# 0 and 1 0 1 2 0 multiply 6

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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]...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online