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  January 2021 Early access  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 and Continuous Dynamical Systems - S, 2021, 14 (1) : 177-203. doi: 10.3934/dcdss.2020344
References:
[1]

J. C. Alexander and J. A. Yorke, Global bifurcation of periodic orbits, Amer. J. Math., 100 (1978), 263-292.  doi: 10.2307/2373851.

[2]

N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 33 (1979), 201-225.  doi: 10.1016/0022-0396(79)90088-3.

[3]

K. T. Alligood and J. A. Yorke, Families of periodic orbits: Virtual period and global continuability, J. Differential Equations, 55 (1984), 59-71.  doi: 10.1016/0022-0396(84)90088-3.

[4] R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. Vol. Ⅰ, Ⅱ, Clarendon Press, Oxford, 1975. 
[5]

J. M. Berg, J. L. Tymoczko, G. J. Gatto and L. Stryer, Biochemistry, W. H. Freeman, New York 2015.

[6]

W. C. Bray, A periodic reaction in homogeneous solution and its relation to catalysis, J. Amer. Chem. Soc., 43 (1921), 1262-1267.  doi: 10.1021/ja01439a007.

[7]

, Comprehensive Chemical Kinetics, 1-43, Elsevier, 1969-2019.

[8]

B. Brehm and B. Fiedler, Sensitivity of chemical reaction networks: A structural approach. 3. Regular multimolecular systems, Math. Methods Appl. Sci., 41 (2018), 1344-1376.  doi: 10.1002/mma.4668.

[9]

M. G. Crandall and P. H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal., 67 (1977), 53-72.  doi: 10.1007/BF00280827.

[10]

G. Dangelmayr, B. Fiedler, K. Kirchgässner and A. Mielke, Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation and Stability, Research Notes in Mathematics Series, 352, Longman, Harlow, 1996.

[11]

M. Eigen, Selforganization of matter and the evolution of biological macromolecules, Naturwissenschaften, 58 (1971), 465-523.  doi: 10.1007/BF00623322.

[12]

H. Errami and M. Eiswirth, et al., Detection of Hopf bifurcations in chemical reaction networks using convex coordinates, J. Comput. Phys., 291 (2015), 279-302. doi: 10.1016/j.jcp.2015.02.050.

[13]

M. G. T. Fechner, Über Umkehrungen der Polarität in der einfachen Kette, J. Chem. Phys., 23 (1828), 61-77. 

[14]

M. Feinberg, Foundations of Chemical Reaction Network Theory, Applied Mathematical Sciences, 202, Springer, Cham, 2019. doi: 10.1007/978-3-030-03858-8.

[15]

R. J. Field, Oregonator, Scholarpedia, 2 (2007). doi: 10.4249/scholarpedia.1386.

[16]

R. J. Field and R. M. Noyes, Oscillations in chemical systems. Ⅳ. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys., 60 (1974), 1877-1884.  doi: 10.1063/1.1681288.

[17]

B. Fiedler, Global Hopf bifurcation in porous catalysts, in Equadiff 82, Lecture Notes in Math., 1017, Springer, Berlin, 1983,177-184. doi: 10.1007/BFb0103250.

[18]

B. Fiedler, An index for global Hopf bifurcation in parabolic systems, J. Reine Angew. Math., 359 (1985), 1-36.  doi: 10.1515/crll.1985.359.1.

[19]

B. Fiedler, Global Hopf bifurcation of two-parameter flows, Arch. Rational Mech. Anal., 94 (1986), 59-81.  doi: 10.1007/BF00278243.

[20]

B. Fiedler, Global Bifurcation of Periodic Solutions with Symmetry, Lecture Notes in Mathematics, 1309, Springer-Verlag, Berlin, 1988. doi: 10.1007/BFb0082943.

[21]

B. Fiedler, Global pathfollowing of homoclinic orbits in two-parameter flows, in Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation and Stability, Research Notes in Mathematics Series, 352, Longman, Harlow, 1996, 79-147.

[22]

B. FiedlerA. MochizukiG. Kurosawa and D. Saito, Dynamics and control at feedback vertex sets. Ⅰ: Informative and determining nodes in regulatory networks, J. Dynam. Differential Equations, 25 (2013), 563-604.  doi: 10.1007/s10884-013-9312-7.

[23]

G. Fusco and W. Oliva, Jacobi matrices and transversality, Proc. Roy. Soc. Edinburgh Sect. A, 109 (1988), 231-243.  doi: 10.1017/S0308210500027748.

[24]

K. GatermannM. Eiswirth and A. Sensse, Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems, J. Symbolic Comput., 40 (2005), 1361-1382.  doi: 10.1016/j.jsc.2005.07.002.

[25]

P. Gurevich, J. Hell, B. Sandstede and A. Scheel, Patterns of Dynamics, Springer Proceedings in Mathematics & Statistics, 205, Springer, Cham, 2017. doi: 10.1007/978-3-319-64173-7.

[26]

H. L. Heathcoate, Vorläufiger Bericht über Passivierung. Passivität und Aktivierung des Eisens, Z. Phys. Chem., 37U (1901), 368-373.  doi: 10.1515/zpch-1901-3720.

[27]

J. Higgins, The theory of oscillating reactions, Ind. Eng. Chem., 59 (1967), 18-62.  doi: 10.1021/ie50689a006.

[28]

J. Hirniak, Zur Frage der periodischen Reaktionen, Z. Phys. Chem., 75U (1910), 675-680.  doi: 10.1515/zpch-1911-7549.

[29] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179.
[30]

E. Hopf, Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Nat. Kl., 95 (1943), 3-22. 

[31]

F. Horn and R. Jackson, General mass action kinetics, Arch. Rational Mech. Anal., 47 (1972), 81-116.  doi: 10.1007/BF00251225.

[32]

R. Imbihl and G. Ertl, Oscillatory kinetics in heterogeneous catalysis, Chem. Rev., 95 (1995), 697-733.  doi: 10.1021/cr00035a012.

[33]

J. Ize and A. Vignoli, Equivariant Degree Theory, De Gruyter Series in Nonlinear Analysis and Applications, 8, Walter de Gruyter & Co., Berlin, 2003. doi: 10.1515/9783110200027.

[34]

C. JeffriesV. Klee and P. van den Driessche, When is a matrix sign stable?, Canadian J. Math., 29 (1977), 315-326.  doi: 10.4153/CJM-1977-035-3.

[35]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, 132, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-66282-9.

[36]

K. Lalchhandama, The path to the 2017 Nobel Prize in physiology or medicine, Sci. Vis., 17 (2017), 1-13. 

[37]

R. Lefever, Dissipative structures in chemical systems, J. Chem. Phys., 49 (1968), 4977-4978.  doi: 10.1063/1.1669986.

[38]

A. J. Lotka, Analytical note on certain rhythmic relations in organic systems, Proc. Nat. Acad. Sc. 6 (1920), 410-415. doi: 10.1073/pnas.6.7.410.

[39]

M. J. Macdonald and L. A. Fahien, et al., Citrate oscillates in liver and pancreatic beta cell mitochondria and in INS-1 insulinoma cells, J. Biol. Chem., 278 (2003), 51894-51900. doi: 10.1074/jbc.M309038200.

[40]

J. Mallet-Paret and H. L. Smith, The Poincaré-Bendixson theorem for monotone cyclic feedback systems, J. Dynam. Differential Equations, 2 (1990), 367-421.  doi: 10.1007/BF01054041.

[41]

J. Mallet-Paret and J. A. Yorke, Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation, J. Differential Equations, 43 (1982), 419-450.  doi: 10.1016/0022-0396(82)90085-7.

[42]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, 19, Springer-Verlag, New York, 1976. doi: 10.1007/978-1-4612-6374-6.

[43]

A. Mielke, Uniform exponential decay for reaction-diffusion systems with complex-balanced mass-action kinetics, in Patterns of Dynamics, Springer Proc. Math. Stat., 205, Springer, Cham, 2017,149--171. doi: 10.1007/978-3-319-64173-7_10.

[44]

H. P. Mirsky and A. C. Liu, et al., A model of the cell-autonomous mammalian circadian clock, PNAS, 27 (2009), 11107-11112. doi: 10.1073/pnas.0904837106.

[45]

A. MochizukiB. FiedlerG. Kurosawa and D. Saito, Dynamics and control at feedback vertex sets. Ⅱ: A faithful monitor to determine the diversity of molecular activities in regulatory networks, J. Theoret. Biol., 335 (2013), 130-146.  doi: 10.1016/j.jtbi.2013.06.009.

[46]

W. M. Oliva, A survey on stably dissipative Lotka-Volterra systems with an application to infinite dimensional Volterra equations, Publ. Mat., 58 (2014), 421-452.  doi: 10.5565/PUBLMAT_Extra14_21.

[47]

I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems. Ⅱ, J. Chem. Phys., 48 (1968), 1695-1700.  doi: 10.1063/1.1668896.

[48] E. Schrödinger, What is Life?, Cambridge University Press, 1944. 
[49]

E. E. Sel'kov, Self-oscillations in glycolysis. 1. A simple kinetic model, European J. Biochem., 4 (1968), 79-86.  doi: 10.1111/j.1432-1033.1968.tb00175.x.

[50]

L. P. Shilnikov, A. L. Shilnikov and D. V. Turaev, Showcase of blue sky catastrophes, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014), 10pp. doi: 10.1142/S0218127414400033.

[51]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[52]

N. Vassena, Sensitivity of Metabolic Networks, Ph.D thesis, Freie Universität in Berlin, 2020.

[53]

V. Volterra, Leçons sur la Théorie Mathématique de la Lutte pour la Vie, Éditions Jacques Gabay, Sceaux, 1990.

[54]

R. Wegscheider, Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reaktionskinetik homogener Systeme, Z. Phys. Chem., 39U (1902), 257-303.  doi: 10.1515/zpch-1902-3919.

[55]

A. M. Zhabotinsky, Belousov-Zhabotinsky reaction, Scholarpedia, 2 (2007). doi: 10.4249/scholarpedia.1435.

[56]

A. M. Zhabotinsky, Periodical oxidation of malonic acid in solution (a study of the Belousov reaction kinetics), Biofizika, 9 (1964), 306-311. 

[57]

A. M. Zhabotinsky, A history of chemical oscillations and waves, Chaos, 1 (1991), 379-386.  doi: 10.1063/1.165848.

show all references

References:
[1]

J. C. Alexander and J. A. Yorke, Global bifurcation of periodic orbits, Amer. J. Math., 100 (1978), 263-292.  doi: 10.2307/2373851.

[2]

N. D. Alikakos, An application of the invariance principle to reaction-diffusion equations, J. Differential Equations, 33 (1979), 201-225.  doi: 10.1016/0022-0396(79)90088-3.

[3]

K. T. Alligood and J. A. Yorke, Families of periodic orbits: Virtual period and global continuability, J. Differential Equations, 55 (1984), 59-71.  doi: 10.1016/0022-0396(84)90088-3.

[4] R. Aris, The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. Vol. Ⅰ, Ⅱ, Clarendon Press, Oxford, 1975. 
[5]

J. M. Berg, J. L. Tymoczko, G. J. Gatto and L. Stryer, Biochemistry, W. H. Freeman, New York 2015.

[6]

W. C. Bray, A periodic reaction in homogeneous solution and its relation to catalysis, J. Amer. Chem. Soc., 43 (1921), 1262-1267.  doi: 10.1021/ja01439a007.

[7]

, Comprehensive Chemical Kinetics, 1-43, Elsevier, 1969-2019.

[8]

B. Brehm and B. Fiedler, Sensitivity of chemical reaction networks: A structural approach. 3. Regular multimolecular systems, Math. Methods Appl. Sci., 41 (2018), 1344-1376.  doi: 10.1002/mma.4668.

[9]

M. G. Crandall and P. H. Rabinowitz, The Hopf bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal., 67 (1977), 53-72.  doi: 10.1007/BF00280827.

[10]

G. Dangelmayr, B. Fiedler, K. Kirchgässner and A. Mielke, Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation and Stability, Research Notes in Mathematics Series, 352, Longman, Harlow, 1996.

[11]

M. Eigen, Selforganization of matter and the evolution of biological macromolecules, Naturwissenschaften, 58 (1971), 465-523.  doi: 10.1007/BF00623322.

[12]

H. Errami and M. Eiswirth, et al., Detection of Hopf bifurcations in chemical reaction networks using convex coordinates, J. Comput. Phys., 291 (2015), 279-302. doi: 10.1016/j.jcp.2015.02.050.

[13]

M. G. T. Fechner, Über Umkehrungen der Polarität in der einfachen Kette, J. Chem. Phys., 23 (1828), 61-77. 

[14]

M. Feinberg, Foundations of Chemical Reaction Network Theory, Applied Mathematical Sciences, 202, Springer, Cham, 2019. doi: 10.1007/978-3-030-03858-8.

[15]

R. J. Field, Oregonator, Scholarpedia, 2 (2007). doi: 10.4249/scholarpedia.1386.

[16]

R. J. Field and R. M. Noyes, Oscillations in chemical systems. Ⅳ. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys., 60 (1974), 1877-1884.  doi: 10.1063/1.1681288.

[17]

B. Fiedler, Global Hopf bifurcation in porous catalysts, in Equadiff 82, Lecture Notes in Math., 1017, Springer, Berlin, 1983,177-184. doi: 10.1007/BFb0103250.

[18]

B. Fiedler, An index for global Hopf bifurcation in parabolic systems, J. Reine Angew. Math., 359 (1985), 1-36.  doi: 10.1515/crll.1985.359.1.

[19]

B. Fiedler, Global Hopf bifurcation of two-parameter flows, Arch. Rational Mech. Anal., 94 (1986), 59-81.  doi: 10.1007/BF00278243.

[20]

B. Fiedler, Global Bifurcation of Periodic Solutions with Symmetry, Lecture Notes in Mathematics, 1309, Springer-Verlag, Berlin, 1988. doi: 10.1007/BFb0082943.

[21]

B. Fiedler, Global pathfollowing of homoclinic orbits in two-parameter flows, in Dynamics of Nonlinear Waves in Dissipative Systems: Reduction, Bifurcation and Stability, Research Notes in Mathematics Series, 352, Longman, Harlow, 1996, 79-147.

[22]

B. FiedlerA. MochizukiG. Kurosawa and D. Saito, Dynamics and control at feedback vertex sets. Ⅰ: Informative and determining nodes in regulatory networks, J. Dynam. Differential Equations, 25 (2013), 563-604.  doi: 10.1007/s10884-013-9312-7.

[23]

G. Fusco and W. Oliva, Jacobi matrices and transversality, Proc. Roy. Soc. Edinburgh Sect. A, 109 (1988), 231-243.  doi: 10.1017/S0308210500027748.

[24]

K. GatermannM. Eiswirth and A. Sensse, Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems, J. Symbolic Comput., 40 (2005), 1361-1382.  doi: 10.1016/j.jsc.2005.07.002.

[25]

P. Gurevich, J. Hell, B. Sandstede and A. Scheel, Patterns of Dynamics, Springer Proceedings in Mathematics & Statistics, 205, Springer, Cham, 2017. doi: 10.1007/978-3-319-64173-7.

[26]

H. L. Heathcoate, Vorläufiger Bericht über Passivierung. Passivität und Aktivierung des Eisens, Z. Phys. Chem., 37U (1901), 368-373.  doi: 10.1515/zpch-1901-3720.

[27]

J. Higgins, The theory of oscillating reactions, Ind. Eng. Chem., 59 (1967), 18-62.  doi: 10.1021/ie50689a006.

[28]

J. Hirniak, Zur Frage der periodischen Reaktionen, Z. Phys. Chem., 75U (1910), 675-680.  doi: 10.1515/zpch-1911-7549.

[29] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, Cambridge, 1998.  doi: 10.1017/CBO9781139173179.
[30]

E. Hopf, Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems, Ber. Verh. Sächs. Akad. Wiss. Leipzig Math.-Nat. Kl., 95 (1943), 3-22. 

[31]

F. Horn and R. Jackson, General mass action kinetics, Arch. Rational Mech. Anal., 47 (1972), 81-116.  doi: 10.1007/BF00251225.

[32]

R. Imbihl and G. Ertl, Oscillatory kinetics in heterogeneous catalysis, Chem. Rev., 95 (1995), 697-733.  doi: 10.1021/cr00035a012.

[33]

J. Ize and A. Vignoli, Equivariant Degree Theory, De Gruyter Series in Nonlinear Analysis and Applications, 8, Walter de Gruyter & Co., Berlin, 2003. doi: 10.1515/9783110200027.

[34]

C. JeffriesV. Klee and P. van den Driessche, When is a matrix sign stable?, Canadian J. Math., 29 (1977), 315-326.  doi: 10.4153/CJM-1977-035-3.

[35]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, 132, Springer-Verlag, Berlin, 1995. doi: 10.1007/978-3-642-66282-9.

[36]

K. Lalchhandama, The path to the 2017 Nobel Prize in physiology or medicine, Sci. Vis., 17 (2017), 1-13. 

[37]

R. Lefever, Dissipative structures in chemical systems, J. Chem. Phys., 49 (1968), 4977-4978.  doi: 10.1063/1.1669986.

[38]

A. J. Lotka, Analytical note on certain rhythmic relations in organic systems, Proc. Nat. Acad. Sc. 6 (1920), 410-415. doi: 10.1073/pnas.6.7.410.

[39]

M. J. Macdonald and L. A. Fahien, et al., Citrate oscillates in liver and pancreatic beta cell mitochondria and in INS-1 insulinoma cells, J. Biol. Chem., 278 (2003), 51894-51900. doi: 10.1074/jbc.M309038200.

[40]

J. Mallet-Paret and H. L. Smith, The Poincaré-Bendixson theorem for monotone cyclic feedback systems, J. Dynam. Differential Equations, 2 (1990), 367-421.  doi: 10.1007/BF01054041.

[41]

J. Mallet-Paret and J. A. Yorke, Snakes: Oriented families of periodic orbits, their sources, sinks, and continuation, J. Differential Equations, 43 (1982), 419-450.  doi: 10.1016/0022-0396(82)90085-7.

[42]

J. E. Marsden and M. McCracken, The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, 19, Springer-Verlag, New York, 1976. doi: 10.1007/978-1-4612-6374-6.

[43]

A. Mielke, Uniform exponential decay for reaction-diffusion systems with complex-balanced mass-action kinetics, in Patterns of Dynamics, Springer Proc. Math. Stat., 205, Springer, Cham, 2017,149--171. doi: 10.1007/978-3-319-64173-7_10.

[44]

H. P. Mirsky and A. C. Liu, et al., A model of the cell-autonomous mammalian circadian clock, PNAS, 27 (2009), 11107-11112. doi: 10.1073/pnas.0904837106.

[45]

A. MochizukiB. FiedlerG. Kurosawa and D. Saito, Dynamics and control at feedback vertex sets. Ⅱ: A faithful monitor to determine the diversity of molecular activities in regulatory networks, J. Theoret. Biol., 335 (2013), 130-146.  doi: 10.1016/j.jtbi.2013.06.009.

[46]

W. M. Oliva, A survey on stably dissipative Lotka-Volterra systems with an application to infinite dimensional Volterra equations, Publ. Mat., 58 (2014), 421-452.  doi: 10.5565/PUBLMAT_Extra14_21.

[47]

I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems. Ⅱ, J. Chem. Phys., 48 (1968), 1695-1700.  doi: 10.1063/1.1668896.

[48] E. Schrödinger, What is Life?, Cambridge University Press, 1944. 
[49]

E. E. Sel'kov, Self-oscillations in glycolysis. 1. A simple kinetic model, European J. Biochem., 4 (1968), 79-86.  doi: 10.1111/j.1432-1033.1968.tb00175.x.

[50]

L. P. Shilnikov, A. L. Shilnikov and D. V. Turaev, Showcase of blue sky catastrophes, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 24 (2014), 10pp. doi: 10.1142/S0218127414400033.

[51]

A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.

[52]

N. Vassena, Sensitivity of Metabolic Networks, Ph.D thesis, Freie Universität in Berlin, 2020.

[53]

V. Volterra, Leçons sur la Théorie Mathématique de la Lutte pour la Vie, Éditions Jacques Gabay, Sceaux, 1990.

[54]

R. Wegscheider, Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reaktionskinetik homogener Systeme, Z. Phys. Chem., 39U (1902), 257-303.  doi: 10.1515/zpch-1902-3919.

[55]

A. M. Zhabotinsky, Belousov-Zhabotinsky reaction, Scholarpedia, 2 (2007). doi: 10.4249/scholarpedia.1435.

[56]

A. M. Zhabotinsky, Periodical oxidation of malonic acid in solution (a study of the Belousov reaction kinetics), Biofizika, 9 (1964), 306-311. 

[57]

A. M. Zhabotinsky, A history of chemical oscillations and waves, Chaos, 1 (1991), 379-386.  doi: 10.1063/1.165848.

[1]

Casian Pantea, Heinz Koeppl, Gheorghe Craciun. Global injectivity and multiple equilibria in uni- and bi-molecular reaction networks. Discrete and 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 and Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523

[4]

Ginestra Bianconi, Riccardo Zecchina. Viable flux distribution in metabolic networks. Networks and Heterogeneous Media, 2008, 3 (2) : 361-369. doi: 10.3934/nhm.2008.3.361

[5]

Klemens Fellner, Wolfang Prager, Bao Q. Tang. The entropy method for reaction-diffusion systems without detailed balance: First order chemical reaction networks. Kinetic and Related Models, 2017, 10 (4) : 1055-1087. doi: 10.3934/krm.2017042

[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 and 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and 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 and Continuous Dynamical Systems, 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 and 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 and 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 and Continuous Dynamical Systems, 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 and Applied Analysis, 2018, 17 (5) : 2139-2147. doi: 10.3934/cpaa.2018102

[19]

Peng Feng, Menaka Navaratna. Modelling periodic oscillations during somitogenesis. Mathematical Biosciences & Engineering, 2007, 4 (4) : 661-673. doi: 10.3934/mbe.2007.4.661

[20]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (238)
  • HTML views (265)
  • Cited by (0)

Other articles
by authors

[Back to Top]