Metrization and Liapunov functions. I
Copyright policy )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.
- Budapest University of Technology and Economics Hungary
- Eötvös Loránd University Hungary
- University of Technology Russian Federation
remetrizations of the phase spaces, Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces, nonequilibrium recurrent trajectories, local finiteness condition, Topological dynamics, compact isolated invariant sets, Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, asymptotically stable closed invariant set, trajectories, Morse- Smale dynamical systems, retraction, Lyapunov functions, metrics of Liapunov type, region of attraction, monotone Lyapunov function, energy functions, attraction and repulsion properties, Morse-Smale flows, asymptotically stable equilibrium points, Metric spaces, metrizability, equilibrium points, metrics of Lyapunov, compact oriented surfaces, attractor-repeller pairs, Entire and meromorphic functions of one complex variable, and related topics, remetrizations of the phase space
remetrizations of the phase spaces, Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces, nonequilibrium recurrent trajectories, local finiteness condition, Topological dynamics, compact isolated invariant sets, Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, asymptotically stable closed invariant set, trajectories, Morse- Smale dynamical systems, retraction, Lyapunov functions, metrics of Liapunov type, region of attraction, monotone Lyapunov function, energy functions, attraction and repulsion properties, Morse-Smale flows, asymptotically stable equilibrium points, Metric spaces, metrizability, equilibrium points, metrics of Lyapunov, compact oriented surfaces, attractor-repeller pairs, Entire and meromorphic functions of one complex variable, and related topics, remetrizations of the phase space
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