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Unformatted text preview: t I . Since any point on the periodic orbit has an innite solution lifetime, ([0 , ]) R D X . Moreover, D X is open in U R by Proposition 1.3.10(ii). Fix a number T > 0. Then for any t [0 , ], there exists a nite number b ( t ) > 0 that depends continuously on the parameter t such that B b ( t ) ( ( t )) { T } D X , where B b ( t ) ( ( t )) denotes the open ball of radius b ( t ) centered at ( t ). Dene = inf t [0 , ] b ( t ). Since [0 , ] is a compact interval, b ( t ) achieves its inmum on [0 , ], so must be nonzero. This proves that there is a positive number such that any point lying within a distance from the periodic orbit has a solution lifetime of at least T . 1...
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This note was uploaded on 02/28/2011 for the course MATH 140A taught by Professor Marsden during the Fall '09 term at Caltech.
 Fall '09
 Marsden

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