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Unformatted text preview: perties of the local unstable manifold !!"# (0) are established by considering the system [1] with ! → −!, that is ! = −!" − !(! ). The stable manifold for this time reversed system is the unstable manifold for [1]. We first prove that [1] has a unique, forward bounded solution for each point ! ∈ ! ! sufficiently close to the origin. Proof (Part 1). Let !! = ! ! ℝ! , !! 0 with the sup norm metric. ! will be determined below in such a way that !! 0 ⊂ !. Consider a Cauchy sequence of functions in !! . Since we are using the sup norm, the limit is continuous. Since !! (0) is closed, the limit will be in !! (0). Therefore !! is complete. Since ! ∈ ! ! on !! (0), lemma 3.6 implies that ! is Lipschitz on !! (0), with constant ! = max!∈!! (!) !"(! ) . In particular, ! ! − !(0) ≤ ! ! − 0 or ! ! ≤ ! ! . Now consider the integral operator [10]. To make sure its image stays in !! let’s first estimate ! ! ! : !! ! ≤ ! ! !!" ! + !" ≤ ! ! + !"# ! ! ! !! !
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! ! ! ! !! !
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! ! ! !" + !" !" + !"# ! ! !! !
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! ! ! !! !
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!" ! ! !" [11] 11 Ch5Lecs ≤ ! ! + 2!"# /! where we have used the Lipschitz property of ! and the estimates [2] and [3]. The condition ! ! ! ≤ ! can be satisfied by requiring ! ≤ ! /2! and ! ≤ !/4! . The latter requirement determines the neighborhood ! ≤ !! ! ∩ !. ! = !: ! ! This effectively defines ! since ! can be made arbitrarily small by making ! sufficiently small. Finally show that ! is a contraction: ! ! −! ! ! ! ! !! !
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! ≤ !" ! − ! + ! ! !! !
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 Fall '14
 MarcEvans
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