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  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, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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. Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

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

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[52]

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

[53]

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

[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.  Google Scholar

[55]

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

[56]

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

[57]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[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.  Google Scholar

[13]

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

[14]

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

[15]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

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

[20]

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

[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. Google Scholar

[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.  Google Scholar

[23]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

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

[28]

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

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

[31]

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

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[36]

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

[37]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

[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.  Google Scholar

[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.  Google Scholar

[52]

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

[53]

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

[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.  Google Scholar

[55]

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

[56]

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

[57]

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

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