Planar S-systems: Global stability and the center problem

Balázs Boros; Josef Hofbauer; Stefan Müller; Georg Regensburger

doi:10.3934/dcds.2019029

Article Contents

2019, Volume 39, Issue 2: 707-727. Doi: 10.3934/dcds.2019029

This issue Previous Article Random attractors for non-autonomous fractional stochastic parabolic equations on unbounded domains Next Article Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems

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
* Corresponding author

Received: July 2017

Revised: December 2017

Published: February 2019

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Abstract

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.

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Mathematics Subject Classification: 34C05, 34C07, 34C14, 34C23, 80A30, 92C42.

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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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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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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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