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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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 Fall '14
 MarcEvans
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