Metrization and Liapunov functions. V
type: Item
instance of: scholarly article
title: Metrization and Liapunov functions. V
zbMATH Open document ID: 0658.54027
DOI: 10.1007/BF01903369
published in: Acta Mathematica Hungarica
publication date: 1987-01-01T00:00:00Z
review text: This paper is the continuation of the papers reviewed in Zbl 0548.54028; Zbl 0562.54060 and Zbl 0658.54026. The problem we deal with is the possibility of constructing remetrizations of the phase space such that the new metrics describe attraction and repulsion properties of the trajectories. An obstacle to this approach is the presence of nonequilibrium recurrent trajectories (precluding the existence of global Liapunov functions). This difficulty can be overcome by factorization according to Auslander recurrence classes. Now we concern ourselves with an additional obstacle. Replacing the finiteness condition of a previous result by local finitness, we prove a theorem on the existence of metrics of Liapunov type. On the other hand, we point out by examples that, in general, local finiteness can not be weakened further. The dynamical system defined in Example 1 has no recurrent trajectories but equilibrium points. Example 2 shows that the local finiteness condition can not be dropped even if compact isolated invariant sets are replaced by asymptotically stable equilibrium points.
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 54E35
zbMATH DE Number: 4076055
zbMATH Keywords: remetrizations of the phase space
zbMATH Keywords: attraction and repulsion properties
zbMATH Keywords: trajectories
zbMATH Keywords: nonequilibrium recurrent trajectories
zbMATH Keywords: metrics of Liapunov type
zbMATH Keywords: equilibrium points
zbMATH Keywords: local finiteness condition
zbMATH Keywords: compact isolated invariant sets
zbMATH Keywords: asymptotically stable equilibrium points
author: Barnabas M. Garay
MaRDI profile type: Publication
cites work: Dynamical systems: Stability theory and applications
cites work: Metrization and Liapunov functions. I
cites work: On dynamical systems in metric spaces: Metrics as Liapunov functions
cites work: Q3713170
full work available at URL: https://doi.org/10.1007/bf01903369
OpenAlex ID: W2088526894
description: scientific article; zbMATH DE number 4076055