A trajectory's limit set consists of its accumulation points along sequences of times tending to positive or negative infinity. The forward-time set is its omega-limit set and the backward-time set its alpha-limit set.
The alpha-limit set of a trajectory consists of the points approached along sequences of times tending to negative infinity. It is the omega-limit set for the time-reversed flow.
The omega-limit set of a trajectory consists of the points approached along sequences of times tending to positive infinity. For a bounded continuous flow it is nonempty, compact, connected, and invariant.
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In the context of mathematical set theory and topology, the concept of a "limit set" can refer to different ideas depending on the specific area of study. Here are a few interpretations: 1. **Limit Set in Topology**: In topology, the limit set of a sequence of points refers to the set of all limit points of that sequence.