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

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 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.
Citation: Jaume Llibre, Marzieh Mousavi. Phase portraits of the Higgins–Selkov system. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 245-256. 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.

[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.

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

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.

[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.

[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.

[4]

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

[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.

[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.

[7]

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

[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.

[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.

[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.

[11]

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

[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.

[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.

[14]

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

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
[1]

Jaume Llibre, Arefeh Nabavi. Phase portraits of the Selkov model in the Poincaré disc. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022056

[2]

Hebai Chen, Xingwu Chen, Jianhua Xie. Global phase portrait of a degenerate Bogdanov-Takens system with symmetry. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1273-1293. doi: 10.3934/dcdsb.2017062

[3]

Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047

[4]

Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete and Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893

[5]

Antonio Garijo, Armengol Gasull, Xavier Jarque. Local and global phase portrait of equation $\dot z=f(z)$. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 309-329. doi: 10.3934/dcds.2007.17.309

[6]

Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114

[7]

Hayato Chiba. Continuous limit and the moments system for the globally coupled phase oscillators. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1891-1903. doi: 10.3934/dcds.2013.33.1891

[8]

Pierluigi Colli, Gianni Gilardi, Pavel Krejčí, Jürgen Sprekels. A vanishing diffusion limit in a nonstandard system of phase field equations. Evolution Equations and Control Theory, 2014, 3 (2) : 257-275. doi: 10.3934/eect.2014.3.257

[9]

Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete and Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439

[10]

Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447

[11]

Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure and Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1

[12]

Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123

[13]

Stijn Luca, Freddy Dumortier, Magdalena Caubergh, Robert Roussarie. Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. Discrete and Continuous Dynamical Systems, 2009, 25 (4) : 1081-1108. doi: 10.3934/dcds.2009.25.1081

[14]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure and Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257

[15]

Yuncheng You. Asymptotical dynamics of Selkov equations. Discrete and Continuous Dynamical Systems - S, 2009, 2 (1) : 193-219. doi: 10.3934/dcdss.2009.2.193

[16]

Fang Wu, Lihong Huang, Jiafu Wang. Bifurcation of the critical crossing cycle in a planar piecewise smooth system with two zones. Discrete and Continuous Dynamical Systems - B, 2022, 27 (9) : 5047-5083. doi: 10.3934/dcdsb.2021264

[17]

Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803

[18]

Bourama Toni. Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization. Conference Publications, 2005, 2005 (Special) : 846-853. doi: 10.3934/proc.2005.2005.846

[19]

Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129

[20]

Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5581-5599. doi: 10.3934/dcdsb.2020368

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (393)
  • HTML views (489)
  • Cited by (0)

Other articles
by authors

[Back to Top]