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January  2022, 27(1): 245-256. doi: 10.3934/dcdsb.2021039

Phase portraits of the Higgins–Selkov system

Jaume Llibre 1, and  Marzieh Mousavi 2,,

1. 

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain
2. 

Department of Mathematical Sciences, Isfahan University of Technology, Isfahan 84156-83111, Iran

* Corresponding author: Marzieh Mousavi

Received  August 2020 Revised  November 2020 Published  January 2022 Early access  January 2021

Fund Project: The first author is partially supported by the Ministerio de Economìa, Industria y competitividad, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d'Ajusts Universitaris i de Recerca grant 2017 SGR 1617, and the European project Dynamics-H2020-MSCA-RISE-2017-777911. The second author is supported by Isfahan University of Technology (IUT)

In this paper we study the dynamics of the Higgins–Selkov system
$ \begin{equation*} \dot{x} = 1-xy^\gamma, \quad\dot{y} = \alpha y(xy^{\gamma -1}-1), \end{equation*} $
where
$ \alpha $
 is a real parameter and
$ \gamma>1 $
 is an integer. We classify the phase portraits of this system for
$ \gamma = 3, 4, 5, 6, $
 in the Poincaré disc for all the values of the parameter
$ \alpha $
. Moreover, we determine in function of the parameter
$ \alpha $
 the regions of the phase space with biological meaning.
Keywords: Higgins–Selkov system, limit cycle, phase portrait, poincaré compactification.
Mathematics Subject Classification: Primary: 34C05, 34C07, 34C08.
Citation: Jaume Llibre, Marzieh Mousavi. Phase portraits of the Higgins–Selkov system. Discrete & Continuous Dynamical Systems - B, 2022, 27 (1) : 245-256. doi: 10.3934/dcdsb.2021039
References:
[1]

M. J. Álvarez, A. Ferragut and X. Jarque, A survey on the blow up technique, Int. J. Bifurcation and Chaos, 21 (2011), 3103-3118.  doi: 10.1142/S0218127411030416.  Google Scholar

[2]

J. C. Arteś, J. Llibre and C. Valls, Dynamics of the Higgins–Selkov and Selkov system, Chaos. Sol. Frac., 114 (2018), 145-150.  doi: 10.1016/j.chaos.2018.07.007.  Google Scholar

[3]

P. Brechmann and A. D. Rendall, Dynamics of the Selkov oscillator, Mathematical Biosciences, 306 (2018), 152-159.  doi: 10.1016/j.mbs.2018.09.012.  Google Scholar

[4]

P. Brechmann and A. D. Rendall, Unbounded solutions of models for glycolysis, preprint, arXiv: 2003.07140. Google Scholar

[5]

H. Chen, J. Llibre and Y. Tang, Global dynamics of a SD oscillator, Nonlinear Dynamics, 91 (2018), 1755-1777.  doi: 10.1007/s11071-017-3979-y.  Google Scholar

[6]

H. Chen and Y. Tang, Proof of Arté-Llibre-Valls's conjectures for the Higgins–Selkov and Selkov systems, J. Differential Equations, 266 (2019), 7638-7657.  doi: 10.1016/j.jde.2018.12.011.  Google Scholar

[7]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, Berlin, 2006.  Google Scholar

[8]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, in: Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

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J. Higgins, A chemical mechanism for oscillation of glycolytic intermediates in yeast cells, Proc. Natl. Acad. Sci. (USA), 51 (1964), 989-994.  doi: 10.1073/pnas.51.6.989.  Google Scholar

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T.-W. Hwang and H.-J. Tsai, Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions, J. Phys. A, 38 (2005), 8211-8223.  doi: 10.1088/0305-4470/38/38/003.  Google Scholar

[11]

L. Perko, Differential Equations and Dynamical Systems, 3rd Ed., Springer, 2001. doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[12]

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

[13]

L. Yang, Recent advances on determining the number of real roots of parametric polynomials, J. Symb. Comput., 28 (1999), 225-242.  doi: 10.1006/jsco.1998.0274.  Google Scholar

[14]

Z. Zhang, T. Ding and W. Huang, Qualitative Theory of Differential Equations, Transl. Math. Monogr. Amer. Soc., Providence, RI, 1992. Google Scholar

show all references

References:
[1]

M. J. Álvarez, A. Ferragut and X. Jarque, A survey on the blow up technique, Int. J. Bifurcation and Chaos, 21 (2011), 3103-3118.  doi: 10.1142/S0218127411030416.  Google Scholar

[2]

J. C. Arteś, J. Llibre and C. Valls, Dynamics of the Higgins–Selkov and Selkov system, Chaos. Sol. Frac., 114 (2018), 145-150.  doi: 10.1016/j.chaos.2018.07.007.  Google Scholar

[3]

P. Brechmann and A. D. Rendall, Dynamics of the Selkov oscillator, Mathematical Biosciences, 306 (2018), 152-159.  doi: 10.1016/j.mbs.2018.09.012.  Google Scholar

[4]

P. Brechmann and A. D. Rendall, Unbounded solutions of models for glycolysis, preprint, arXiv: 2003.07140. Google Scholar

[5]

H. Chen, J. Llibre and Y. Tang, Global dynamics of a SD oscillator, Nonlinear Dynamics, 91 (2018), 1755-1777.  doi: 10.1007/s11071-017-3979-y.  Google Scholar

[6]

H. Chen and Y. Tang, Proof of Arté-Llibre-Valls's conjectures for the Higgins–Selkov and Selkov systems, J. Differential Equations, 266 (2019), 7638-7657.  doi: 10.1016/j.jde.2018.12.011.  Google Scholar

[7]

F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, Berlin, 2006.  Google Scholar

[8]

J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, in: Applied Mathematical Sciences, vol. 42, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.  Google Scholar

[9]

J. Higgins, A chemical mechanism for oscillation of glycolytic intermediates in yeast cells, Proc. Natl. Acad. Sci. (USA), 51 (1964), 989-994.  doi: 10.1073/pnas.51.6.989.  Google Scholar

[10]

T.-W. Hwang and H.-J. Tsai, Uniqueness of limit cycles in theoretical models of certain oscillating chemical reactions, J. Phys. A, 38 (2005), 8211-8223.  doi: 10.1088/0305-4470/38/38/003.  Google Scholar

[11]

L. Perko, Differential Equations and Dynamical Systems, 3rd Ed., Springer, 2001. doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[12]

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

[13]

L. Yang, Recent advances on determining the number of real roots of parametric polynomials, J. Symb. Comput., 28 (1999), 225-242.  doi: 10.1006/jsco.1998.0274.  Google Scholar

[14]

Z. Zhang, T. Ding and W. Huang, Qualitative Theory of Differential Equations, Transl. Math. Monogr. Amer. Soc., Providence, RI, 1992. Google Scholar

Figure 1.  The phase portraits of system (1) for $ \gamma = 3\; \text{and}\; 5 $ in the Poincaré disc. The shaded areas correspond to the initial conditions of the orbits having a final finite evolution, so these are the initial conditions with biological meaning. In the phase portrait (c) the final behaviour is a stable singular point, and in the phase portrait (d) is a stable limit cycle
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Figure 2.  The phase portraits of system (1) for $ \gamma = 4\; \text{and}\; 6 $ in the Poincaré disc. The shaded areas correspond to the initial conditions of the orbits having a final finite evolution, so these are the initial conditions with biological meaning. In the phase portrait (c) the final behaviour is a stable singular point, and in the phase portrait (d) is a stable limit cycle
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