Phase portraits of the Selkov model in the Poincaré disc

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doi: 10.3934/dcdsb.2022056

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Phase portraits of the Selkov model in the Poincaré disc

Jaume Llibre ^1,, and Arefeh Nabavi ^2,

1.	Departament de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
2.	Department of Mathematical Sciences, Isfahan University of Technology, 84156-83111, Isfahan, Iran

^*Corresponding author: Jaume Llibre

Received November 2021 Early access March 2022

Fund Project: This work was partially supported by the Agencia Estatal de Investigación grants PID2019-104658GB-I00, and the H2020 European Research Council grant MSCA-RISE-2017-777911

In this paper we classify the phase portraits in the Poincaré disc of the Selkov model for the glycolysis process

$ \begin{equation*} \dot{x}\, = \, -x+a y+x^2 y, \quad \dot{y} \, = \, b-a y-x^2 y, \end{equation*} $

in function of its parameters

$ a, b \in \mathbb{R} $

. In particular we determine the regions in the parameter plane with biological meaning, i.e. with

$ a $

,

$ b $

,

$ x $

and

$ y $

positive.

Keywords: Selkov model, Phase portrait, Poincaré compactification.

Mathematics Subject Classification: Primary 34C05; Secondary 34C07, 37C08.

Citation: Jaume Llibre, Arefeh Nabavi. Phase portraits of the Selkov model in the Poincaré disc. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022056

References:

[1]	K. S. Al Noufaey and T. R. Marchant, Semi-analytical solutions for the reversible Selkov model withfeedback delay, Appl. Math. Comput., 232 (2014), 49–59. doi: 10.1016/j.amc.2014.01.059.
[2]	M. J. Álvarez, A. Ferragut and X. Jarque, A survey on the blow up technique, Int. J. Bifurc Chaos., 21 (2011), 3103–3118. doi: 10.1142/S0218127411030416.
[3]	J. C. Artés, J. Llibre and C. Valls, Dynamics of the Higgings–Selkov and Selkov systems, Chaos, Solitons and Fractals, 114 (2018), 145–150. doi: 10.1016/j.chaos.2018.07.007.
[4]	P. Brechmann and A. D. Rendall, Dynamics of the Selkov oscillator, Math. Biosci., 306 (2018), 152–159. doi: 10.1016/j.mbs.2018.09.012.
[5]	H. Chen and Y. Tang, Proof of Artés-Llibre-Valls's conjectures for the Higgins-Selkov systems, J. Differential Equations, 266 (2019), 7638–7657. doi: 10.1016/j.jde.2018.12.011.
[6]	Q. Din, Bifurcation analysis and chaos control in discrete-time glycolysis models, J. Math. Chem., 56 (2018), 904–931. doi: 10.1007/s10910-017-0839-4.
[7]	F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, New York, 2006.
[8]	Y. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, Vol. 112, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.
[9]	L. Markus, Global structure of ordinary differential equations in the plane, Trans. Amer. Math. Soc., 76 (1954), 127–148. doi: 10.1090/S0002-9947-1954-0060657-0.
[10]	D. A. Neumann, Classification of continuous flows on 2-manifolds, Proc. Amer. Math. Soc., 48 (1975), 73–81. doi: 10.1090/S0002-9939-1975-0356138-6.
[11]	M. M. Peixoto, Proccedings of a simposium held at the university of Bahia, Acad. Press, New York, 1973,349–420.
[12]	P. C. Rech, Organization of the periodicity in the parameter-space of a glycolysis discrete-time mathematical model, J. Math. Chem., 57 (2019), 632–637. doi: 10.1007/s10910-018-0976-4.
[13]	E. Selkov, Model of glycolytic oscillations, Eur. J. Biochem., 4 (1968), 79–86.
[14]	S. H. Strogatz, Nonlinear Dynamics and Chaos, Mathematics & Statistics, 2019. doi: 10.1201/9780429492563.

show all references

References:

[1]	K. S. Al Noufaey and T. R. Marchant, Semi-analytical solutions for the reversible Selkov model withfeedback delay, Appl. Math. Comput., 232 (2014), 49–59. doi: 10.1016/j.amc.2014.01.059.
[2]	M. J. Álvarez, A. Ferragut and X. Jarque, A survey on the blow up technique, Int. J. Bifurc Chaos., 21 (2011), 3103–3118. doi: 10.1142/S0218127411030416.
[3]	J. C. Artés, J. Llibre and C. Valls, Dynamics of the Higgings–Selkov and Selkov systems, Chaos, Solitons and Fractals, 114 (2018), 145–150. doi: 10.1016/j.chaos.2018.07.007.
[4]	P. Brechmann and A. D. Rendall, Dynamics of the Selkov oscillator, Math. Biosci., 306 (2018), 152–159. doi: 10.1016/j.mbs.2018.09.012.
[5]	H. Chen and Y. Tang, Proof of Artés-Llibre-Valls's conjectures for the Higgins-Selkov systems, J. Differential Equations, 266 (2019), 7638–7657. doi: 10.1016/j.jde.2018.12.011.
[6]	Q. Din, Bifurcation analysis and chaos control in discrete-time glycolysis models, J. Math. Chem., 56 (2018), 904–931. doi: 10.1007/s10910-017-0839-4.
[7]	F. Dumortier, J. Llibre and J. C. Artés, Qualitative Theory of Planar Differential Systems, Springer Verlag, New York, 2006.
[8]	Y. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, Vol. 112, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.
[9]	L. Markus, Global structure of ordinary differential equations in the plane, Trans. Amer. Math. Soc., 76 (1954), 127–148. doi: 10.1090/S0002-9947-1954-0060657-0.
[10]	D. A. Neumann, Classification of continuous flows on 2-manifolds, Proc. Amer. Math. Soc., 48 (1975), 73–81. doi: 10.1090/S0002-9939-1975-0356138-6.
[11]	M. M. Peixoto, Proccedings of a simposium held at the university of Bahia, Acad. Press, New York, 1973,349–420.
[12]	P. C. Rech, Organization of the periodicity in the parameter-space of a glycolysis discrete-time mathematical model, J. Math. Chem., 57 (2019), 632–637. doi: 10.1007/s10910-018-0976-4.
[13]	E. Selkov, Model of glycolytic oscillations, Eur. J. Biochem., 4 (1968), 79–86.
[14]	S. H. Strogatz, Nonlinear Dynamics and Chaos, Mathematics & Statistics, 2019. doi: 10.1201/9780429492563.

Figure 1. Phase portraits of system (1) with $ a\in \mathbb{R} $ and $ b\ge 0 $ in the Poincaré disc. The shaded areas correspond to the initial conditions of the orbits having a finite final evolution, so these are the initial conditions with biological meaning. In the phase portraits $ E $ and $ I $ the final behaviour is a stable limit cycle, while in the phase portraits $ F $, $ G $ and $ H $ the final behaviour is a singular point

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Figure 2. The regions $ R_1 $ and $ R_2 $ where the differential system (1) has limit cycles computed numerically

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Figure 3. Regions and curves separating the different local phase portraits of the finite singular point

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Figure 4. Blowing down of the origin of the local chart $ U_2 $ for $ a \ge 0 $

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Figure 5. Blowing down of the origin of the local chart $ U_2 $ for $ a<-b^2 $, $ b \ge 0 $

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Figure 6. Blowing down of the origin of the local chart $ U_2 $ for $ a = -b^2 $, $ b \neq 0 $

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Figure 7. Blowing down of the origin of the local chart $ U_2 $ for $ a \in (-b^2, 0) $

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Figure 8. The regions $ I $, $ II $, $ III $, the point $ (b, a) = (0, 0) $, and the curve $ \gamma:\lbrace a = -b^2, b\neq 0\rbrace $ separating the different local phase portraits at the origin of the local chart $ U_2 $ in the half plane $ \{(b, a): b\ge 0, a\in {\mathbb{R}}\} $

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Figure 9. Bifurcation diagram

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Figure 10. Local phase portraits at the finite and infinite singular points in the Poincaré disc

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Figure 11. The phase portrait of system (4)

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Figure 12. The possible global phase portraits taken from the local phase portrait in the region $ R_{12} $

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Figure 13. The direction of vector field for the values of the parameters located in the region $ R_{12} $. The red and blue curves are respectively the graphs of $ y = b/(a+x^2) $ and $ y = x/(a+x^2) $

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Figure 14. The possible global phase portraits taken from the local phase portrait (Figure 10), for the values of parameters on $ P_3 $, $ P_4 $, $ L_4 $, $ L_6 $, $ L_8 $ and $ R_4 $

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Figure 15. The possible global phase portraits taken from the local phase portrait in the region $ R_6 $ (Figure 10)

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Figure 16. The direction of vector field for the values of parameters located in the region $ R_6 $. The red and blue curves are respectively the graphs of $ y = b/(a+x^2) $ and $ y = x/(a+x^2) $

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Table 1. We provide the local phase portrait of the finite singular point when the parameters of the differential system (1) are on the curves or in the regions of Figure 3

Curve	Region	finite singular point
$ \alpha_1 $, $ \alpha_2 $	(1), (2)	stable node
$ \beta $ ($ b\le b_2 $)	(3)	stable focus
$ \beta $ ($ b>b_2 $)	(4)	unstable focus
$ \alpha_3 $	(5)	unstable node
$ \gamma $		empty
	(6)	saddle

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Curve	Region	finite singular point
$ \alpha_1 $, $ \alpha_2 $	(1), (2)	stable node
$ \beta $ ($ b\le b_2 $)	(3)	stable focus
$ \beta $ ($ b>b_2 $)	(4)	unstable focus
$ \alpha_3 $	(5)	unstable node
$ \gamma $		empty
	(6)	saddle

Table 2. Some approximation of the points on the curves $ \delta_1 $ and $ \delta_2 $

a	b	on the curve
$ 0 $	$ (0.9, 0.91):=b_1 $	$ \delta_1 $
$ -0.001 $	$ (0.904, 0.905) $	$ \delta_1 $
$ -0.15 $	$ (1.168, 1.169) $	$ \delta_1 $
$ -1 $	$ (1.749, 1.75) $	$ \delta_2 $
$ -2 $	$ (2.154, 2.155) $	$ \delta_2 $
$ -3 $	$ (2.467, 2.468) $	$ \delta_2 $

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a	b	on the curve
$ 0 $	$ (0.9, 0.91):=b_1 $	$ \delta_1 $
$ -0.001 $	$ (0.904, 0.905) $	$ \delta_1 $
$ -0.15 $	$ (1.168, 1.169) $	$ \delta_1 $
$ -1 $	$ (1.749, 1.75) $	$ \delta_2 $
$ -2 $	$ (2.154, 2.155) $	$ \delta_2 $
$ -3 $	$ (2.467, 2.468) $	$ \delta_2 $

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