Difference between revisions of "Integral funnel"
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− | + | ''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|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|Differential inclusion]]) | ||
+ | |||
+ | $$ | ||
+ | \dot{x} \in F ( t , x ) | ||
+ | $$ | ||
+ | |||
+ | under specified hypotheses concerning the set $ F ( t , x ) $. | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Kamke, "Zur Theorie der Systeme gewöhnlicher Differentialgleichungen. II" ''Acta Math.'' , '''58''' (1932) pp. 57–85</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.F. Bokstein, ''Uchen. Zap. Moskov. Gos. Univ. Ser. Mat.'' , '''15''' (1939) pp. 3–72</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> C.C. Pugh, "Funnel sections" ''J. Differential Eq.'' , '''19''' : 2 (1975) pp. 270–295</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> E. Kamke, "Zur Theorie der Systeme gewöhnlicher Differentialgleichungen. II" ''Acta Math.'' , '''58''' (1932) pp. 57–85</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M.F. Bokstein, ''Uchen. Zap. Moskov. Gos. Univ. Ser. Mat.'' , '''15''' (1939) pp. 3–72</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> C.C. Pugh, "Funnel sections" ''J. Differential Eq.'' , '''19''' : 2 (1975) pp. 270–295</TD></TR></table> | ||
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====Comments==== | ====Comments==== | ||
− | |||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.F. Filippov, "Differential equations with discontinuous righthand sides" , Kluwer (1988) pp. 16 (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A.F. Filippov, "Differential equations with discontinuous righthand sides" , Kluwer (1988) pp. 16 (Translated from Russian)</TD></TR></table> |
Latest revision as of 22:12, 5 June 2020
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 |
Comments
References
[a1] | A.F. Filippov, "Differential equations with discontinuous righthand sides" , Kluwer (1988) pp. 16 (Translated from Russian) |
Integral funnel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_funnel&oldid=19086