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Planar S-systems: Global stability and the center problem

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  • 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.

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

    Citation:

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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.

    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).

    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).

    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).

    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 $\dot u - \dot v$.

    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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    [2] B. BorosJ. 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. BorosJ. HofbauerS. 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.
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