
-
Previous Article
The wave interactions of an improved Aw-Rascle-Zhang model with a non-genuinely nonlinear field
- DCDS-B Home
- This Issue
-
Next Article
Confining integro-differential equations originating from evolutionary biology: Ground states and long time dynamics
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 Selkov model in the Poincaré disc
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 |
$ \begin{equation*} \dot{x}\, = \, -x+a y+x^2 y, \quad \dot{y} \, = \, b-a y-x^2 y, \end{equation*} $ |
$ a, b \in \mathbb{R} $ |
$ a $ |
$ b $ |
$ x $ |
$ y $ |
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. |












Curve | Region | finite singular point |
(1), (2) | stable node | |
(3) | stable focus | |
(4) | unstable focus | |
(5) | unstable node | |
empty | ||
(6) | saddle |
Curve | Region | finite singular point |
(1), (2) | stable node | |
(3) | stable focus | |
(4) | unstable focus | |
(5) | unstable node | |
empty | ||
(6) | saddle |
a | b | on the curve |
a | b | on the curve |
[1] |
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 |
[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] |
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 |
[4] |
Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete and Continuous Dynamical Systems, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229 |
[5] |
Honghu Liu. Phase transitions of a phase field model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 883-894. doi: 10.3934/dcdsb.2011.16.883 |
[6] |
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 |
[7] |
Shaoqiang Tang, Huijiang Zhao. Stability of Suliciu model for phase transitions. Communications on Pure and Applied Analysis, 2004, 3 (4) : 545-556. doi: 10.3934/cpaa.2004.3.545 |
[8] |
Xinfu Chen, G. Caginalp, Christof Eck. A rapidly converging phase field model. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1017-1034. doi: 10.3934/dcds.2006.15.1017 |
[9] |
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 |
[10] |
Runlin Zhang. Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 2053-2071. doi: 10.3934/dcds.2021183 |
[11] |
Sylvie Benzoni-Gavage, Laurent Chupin, Didier Jamet, Julien Vovelle. On a phase field model for solid-liquid phase transitions. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 1997-2025. doi: 10.3934/dcds.2012.32.1997 |
[12] |
Pierluigi Colli, Antonio Segatti. Uniform attractors for a phase transition model coupling momentum balance and phase dynamics. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 909-932. doi: 10.3934/dcds.2008.22.909 |
[13] |
José Luiz Boldrini, Gabriela Planas. A tridimensional phase-field model with convection for phase change of an alloy. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 429-450. doi: 10.3934/dcds.2005.13.429 |
[14] |
Valeria Berti, Mauro Fabrizio, Diego Grandi. A phase field model for liquid-vapour phase transitions. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 317-330. doi: 10.3934/dcdss.2013.6.317 |
[15] |
Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089 |
[16] |
Theodore Tachim Medjo. A two-phase flow model with delays. Discrete and Continuous Dynamical Systems - B, 2017, 22 (9) : 3273-3294. doi: 10.3934/dcdsb.2017137 |
[17] |
Pavel Krejčí, Songmu Zheng. Pointwise asymptotic convergence of solutions for a phase separation model. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 1-18. doi: 10.3934/dcds.2006.16.1 |
[18] |
Pavel Drábek, Stephen Robinson. Continua of local minimizers in a quasilinear model of phase transitions. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 163-172. doi: 10.3934/dcds.2013.33.163 |
[19] |
A. Jiménez-Casas. Invariant regions and global existence for a phase field model. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 273-281. doi: 10.3934/dcdss.2008.1.273 |
[20] |
Pavel Krejčí, Jürgen Sprekels. Long time behaviour of a singular phase transition model. Discrete and Continuous Dynamical Systems, 2006, 15 (4) : 1119-1135. doi: 10.3934/dcds.2006.15.1119 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]