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Difference between revisions of "Complete instability"

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A property of a [[Dynamical system|dynamical system]]. A dynamical system is called completely unstable if all its points are wandering points (cf. [[Wandering point|Wandering point]]).
 
A property of a [[Dynamical system|dynamical system]]. A dynamical system is called completely unstable if all its points are wandering points (cf. [[Wandering point|Wandering point]]).
  
For a dynamical system given in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023770/c0237701.png" /> to be globally straightenable (or globally rectifiable) (i.e. there exists a topological imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023770/c0237702.png" /> that maps each trajectory of the system into some straight line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023770/c0237703.png" />, where the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023770/c0237704.png" /> depends on the trajectory) it is necessary and sufficient that it is completely unstable and has no [[Saddle at infinity|saddle at infinity]] (Nemytskii's theorem [[#References|[1]]]).
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For a dynamical system given in $\mathbf R^n$ to be globally straightenable (or globally rectifiable) (i.e. there exists a topological imbedding $\mathbf R^n\to\mathbf R^n\times\mathbf R^n$ that maps each trajectory of the system into some straight line $\{a\}\times\mathbf R$, where the point $a\in\mathbf R^n$ depends on the trajectory) it is necessary and sufficient that it is completely unstable and has no [[Saddle at infinity|saddle at infinity]] (Nemytskii's theorem [[#References|[1]]]).
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
For a slightly different formulation of Nemytskii's theorem (avoiding the notion of saddle at infinity), see [[#References|[a2]]]. An easily accessible proof is given in [[#References|[a1]]]. The property of being globally straightenable is closely related to that of being globally parallelizable: A dynamical system is said to be (globally) parallelizable whenever it is isomorphic to a system of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023770/c0237705.png" /> where all points move with speed 1 along the lines <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023770/c0237706.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023770/c0237707.png" />).
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For a slightly different formulation of Nemytskii's theorem (avoiding the notion of saddle at infinity), see [[#References|[a2]]]. An easily accessible proof is given in [[#References|[a1]]]. The property of being globally straightenable is closely related to that of being globally parallelizable: A dynamical system is said to be (globally) parallelizable whenever it is isomorphic to a system of the form $S\times\mathbf R$ where all points move with speed 1 along the lines $\{x\}\times\mathbf R$ ($x\in S$).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,  H.A. Antosiewicz,  "Parallelizable flows and Liapunov's second method"  ''Ann. of Math.'' , '''73'''  (1961)  pp. 543–555</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.V. Nemytskii,  "Topological problems in the theory of dynamical systems"  ''AMS Transl. Series 1'' , '''5'''  (1954)  pp. 414–497  ''Uspekhi Mat. Nauk'' , '''4'''  (1949)  pp. 91–153</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Dugundji,  H.A. Antosiewicz,  "Parallelizable flows and Liapunov's second method"  ''Ann. of Math.'' , '''73'''  (1961)  pp. 543–555</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  V.V. Nemytskii,  "Topological problems in the theory of dynamical systems"  ''AMS Transl. Series 1'' , '''5'''  (1954)  pp. 414–497  ''Uspekhi Mat. Nauk'' , '''4'''  (1949)  pp. 91–153</TD></TR></table>

Latest revision as of 08:45, 29 August 2014

A property of a dynamical system. A dynamical system is called completely unstable if all its points are wandering points (cf. Wandering point).

For a dynamical system given in $\mathbf R^n$ to be globally straightenable (or globally rectifiable) (i.e. there exists a topological imbedding $\mathbf R^n\to\mathbf R^n\times\mathbf R^n$ that maps each trajectory of the system into some straight line $\{a\}\times\mathbf R$, where the point $a\in\mathbf R^n$ depends on the trajectory) it is necessary and sufficient that it is completely unstable and has no saddle at infinity (Nemytskii's theorem [1]).

References

[1] V.V. Nemytskii, V.V. Stepanov, "Qualitative theory of differential equations" , Princeton Univ. Press (1960) (Translated from Russian)


Comments

For a slightly different formulation of Nemytskii's theorem (avoiding the notion of saddle at infinity), see [a2]. An easily accessible proof is given in [a1]. The property of being globally straightenable is closely related to that of being globally parallelizable: A dynamical system is said to be (globally) parallelizable whenever it is isomorphic to a system of the form $S\times\mathbf R$ where all points move with speed 1 along the lines $\{x\}\times\mathbf R$ ($x\in S$).

References

[a1] J. Dugundji, H.A. Antosiewicz, "Parallelizable flows and Liapunov's second method" Ann. of Math. , 73 (1961) pp. 543–555
[a2] V.V. Nemytskii, "Topological problems in the theory of dynamical systems" AMS Transl. Series 1 , 5 (1954) pp. 414–497 Uspekhi Mat. Nauk , 4 (1949) pp. 91–153
How to Cite This Entry:
Complete instability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_instability&oldid=33194
This article was adapted from an original article by V.M. Millionshchikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article