Metrization and Liapunov functions. I
Let (X,d) be a metric space and \(T_ t:X\to X\) (\(t\in {\mathbb{R}})\) be a dynamical system. Let M, \(\emptyset\neq M\subset X\), be an asymptotically stable closed invariant set and A(M) be its region of attraction. It is well known that there exists a continuous function \(V:A(M)\to {\mathbb{R}}\) satisfying the following conditions: \((i)\quad V(x)=0\) (\(x\in M)\) and \(V(x)>0\quad (x\in A(M)\backslash M);\) (ii) \(V(T_ tx)\to 0\) as \(t\to +\infty\) (\(x\in A(M))\); \((iii)\quad V(x)\geq d(x,M)\) (\(x\in A(M))\). Theorem 1. Assume that there is a retraction \(r:A(M)\to M\) such that for any \(\{y_ n\}^{\infty}_{n=1}\subset A(M),\) \(x\in M\) the conditions \(d(y_ n,M)\to 0\) and \(r(y_ n)\to X\) imply \(y_ n\to x\). Then there exists a metric \(\rho\) on X such that \(\rho| M\times M=d| M\times M\) and \(V(x)=\rho (x,M)=\rho (x,r(x))\) for \(x\in A(M)\). It is also shown that an analogous assertion holds for Morse-Smale flows on compact oriented surfaces.
asymptotically stable closed invariant set
scientific article; zbMATH DE number 3874198
scientific article