Ushiki's theorem: Difference between revisions
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== The theorem == |
== The theorem == |
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A mapping <math>F:\mathbb{C}^n\to\mathbb{C}^n</math> does not have a 1-dimensional compact smooth [[invariant manifold]]. |
A [[mapping (mathematics)|mapping]] <math>F:\mathbb{C}^n\to\mathbb{C}^n</math> does not have a 1-dimensional compact smooth [[invariant manifold]]. |
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In particular, such a map cannot have a [[homoclinic connection]] or [[heteroclinic connection]]. |
In particular, such a map cannot have a [[homoclinic connection]] or [[heteroclinic connection]]. |
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== Comentary == |
== Comentary == |
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Invariant manifolds typically appear as solutions of certain asymptotic problems in [[dynamical systems]]. |
Invariant manifolds typically appear as solutions of certain asymptotic problems in [[dynamical systems]]. |
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Revision as of 19:33, 26 November 2010
The theorem
A mapping does not have a 1-dimensional compact smooth invariant manifold. In particular, such a map cannot have a homoclinic connection or heteroclinic connection.
Comentary
Invariant manifolds typically appear as solutions of certain asymptotic problems in dynamical systems. The most common is the stable manifold or its kin, the unstable manifold.
The publication
Ushiki's theorem was published in 1980[1]. Interestingly, the theorem appeared in print again several years later, in a certain Russian journal, by an author apparently unaware of Ushiki's work.
An application
The standard map cannot have a homoclinic or heteroclinic connection. The practical consequence is that one cannot show the existence of a Smale's horseshoe in this system by the method invented by Stephen Smale.
See also
References
- ^ S. Ushiki. Sur les liasons-cols des systemes dynamiques analytiques. C. R. Acad. Sci. Paris, 291(7):447-449, 1980