto Piccoli. Stability of metabolic networks via Linear-in-Flux-Expressions. Networks & Heterogeneous Media, 2019, 14 (1) : 101-130. doi: 10.3934/nhm.2019006 [14] Congming Li, Eric S. Wright. Modeling chemical reactions in rivers: A three component reaction. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 377-384. doi: 10.3934/dcds.2001.7.373 [15] Sanling Yuan, Yongli Song, Junhui Li. Oscillations in a plasmid turbidostat model with delayed feedback control. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 893-914. doi: 10.3934/dcdsb.2011.15.893 [16] Willard S. Keeran, Patrick D. Leenheer, Sergei S. Pilyugin. Feedback-mediated coexistence and oscillations in the chemostat. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 321-351. doi: 10.3934/dcdsb.2008.9.321 [17] Samuel Walsh. Steady stratified periodic gravity waves with surface tension II: Global bifurcation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3287-3315. doi: 10.3934/dcds.2014.34.3287 [18] Tetsutaro Shibata. Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2139-2147. doi: 10.3934/cpaa.2018102 [19] Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems - A, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245 [20] Peng Feng, Menaka Navaratna. Modelling periodic oscillations during somitogenesis. Mathematical Biosciences & Engineering, 2007, 4 (4) : 661-673. doi: 10.3934/mbe.2007.4.661

2019 Impact Factor: 1.233