Properties of basins of attraction for planar discrete cooperative maps

Abstract

<p style='text-indent:20px;'>It is shown that locally asymptotically stable equilibria of planar cooperative or competitive maps have basin of attraction <inline-formula><tex-math id="M1">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula> with relatively simple geometry: the boundary of each component of <inline-formula><tex-math id="M2">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula> consists of the union of two unordered curves, and the components of <inline-formula><tex-math id="M3">\begin{document}$ \mathcal{B} $\end{document}</tex-math></inline-formula> are not comparable as sets. The boundary curves are Lipschitz if the map is of class <inline-formula><tex-math id="M4">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula>. Further, if a periodic point is in <inline-formula><tex-math id="M5">\begin{document}$ \partial \mathcal{B} $\end{document}</tex-math></inline-formula>, then <inline-formula><tex-math id="M6">\begin{document}$ \partial\mathcal{B} $\end{document}</tex-math></inline-formula> is tangential to the line through the point with direction given by the eigenvector associated with the smaller characteristic value of the map at the point. Examples are given.

Locations

• Discrete and Continuous Dynamical Systems - B - View - PDF