doi: 10.3934/dcdsb.2021039
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Phase portraits of the Higgins–Selkov system

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 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.
Citation: Jaume Llibre, Marzieh Mousavi. Phase portraits of the Higgins–Selkov system. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021039
References:
[1]

M. J. ÁlvarezA. 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. ArteśJ. 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

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H. ChenJ. 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

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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. ÁlvarezA. 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. ArteśJ. 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. ChenJ. 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
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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