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Planar S-systems: Global stability and the center problem
1. | Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria |
2. | Johannes Kepler University Linz, Altenberger Straße 69, 4040 Linz, Austria |
S-systems are simple examples of power-law dynamical systems (polynomial systems with real exponents). For planar S-systems, we study global stability of the unique positive equilibrium and solve the center problem. Further, we construct a planar S-system with two limit cycles.
References:
[1] |
B. Boros and J. Hofbauer, Planar S-systems: Permanence, J. Differential Equations (2018).
doi: 10.1016/j.jde.2018.09.016. |
[2] |
B. Boros, J. Hofbauer and S. Müller,
On global stability of the Lotka reactions with generalized mass-action kinetics, Acta Appl. Math., 151 (2017), 53-80.
doi: 10.1007/s10440-017-0102-9. |
[3] |
B. Boros, J. Hofbauer, S. Müller and G. Regensburger,
The center problem for the Lotka reactions with generalized mass-action kinetics, Qual. Theory Dyn. Syst., 17 (2018), 403-410.
doi: 10.1007/s12346-017-0243-2. |
[4] |
R. L. Devaney,
Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113.
doi: 10.2307/1997429. |
[5] |
A. G. Khovanskiĭ, Fewnomials, American Mathematical Society, Providence, RI, 1991. |
[6] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, vol. 112 of Applied Mathematical Sciences, 3rd edition, Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[7] |
O. A. Kuznetsova, An example of symbolic computation of Lyapunov quantities in Maple, in Proceedings of the 5th WSEAS Congress on Applied Computing Conference, and Proceedings of the 1st International Conference on Biologically Inspired Computation, BICA'12, World Scientific and Engineering Academy and Society (WSEAS), Stevens Point, Wisconsin, USA, 2012,195–198. |
[8] |
D. C. Lewis, A qualitative analysis of S-systems: Hopf bifurcations, in Canonical Nonlinear Modeling (ed. E. Voit), Van Nostrand Reinhold, 1991,304–344. |
[9] |
S. Müller and G. Regensburger,
Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces, SIAM J. Appl. Math., 72 (2012), 1926-1947.
doi: 10.1137/110847056. |
[10] |
S. Müller and G. Regensburger, Generalized mass-action systems and positive solutions of polynomial equations with real and symbolic exponents, in Computer Algebra in Scientific Computing. Proceedings of the 16th International Workshop (CASC 2014) (eds. V. P. Gerdt, W. Koepf, E. W. Mayr and E. H. Vorozhtsov), vol. 8660 of Lecture Notes in Comput. Sci., Springer, Cham, 2014,302–323. |
[11] |
V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, 1960. |
[12] |
V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser Boston, Inc., Boston, MA, 2009.
doi: 10.1007/978-0-8176-4727-8. |
[13] |
M. A. Savageau,
Biochemical systems analysis: Ⅰ. Some mathematical properties of the rate law for the component enzymatic reactions, J. Theor. Biol., 25 (1969), 365-369.
|
[14] |
M. A. Savageau,
Biochemical systems analysis: Ⅱ. The steady state solutions for an n-pool system using a power-law approximation, J. Theor. Biol., 25 (1969), 370-379.
|
[15] |
E. E. Sel'kov,
Self-oscillations in glycolysis, Eur. J. Biochem., 4 (1968), 79-86.
|
[16] |
F. Sottile, Real Solutions to Equations from Geometry, American Mathematical Society, Providence, RI, 2011.
doi: 10.1090/ulect/057. |
[17] |
E. O. Voit, Biochemical systems theory: A review, ISRN Biomath., (2013), Article ID 897658. |
[18] |
W. Yin and E. O. Voit,
Construction and customization of stable oscillation models in biology, J. Biol. Syst., 16 (2008), 463-478.
|
show all references
References:
[1] |
B. Boros and J. Hofbauer, Planar S-systems: Permanence, J. Differential Equations (2018).
doi: 10.1016/j.jde.2018.09.016. |
[2] |
B. Boros, J. Hofbauer and S. Müller,
On global stability of the Lotka reactions with generalized mass-action kinetics, Acta Appl. Math., 151 (2017), 53-80.
doi: 10.1007/s10440-017-0102-9. |
[3] |
B. Boros, J. Hofbauer, S. Müller and G. Regensburger,
The center problem for the Lotka reactions with generalized mass-action kinetics, Qual. Theory Dyn. Syst., 17 (2018), 403-410.
doi: 10.1007/s12346-017-0243-2. |
[4] |
R. L. Devaney,
Reversible diffeomorphisms and flows, Trans. Amer. Math. Soc., 218 (1976), 89-113.
doi: 10.2307/1997429. |
[5] |
A. G. Khovanskiĭ, Fewnomials, American Mathematical Society, Providence, RI, 1991. |
[6] |
Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, vol. 112 of Applied Mathematical Sciences, 3rd edition, Springer-Verlag, New York, 2004.
doi: 10.1007/978-1-4757-3978-7. |
[7] |
O. A. Kuznetsova, An example of symbolic computation of Lyapunov quantities in Maple, in Proceedings of the 5th WSEAS Congress on Applied Computing Conference, and Proceedings of the 1st International Conference on Biologically Inspired Computation, BICA'12, World Scientific and Engineering Academy and Society (WSEAS), Stevens Point, Wisconsin, USA, 2012,195–198. |
[8] |
D. C. Lewis, A qualitative analysis of S-systems: Hopf bifurcations, in Canonical Nonlinear Modeling (ed. E. Voit), Van Nostrand Reinhold, 1991,304–344. |
[9] |
S. Müller and G. Regensburger,
Generalized mass action systems: Complex balancing equilibria and sign vectors of the stoichiometric and kinetic-order subspaces, SIAM J. Appl. Math., 72 (2012), 1926-1947.
doi: 10.1137/110847056. |
[10] |
S. Müller and G. Regensburger, Generalized mass-action systems and positive solutions of polynomial equations with real and symbolic exponents, in Computer Algebra in Scientific Computing. Proceedings of the 16th International Workshop (CASC 2014) (eds. V. P. Gerdt, W. Koepf, E. W. Mayr and E. H. Vorozhtsov), vol. 8660 of Lecture Notes in Comput. Sci., Springer, Cham, 2014,302–323. |
[11] |
V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, 1960. |
[12] |
V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser Boston, Inc., Boston, MA, 2009.
doi: 10.1007/978-0-8176-4727-8. |
[13] |
M. A. Savageau,
Biochemical systems analysis: Ⅰ. Some mathematical properties of the rate law for the component enzymatic reactions, J. Theor. Biol., 25 (1969), 365-369.
|
[14] |
M. A. Savageau,
Biochemical systems analysis: Ⅱ. The steady state solutions for an n-pool system using a power-law approximation, J. Theor. Biol., 25 (1969), 370-379.
|
[15] |
E. E. Sel'kov,
Self-oscillations in glycolysis, Eur. J. Biochem., 4 (1968), 79-86.
|
[16] |
F. Sottile, Real Solutions to Equations from Geometry, American Mathematical Society, Providence, RI, 2011.
doi: 10.1090/ulect/057. |
[17] |
E. O. Voit, Biochemical systems theory: A review, ISRN Biomath., (2013), Article ID 897658. |
[18] |
W. Yin and E. O. Voit,
Construction and customization of stable oscillation models in biology, J. Biol. Syst., 16 (2008), 463-478.
|





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