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''of a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i0514601.png" /> for a differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i0514602.png" />''
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The set of all points lying on the integral curves (cf. [[Integral curve|Integral curve]]) passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i0514603.png" />. (By an equation one can mean a system of equations in vector notation with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i0514604.png" />.) If only one integral curve passes through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i0514605.png" />, then the integral funnel consists of this single curve. In the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i0514606.png" />, that is, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i0514607.png" /> is scalar, the integral funnel consists of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i0514608.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i0514609.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146011.png" /> are the upper and lower solutions, that is, the largest and smallest solutions passing through <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146012.png" />.
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If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146013.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146014.png" /> exist on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146015.png" />, then this segment of the funnel (the part of the integral funnel defined by the inequalities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146016.png" />) and the section of the integral funnel by any plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146017.png" /> are connected compact sets. Any point on the boundary of the integral funnel can be joined to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146018.png" /> by a piece of the integral curve lying on the boundary of the integral funnel. If the sequence of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146020.png" /> converges to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146021.png" />, then the segments of the funnels of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146022.png" /> converge to the segment of the funnel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146023.png" /> in the sense that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146024.png" /> they are contained in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146025.png" />-neighbourhood of the segment of the funnel of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146026.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146027.png" />. Analogous properties are possessed by integral funnels for differential inclusions (cf. [[Differential inclusion|Differential inclusion]])
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''of a point  $  P ( t _ {0} , x _ {0} ) $
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for a differential equation  $  dx / dt = f ( t , x ) $''
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146028.png" /></td> </tr></table>
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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 ) $
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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 $.
  
under specified hypotheses concerning the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051460/i05146029.png" />.
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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} $
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converge to the segment of the funnel of  $  P $
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in the sense that for any  $  \epsilon > 0 $
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they are contained in an  $  \epsilon $-
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neighbourhood of the segment of the funnel of  $  P $
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if  $  k > k _ {1} ( \epsilon ) $.  
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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>
 
 
  
 
====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)
How to Cite This Entry:
Integral funnel. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integral_funnel&oldid=19086
This article was adapted from an original article by A.F. Filippov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article