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Unformatted text preview: tisfied. Thus ! ! 0,0 = !, ! ! : ! ! = − !
!
! !" !" . What is the slope of the stable manifold at the origin? !"
!" ! !! =− !!
! (!") !!"
!
! !! = !!
!! !!
!
! ! !"! ! !! , where the last expression is obtained by making the substitution ! = !". Use L’Hospital’s rule to find: ! − lim!→! ! ! !!
!
! ! !"! = − lim!→! !! ! !
!" ! = − ! g ! (0). Recall [3] to see that ! ! is tangent to ! ! at the origin. In some neighborhood of the origin, the unstable component of ! ! is a function of the stable component of ! ! . That is, ! ! is a graph over ! ! . We also see that ! ! ∈ ! ! . Example. Consider [1] with ! ! = −sin(! ). Then !! 0 = −1 and from [3]: ! ! ! = { ! , ! : ! = ! ! } , and !! = !
sin
! !" !" = !!
! cos !" !
! = !!!"# !
! . Substitute the Maclaurin series expansion for cos(! ) into the last expression and simplify to get 7 Ch5Lecs ! ! ! ! = ! ! 1 − !" ! ! + ⋯ , Then the geometry of the flow looks like: 〈!"
axes, ! ! , ! ! , ! ! , arrows showing flow along ! ! and ! ! 〉. ■ In preparation for the next section, we now discuss the construction of projection operators onto ! ! and ! ! in ℝ! . Let ! ∈ !(ℝ! ) be hyperbolic. Then ℝ! = ! ! ⊕ ! ! , where ! ! is the stable eigenspace of ! and ! ! is the unstable eigenspace. Let {!! , … , !! } be a basis of generalized eigenvectors such that {!! , … , !! } span ! ! and {!!!! , … , !! } span ! ! . Choose any ! ∈ ℝ! . Then != !
! !! !! !! . [5] Define the projection operators !! and !! by !! ! = !
! !! !! !! and !! !
! !! !! !! !! . != Example. Consider the system [1]. Then the Jacobian is given by [2]...
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 Fall '14
 MarcEvans
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