Quotient space

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of a dynamical system given on a topological space

The quotient space of relative to the equivalence: if the points and belong to the same trajectory. In other words, the points of the quotient space are the trajectories of the dynamical system (in a different notation , see [1]), and the topology is the strongest in which the mapping associating each point of with its trajectory is continuous (thus,

( is a directed set) if and only if there are such that

if is a metric space, then ). The quotient spaces of many dynamical systems do not satisfy any of the separation axioms, even if does. For example, if is a minimal set, then the closure of every non-empty set in the quotient space is the whole quotient space. If a dynamical system given on a metric space is completely unstable (see Complete instability), then for its quotient space to be Hausdorff it is necessary and sufficient that this dynamical system does not have saddles at infinity (cf. Saddle at infinity).

References

 [1] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian) [2] N. Bourbaki, "Elements of mathematics. General topology" , Addison-Wesley (1966) (Translated from French) [3] V.M. Millionshchikov, "A comment on the Nemytskii–Bebutov theorem concerning unstable dynamic system" Differential Eq. , 10 : 12 (1975) pp. 1775–1776 Differensial'nye Uravneniya , 10 : 12 (1975) pp. 2292–2293