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 $F ( t , x )$.

References

[1]	E. Kamke, "Zur Theorie der Systeme gewöhnlicher Differentialgleichungen. II" Acta Math. , 58 (1932) pp. 57–85
[2]	M.F. Bokstein, Uchen. Zap. Moskov. Gos. Univ. Ser. Mat. , 15 (1939) pp. 3–72
[3]	C.C. Pugh, "Funnel sections" J. Differential Eq. , 19 : 2 (1975) pp. 270–295

References

[a1]	A.F. Filippov, "Differential equations with discontinuous righthand sides" , Kluwer (1988) pp. 16 (Translated from Russian)