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Uniform stability and semi-stability of motions in dynamical systems on metric spaces. (English) Zbl 0847.34054

The paper describes basic properties of semistability of orbits of pseudo-dynamical semi-systems, with applications to situations where some or all of the modifiers ‘semi’ and ‘pseudo’ are removed. Here a dynamical system means a flow on a metric space, \(\mathbb{R}\times X\to X\) (i.e., a continuous one-parameter group of homeomorphisms); a pseudo-dynamical system is a flow without the requirement of continuity (a one-parameter group of bijections); a semi-system is one defined only of \([0, \infty)\times X\). Stability of an orbit is understood in the sense of Lyapunov: orbits that start nearby stay nearby for all positive times. Semistability requires only that the orbits stay nearby for all large times. These notions are investigated for individual orbits and for sets of orbits. Implications of semistability for limit sets and connections with Lyapunov functions are dscribed.

MSC:

37C99 Smooth dynamical systems: general theory
54H20 Topological dynamics (MSC2010)
37C75 Stability theory for smooth dynamical systems
37C10 Dynamics induced by flows and semiflows
34D05 Asymptotic properties of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
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