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Unformatted text preview: +!" ! ! ! !! ! ! |! ! ! ! !! ! ! ! !; !! − ! !, !! |!" ! !; !! − ! !; !! !" 13 Ch5Lecs This has the form of the generalized Grönwall’s inequality [12], with ! ! = ! !, !! − ! !; !! , ! = ! |!! − !! | and ! = !" . Then !, ! and ! are positive and ! < !/2 as discussed above [13]. From the form of [13] obtain !" ! !; !! − ! (!; !! ) ≤ 2! !! − !! ! ! ! . !" It follows that !! ! !; !! − !! ! (!; !! ) ≤ 2! !! − !! ! ! ! . Thus !! ! (!; !) is a Lipschitz function of !. □ Differentiability of the local stable manifold and its tangency to ! ! at ! = 0 follow from the following theorem. A uniform contraction map !(! ; !) depends on a parameter ! with a contraction constant ! < 1 that is independent of !. Theorem 5.5 (Uniform Contraction Principle). Let ! and ! be closed subsets of two Banach spaces and let ! ∈ ! ! (!×!, !), ! ≥ 0, be a uniform contraction map. Then there is a unique fixed point ! ! = !(! ! , !), where ! ! ∈ ! is a ! ! function of ! ∈ !. Proof. Do not give. Proof of theorem 5.3 (Part 4). Differentiating [14] with respect to the initial condition ! (0), where ! = !! ! (0), we obtain !! ! ! !, ! = ! !" !! ! + ! ! ! − ! ! !! ! !! !" ! !; ! !! ! !! ! ! !! !" ! !; ! !! ! ! ! !; ! !" ! !; ! !". Evaluate this expression at ! = 0 using ! !, 0 = 0 and !" 0 = 0 to find !!(!) ! !; 0 = ! !" !! . Thus ! !, ! = ! !, 0 + !! ! !, 0 ! 0 + ! ! 0 = ! !" !! ! 0 + ! ! 0 ! ! 0 , and it follows that ! ! is tangent to ! ! at the origin. □ 5.5 Global Stable Manifolds ! Define ! ! by flowing !loc backward in time: ! ! ! = {!! ! : ! ∈ !!"# , ! ∈ ℝ}. ! !!"# is smooth and !! ! ∈ ! ! so ! ! i...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .

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