Proof of Artés-Llibre-Valls’s conjectures for the Higgins-Selkov and the Selkov systems. (English) Zbl 1459.34116
Concerning the Higgins-Selkow system
\[
\dot x=(y-x)x^2-x,\quad \dot y=\frac{1}{\sqrt\alpha}-x\tag{\(*\)}
\]
the authors prove, that
- (i)
- (\(*\)) has no limit cycle for \(\alpha\in(0,1]\), (\(*\)) has at most one limit cycle for \(\alpha\in(1,3)\). If the limit cycle exists, it is hyperbolic and stable.
- (ii)
- There exist \(\alpha^*\in(1,3)\) such that (\(*\)) has a unique limit cycle for \(\alpha\in(1,\alpha^*)\) and no limit cycle for \(\alpha\in(\alpha^*,3)\). The amplitude of the limit cycle increases with \(\alpha\).
- (i)
- (\(**\)) has no limit cycle for \(a=-b^2\).
- (ii)
- (\(**\)) has no limit cycle for \(a\ge\sigma_0\), where \(\sigma_0\) is the unique root of \(1-4a(1+a)b^2-4a(1+a)^2b^4=0\).
- (iii)
- (\(**\)) has no limit cycle for \(b=1\), \(a\ge 0\).
- (iv)
- In case \(b=1\), (\(**\)) has no limit cycle for \(-1<a\le a_0\) and at most one limit cycle for \(a_0<a<0\), where \(a_0\approx-0.11584\).
Reviewer: Klaus R. Schneider (Berlin)
MSC:
| 34C60 | Qualitative investigation and simulation of ordinary differential equation models |
| 34C23 | Bifurcation theory for ordinary differential equations |
| 92C45 | Kinetics in biochemical problems (pharmacokinetics, enzyme kinetics, etc.) |
| 34C05 | Topological structure of integral curves, singular points, limit cycles of ordinary differential equations |
| 34D20 | Stability of solutions to ordinary differential equations |
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