Unformatted text preview: t ∈ I . Since any point on the periodic orbit has an inﬁnite solution lifetime, γ ([0 , τ ]) × 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 , τ ], there exists a ﬁnite number b ( t ) > 0 that depends continuously on the parameter t such that B b ( t ) ( γ ( t )) ×{ T } ⊂ D X , where B b ( t ) ( γ ( t )) denotes the open ball of radius b ( t ) centered at γ ( t ). Deﬁne ε = inf t ∈ [0 ,τ ] b ( t ). Since [0 , τ ] is a compact interval, b ( t ) achieves its inﬁmum on [0 , τ ], so ε must be nonzero. This proves that there is a positive number ε such that any point lying within a distance ε from the periodic orbit has a solution lifetime of at least T . 1...
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 Fall '09
 Marsden
 Topology, Euclidean space, Dynamical systems, nonlinear phase portrait

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