Integral funnel

of a point $ P ( t _ {0} , x _ {0} ) $ for a differential equation $ dx / dt = f ( t , x ) $

The set of all points lying on the integral curves (cf. Integral curve) passing through $ P $. (By an equation one can mean a system of equations in vector notation with $ x = ( x _ {1} \dots x _ {n} ) $.) If only one integral curve passes through $ P $, then the integral funnel consists of this single curve. In the case $ n = 1 $, that is, when $ x $ is scalar, the integral funnel consists of points $ ( t , x ) $ for which $ x _ {*} ( t ) \leq x \leq x ^ {*} ( t ) $, where $ x ^ {*} ( t ) $ and $ x _ {*} ( t ) $ are the upper and lower solutions, that is, the largest and smallest solutions passing through $ P $.

If the function $ f ( t , x ) $ is continuous (or satisfies the conditions of the Carathéodory existence theorem), then the integral funnel is a closed set. Furthermore, if all the solutions passing through $ P $ exist on the interval $ a \leq t \leq b $, then this segment of the funnel (the part of the integral funnel defined by the inequalities $ a \leq t \leq b $) and the section of the integral funnel by any plane $ t = t _ {1} \in [ a , b ] $ are connected compact sets. Any point on the boundary of the integral funnel can be joined to $ P $ by a piece of the integral curve lying on the boundary of the integral funnel. If the sequence of points $ P _ {k} $, $ k = 1 , 2 \dots $ converges to $ P $, then the segments of the funnels of the points $ P _ {k} $ converge to the segment of the funnel of $ P $ in the sense that for any $ \epsilon > 0 $ they are contained in an $ \epsilon $- neighbourhood of the segment of the funnel of $ P $ if $ k > k _ {1} ( \epsilon ) $. Analogous properties are possessed by integral funnels for differential inclusions (cf. Differential inclusion)

$$ \dot{x} \in F ( t , x ) $$

under specified hypotheses concerning the set $ F ( t , x ) $.

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