Proof of Artés-Llibre-Valls's conjectures for the Higgins-Selkov and the Selkov systems
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 \begin{itemize} \item[(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. \item[(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 \(\alpha\). \end{itemize} Concerning the Selkov system \[ \dot x=-x+ay+x^2y,\quad \dot y=b-ay-x^2y\tag{\(**\)} \] they prove \begin{itemize} \item[(i)] (\(**\)) has no limit cycle for \(a=-b^2\). \item[(ii)] (\(**\)) has no limit cycle for \(a\ge\sigma_0\), where \(\sigma_0\) is the unique root of \(1-4a(1+a)b^2-4a(1+a)^2b^4=0\). \item[(iii)] (\(**\)) has no limit cycle for \(b=1\), \(a\ge 0\). \item[(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 \(a_0\approx-0.11584\). \end{itemize} The proofs are based on the transformation of (\(*\)) and (\(**\)) into Liénard systems.
scientific article; zbMATH DE number 7040869
scientific article