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Unformatted text preview: are in the stable subspace of – !: ! !!" !! ! ≤ ! ! !!" |!! ! | for ! ≥ 0. [3] This section constructs the local stable manifold of the equilibrium point of [1] using the Contraction Mapping Theorem. The result of the following lemma helps us to construct a contraction operator with the right form. Lemma 5.2. Consider the affine, nonautonomous initial value problem ! = !" + ! ! ,!! ! 0 = ! ∈ ! ! . [4] 9 Ch5Lecs Suppose ! is hyperbolic and ! ! is bounded and continuous for ! ≥ 0. Then the unique solution ! (!; !) of [4] that is bounded for positive time is ! ! !! ! ! !! ! ! ! !; ! = ! !" ! + ! ! !! ! ! ! ! !" − !! ! ! !". [5] Proof. Let ! ! = ! !" ! (!). Substitute into the ODE to obtain ! ! = ! !!" !(!). Then integrate and solve for ! (!) to obtain ! ! =! ! !! ! !! + ! ! !! ! ! ! ! ! !". [6] Set ! = 0 and take the stable projection, using the initial condition in [4]: !! ! ! = ! !" !+ ! ! !! ! ! !! ! ! ! !". [7] By assumption, there is a ! ≥ 0 such that ! ! can show that |!! ! ! | is bounded as ! → ∞ !! ! ! ≤ ! ! !!" ! + ! ! !! ! ! !! ! ! ≤ ! ! !!" ! + !" ! !! ! !! ! ! ≤ ! for all ! ≥ 0. Then, with the aid of [2], we ! !" !" ≤ ! ! !!" ! + !" /! Projecting [6] onto the unstable subspace yields !! ! ! = ! !" (! !!" !! ! ! + ! !!" ! !! ! ! ! !") [8] In order that ! (!, !) remain bounded as ! → ∞, its unstable projection !! ! (!) certainly must remain bounded. Since the exponential ! !" grows without bound on ! ! , we must have that expression in parentheses vanish as ! → ∞, that is ! !!" !! ! ! = − ! !!" ! !!...
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