is bounded and continuous for 0 then the unique

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Unformatted text preview: with !! = 0,1 ! and !! = 2, −!! 0 ! . [5] becomes ! ! = !! !! + !! !! = !! !! where ! = !! !! = !" , [6] 0 2 ! !! 0 and !!! = ! ! 1 −! 0 1 2 . 0 Multiply [6] on the left by !!! to find ! !! ! !! !! = ! != ! !! 0 ! + ! ! ! ! Then ! !! ! = !! !! = 0 ! 0 !+! = ! ! ! ! 2 ! !! ! = !! !! = ! ! −! ! 0 and thus 0 !! 0 ! ! 1 = − ! !! 0 ! 0 1 0 0 ! ! , ! ! , 8 Ch5Lecs !! = 0 !! 0 ! ! 0 1 and !! = − ! !! 0 1 ! 0 0 . ■ 5.4 Local Stable Manifold Theorem Let !: ℝ! → ℝ! be ! ! and suppose that ! = !(! ) has a hyperbolic equilibrium point ! ∗ . Shift coordinates to place the equilibrium at the origin by replacing ! → ! + ! ∗ . Then obtain ! = !" + !(! ), [1] where ! = !"(! ∗ ) and ! ! = ! ! + ! ∗ − !" . Notice that ! 0 = !" 0 = 0. !(! ) represents the nonlinear terms in the equation. Let ! ! and ! ! be the stable and unstable spaces associated with !. Then ℝ! = ! ! ⊕ ! ! . We can write ! = !! + !! where !! ∈ ! ! and !! ∈ ! ! . Introduce the projection operators !! : ℝ! → ! ! and !! : ℝ! → ! ! whose action is !! ! = !! and !! ! = !! . Since ! leaves ! ! and ! ! invariant !! , ! = !! , ! = 0. For example, if !" = !, then !! !" = !! ! = !! and !!! ! = !!! = !! . We also have !! , ! !" = !! , ! !" = 0. Since ! is hyperbolic there is an ! > 0 such that Re!! > ! for each eigenvalue !! of !. Eq. 2.44 of Meiss bounds how rapidly a point !! ! in the stable subspace of ! evolves under the action of ! !" toward the equilibrium point at the origin ! !" !! ! ≤ ! ! !!" |!! ! | for ! ≥ 0, [2] where ! ≥ 1. A similar bound can be given for points !! ! which...
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