American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2021039
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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, 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

[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

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
Figure Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Hebai Chen, Xingwu Chen, Jianhua Xie. Global phase portrait of a degenerate Bogdanov-Takens system with symmetry. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1273-1293. doi: 10.3934/dcdsb.2017062
[2]

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 & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047
[3]

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

Antonio Garijo, Armengol Gasull, Xavier Jarque. Local and global phase portrait of equation $\dot z=f(z)$. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 309-329. doi: 10.3934/dcds.2007.17.309
[5]

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 & Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114
[6]

Hayato Chiba. Continuous limit and the moments system for the globally coupled phase oscillators. Discrete & Continuous Dynamical Systems, 2013, 33 (5) : 1891-1903. doi: 10.3934/dcds.2013.33.1891
[7]

Pierluigi Colli, Gianni Gilardi, Pavel Krejčí, Jürgen Sprekels. A vanishing diffusion limit in a nonstandard system of phase field equations. Evolution Equations & Control Theory, 2014, 3 (2) : 257-275. doi: 10.3934/eect.2014.3.257
[8]

Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439
[9]

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
[10]

Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure & Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1
[11]

Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123
[12]

Stijn Luca, Freddy Dumortier, Magdalena Caubergh, Robert Roussarie. Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1081-1108. doi: 10.3934/dcds.2009.25.1081
[13]

Meilan Cai, Maoan Han. Limit cycle bifurcations in a class of piecewise smooth cubic systems with multiple parameters. Communications on Pure & Applied Analysis, 2021, 20 (1) : 55-75. doi: 10.3934/cpaa.2020257
[14]

Yuncheng You. Asymptotical dynamics of Selkov equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 193-219. doi: 10.3934/dcdss.2009.2.193
[15]

Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803
[16]

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
[17]

Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129
[18]

Huanhuan Tian, Maoan Han. Limit cycle bifurcations of piecewise smooth near-Hamiltonian systems with a switching curve. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5581-5599. doi: 10.3934/dcdsb.2020368
[19]

Richard Evan Schwartz. Outer billiards on the Penrose kite: Compactification and renormalization. Journal of Modern Dynamics, 2011, 5 (3) : 473-581. doi: 10.3934/jmd.2011.5.473
[20]

Dorota Bors, Robert Stańczy. Dynamical system modeling fermionic limit. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 45-55. doi: 10.3934/dcdsb.2018004

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (125)
  • HTML views (332)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences