# American Institute of Mathematical Sciences

• Previous Article
Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains
• DCDS Home
• This Issue
• Next Article
Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems
February 2019, 39(2): 707-727. doi: 10.3934/dcds.2019029

## 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

* 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.

Citation: Balázs Boros, Josef Hofbauer, Stefan Müller, Georg Regensburger. Planar S-systems: Global stability and the center problem. Discrete & Continuous Dynamical Systems - A, 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.
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.
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).
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).
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).
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 $\dot u - \dot v$.
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 $\begin{cases} p = a_1 - a_2, q = a_4 - a_1, \\ r = b_1 - b_4 \end{cases}$
 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 $\begin{cases} p = a_1 - a_2, q = a_4 - a_1, \\ r = b_1 - b_4 \end{cases}$
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 $a_1-b_1=a_3-b_3$ $a_2-b_2=a_4-b_4$
 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 $a_1-b_1=a_3-b_3$ $a_2-b_2=a_4-b_4$
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 $a_3-b_3=0$ $a_2-b_2=a_4-b_4$
 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 $a_3-b_3=0$ $a_2-b_2=a_4-b_4$
 [1] Fabio Scalco Dias, Luis Fernando Mello. The center--focus problem and small amplitude limit cycles in rigid systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1627-1637. doi: 10.3934/dcds.2012.32.1627 [2] B. Coll, A. Gasull, R. Prohens. Center-focus and isochronous center problems for discontinuous differential equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 609-624. doi: 10.3934/dcds.2000.6.609 [3] Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152 [4] Haihua Liang, Yulin Zhao. Quadratic perturbations of a class of quadratic reversible systems with one center. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 325-335. doi: 10.3934/dcds.2010.27.325 [5] Armengol Gasull, Jaume Giné, Joan Torregrosa. Center problem for systems with two monomial nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (2) : 577-598. doi: 10.3934/cpaa.2016.15.577 [6] Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045 [7] Michal Fečkan, Michal Pospíšil. Discretization of dynamical systems with first integrals. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3543-3554. doi: 10.3934/dcds.2013.33.3543 [8] André Vanderbauwhede. Continuation and bifurcation of multi-symmetric solutions in reversible Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 359-363. doi: 10.3934/dcds.2013.33.359 [9] Rebecca McKay, Theodore Kolokolnikov, Paul Muir. Interface oscillations in reaction-diffusion systems above the Hopf bifurcation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2523-2543. doi: 10.3934/dcdsb.2012.17.2523 [10] Jaume Llibre, Ernesto Pérez-Chavela. Zero-Hopf bifurcation for a class of Lorenz-type systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1731-1736. doi: 10.3934/dcdsb.2014.19.1731 [11] Marilena Filippucci, Andrea Tallarico, Michele Dragoni. Simulation of lava flows with power-law rheology. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 677-685. doi: 10.3934/dcdss.2013.6.677 [12] Lora Billings, Erik M. Bollt, David Morgan, Ira B. Schwartz. Stochastic global bifurcation in perturbed Hamiltonian systems. Conference Publications, 2003, 2003 (Special) : 123-132. doi: 10.3934/proc.2003.2003.123 [13] M. R. S. Kulenović, Orlando Merino. Global bifurcation for discrete competitive systems in the plane. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 133-149. doi: 10.3934/dcdsb.2009.12.133 [14] S. Secchi, C. A. Stuart. Global bifurcation of homoclinic solutions of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1493-1518. doi: 10.3934/dcds.2003.9.1493 [15] Isaac A. García, Claudia Valls. The three-dimensional center problem for the zero-Hopf singularity. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2027-2046. doi: 10.3934/dcds.2016.36.2027 [16] Claudio A. Buzzi, Jeroen S.W. Lamb. Reversible Hamiltonian Liapunov center theorem. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 51-66. doi: 10.3934/dcdsb.2005.5.51 [17] Rafael Ortega, Andrés Rivera. Global bifurcations from the center of mass in the Sitnikov problem. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 719-732. doi: 10.3934/dcdsb.2010.14.719 [18] Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873 [19] Lingling Liu, Bo Gao, Dongmei Xiao, Weinian Zhang. Identification of focus and center in a 3-dimensional system. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 485-522. doi: 10.3934/dcdsb.2014.19.485 [20] Jaume Giné. Center conditions for generalized polynomial kukles systems. Communications on Pure & Applied Analysis, 2017, 16 (2) : 417-426. doi: 10.3934/cpaa.2017021

2017 Impact Factor: 1.179