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February  2019, 39(2): 707-727. doi: 10.3934/dcds.2019029

Planar S-systems: Global stability and the center problem

Balázs Boros 1, , Josef Hofbauer 1, , Stefan Müller 1,, and  Georg Regensburger 2,

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

* Corresponding author

Received  July 2017 Revised  December 2017 Published  November 2018

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.

Keywords: Power-law systems, center-focus problem, first integrals, reversible systems, focal values, global center, Andronov-Hopf bifurcation, Bautin bifurcation.
Mathematics Subject Classification: 34C05, 34C07, 34C14, 34C23, 80A30, 92C42.
Citation: Balázs Boros, Josef Hofbauer, Stefan Müller, Georg Regensburger. Planar S-systems: Global stability and the center problem. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 707-727. doi: 10.3934/dcds.2019029
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. 

Figure 1.  Phase portraits of the ODE (3) in case $\det J > 0$ and both of the diagonal entries of $J$ are negative. As claimed in Lemma 4 (a), all solutions are bounded in positive time. Seven cases are ultimately monotonic, the remaining two (top left and bottom right) can spiral, but only inwards.
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Figure 2.  The forward invariant sets used in the proofs of Lemma 5 (b1) and (b2), respectively, to show the necessity of $a_3 \leq a_2 < a_1 \leq a_4$ (top panel) and $a_3 \leq a_2 = a_1 \leq a_4$ (bottom panel) for the boundedness of the solutions of the ODE (3).
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Figure 3.  The level sets of the Lyapunov function used to show the sufficiency of $a_3 \leq a_2 = a_1 \leq a_4$ for the boundedness of the solutions of the ODE (3) in Lemma 3 (b2).
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Figure 4.  The bounded forward invariant sets used to show the sufficiency of $a_3 \leq a_2 = a_1 \leq a_4$ for the boundedness of the solutions of the ODE (3) in Lemma 5 (b2).
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Figure 5.  Illustration of the proof of Theorem 7, case R1, to show the sufficiency of $a_3 \leq a_2 < a_1 \leq a_4$ (and $a_3 \leq a_1 < a_2 \leq a_4$, respectively) for the origin being a global center of the ODE (3). Both panels display the nullcline geometry, the sign structure of the vector field, the line of reflection, and the signs of $\dot u + \dot v$ and $\dot u - \dot v$.
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Table 1.  First integrals corresponding to cases S, Ⅰ1, Ⅰ2, Ⅰ3, Ⅰ4. If $\alpha$ is zero in $\frac{{\rm{e}}^{\alpha z}}{\alpha}$ (in a first integral), replace $\frac{{\rm{e}}^{\alpha z}}{\alpha}$ by $z$.
case first integral
S $\displaystyle \left(\frac{{\rm{e}}^{pu}}{p} - \frac{{\rm{e}}^{qu}}{q}\right) - \left(\frac{{\rm{e}}^{rv}}{r} - \frac{{\rm{e}}^{sv}}{s}\right)$, where $\begin{cases} p = a_3-a_1, q = a_4-a_1, \\ r = b_1-b_4, s = b_2-b_4 \end{cases}$
Ⅰ1 $\displaystyle +\frac{{\rm{e}}^{p(u-v)}}{p} + \frac{{\rm{e}}^{q u}}{q} - \frac{{\rm{e}}^{r v}}{r}$, where $\begin{cases} p = a_1 - a_2, q = a_4 - a_2, \\ r = b_2 - b_4 \end{cases}$
Ⅰ2 $\displaystyle -\frac{{\rm{e}}^{p(u+v)}}{p} + \frac{{\rm{e}}^{q u}}{q} + \frac{{\rm{e}}^{r v}}{r}$, where $\begin{cases} p = a_1 - a_2, q = a_3 - a_2, \\ r = b_2 - b_3 \end{cases}$
Ⅰ3 $\displaystyle +\frac{{\rm{e}}^{p(-u+v)}}{p} - \frac{{\rm{e}}^{q u}}{q} + \frac{{\rm{e}}^{r v}}{r}$, where $\begin{cases} p = a_1 - a_2, q = a_3 - a_1, \\ r = b_1 - b_3 \end{cases}$
Ⅰ4 $\displaystyle -\frac{{\rm{e}}^{p(-u-v)}}{p} - \frac{{\rm{e}}^{q u}}{q} - \frac{{\rm{e}}^{r v}}{r}$, where $\begin{cases} p = a_1 - a_2, q = a_4 - a_1, \\ r = b_1 - b_4 \end{cases}$
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case first integral
S $\displaystyle \left(\frac{{\rm{e}}^{pu}}{p} - \frac{{\rm{e}}^{qu}}{q}\right) - \left(\frac{{\rm{e}}^{rv}}{r} - \frac{{\rm{e}}^{sv}}{s}\right)$, where $\begin{cases} p = a_3-a_1, q = a_4-a_1, \\ r = b_1-b_4, s = b_2-b_4 \end{cases}$
Ⅰ1 $\displaystyle +\frac{{\rm{e}}^{p(u-v)}}{p} + \frac{{\rm{e}}^{q u}}{q} - \frac{{\rm{e}}^{r v}}{r}$, where $\begin{cases} p = a_1 - a_2, q = a_4 - a_2, \\ r = b_2 - b_4 \end{cases}$
Ⅰ2 $\displaystyle -\frac{{\rm{e}}^{p(u+v)}}{p} + \frac{{\rm{e}}^{q u}}{q} + \frac{{\rm{e}}^{r v}}{r}$, where $\begin{cases} p = a_1 - a_2, q = a_3 - a_2, \\ r = b_2 - b_3 \end{cases}$
Ⅰ3 $\displaystyle +\frac{{\rm{e}}^{p(-u+v)}}{p} - \frac{{\rm{e}}^{q u}}{q} + \frac{{\rm{e}}^{r v}}{r}$, where $\begin{cases} p = a_1 - a_2, q = a_3 - a_1, \\ r = b_1 - b_3 \end{cases}$
Ⅰ4 $\displaystyle -\frac{{\rm{e}}^{p(-u-v)}}{p} - \frac{{\rm{e}}^{q u}}{q} - \frac{{\rm{e}}^{r v}}{r}$, where $\begin{cases} p = a_1 - a_2, q = a_4 - a_1, \\ r = b_1 - b_4 \end{cases}$
Table 2.  Special cases of the ODE (3) having a center. Additionally, in all cases ${\rm{tr}}\;J = a_1 - a_2 + b_3 - b_4 = 0$, which is trivially fulfilled in case S, and $\det J = (a_1-a_2)(b_3-b_4)-(a_3-a_4)(b_1-b_2) > 0$.
case parameters
S $a_1=a_2$ $b_3=b_4$
Ⅰ1 $a_1=a_3$ $b_1=b_3$
Ⅰ2 $a_1=a_4$ $b_1=b_4$
Ⅰ3 $a_2=a_4$ $b_2=b_4$
Ⅰ4 $a_2=a_3$ $b_2=b_3$
R1 $a_1+b_1=a_4+b_4$ $a_2+b_2=a_3+b_3$
R2 $a_1-b_1=a_3-b_3$ $a_2-b_2=a_4-b_4$
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case parameters
S $a_1=a_2$ $b_3=b_4$
Ⅰ1 $a_1=a_3$ $b_1=b_3$
Ⅰ2 $a_1=a_4$ $b_1=b_4$
Ⅰ3 $a_2=a_4$ $b_2=b_4$
Ⅰ4 $a_2=a_3$ $b_2=b_3$
R1 $a_1+b_1=a_4+b_4$ $a_2+b_2=a_3+b_3$
R2 $a_1-b_1=a_3-b_3$ $a_2-b_2=a_4-b_4$
Table 3.  Special cases of the ODE (20) having a center. Additionally, in all cases ${\rm{tr}}\;J = - a_2 + b_3 - b_4 = 0$, which is trivially fulfilled in case S, and $\det J = (a_3-a_4)b_2-(b_3-b_4)^2 > 0$.
case parameters
S $a_2=0$ $b_3=b_4$
Ⅰ1 $a_3=0$ $b_3=0$
Ⅰ2 $a_4=0$ $b_4=0$
Ⅰ3 $a_2=a_4$ $b_2=b_4$
Ⅰ4 $a_2=a_3$ $b_2=b_3$
R1 $a_4+b_4=0$ $a_2+b_2=a_3+b_3$
R2 $a_3-b_3=0$ $a_2-b_2=a_4-b_4$
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case parameters
S $a_2=0$ $b_3=b_4$
Ⅰ1 $a_3=0$ $b_3=0$
Ⅰ2 $a_4=0$ $b_4=0$
Ⅰ3 $a_2=a_4$ $b_2=b_4$
Ⅰ4 $a_2=a_3$ $b_2=b_3$
R1 $a_4+b_4=0$ $a_2+b_2=a_3+b_3$
R2 $a_3-b_3=0$ $a_2-b_2=a_4-b_4$
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