A characterization of the \(\omega\)-limit sets of planar continuous dynamical systems
If \(f\) is a continuous function on an open subset \(E\) of \(\mathbb{R}^2\), it is called trajectory of the system \(x'=f(x)\), \(x\in E\), the image of a nondegenerate maximal solution. The definition of the \(\omega\)-limit set of a trajectory considered in this paper is slightly different from the standard one by allowing it to be a subset of \(\mathbb{R}^2 \cup \{\infty\}\), as opposed to a subset of \(E\) only, such that if \(\varphi:(u,v)\to E\) is the maximal solution associated to the trajectory, its \(\omega\)-limit set is: \(\omega (\varphi)= \{x\in\mathbb{R}^2 \cup\{\infty\}:\exists (t_n)_n\) a sequence of points from \((u,v)\) converging to \(v\) such that \((\varphi (t_n))_n\) converges to \(x\}\). If \(\Gamma\) is a trajectory, any of its associated solutions \(\varphi\) induces a natural orientation of \(\Gamma\) and this is denoted by \([\varphi]\). Given a family of curves \(\{\Gamma_i\}_i\) and a set of points \(S\) from \(\mathbb{R}^2\cup \{\infty\}\), the authors show that the existence of a \({\mathcal C}^k\) map \(f\), \(0\leq k\leq \infty\) (if \(k=0\) it is assumed that \(f\) is locally Lipschitz) from the plane into itself such that each \(\Gamma_i\) is a nondegenerate trajectory and, respectively, the points of \(S\) are singular points of the system \(x'= f(x)\), is equivalent to \((\{(\Gamma_i, [\varphi_i])\}_i,S)\) being a \((k+1)\)-realizable system. In this case, the authors complete the characterization of the \(\omega\)-limit sets with a description of their geometric structure, namely, after contracting each of the connected components of \(S\), \(\omega\)-limit sets are essentially unions of a countable number of Jordan curves with pairwise disjoint interior domains.
scientific article; zbMATH DE number 1206629