Unformatted text preview: s smooth. = ! !! ! + 14 Ch5Lecs 5.6 Center Manifold Theorem 5.8 (Center Manifold) Suppose that ! is a ! ! vector field, ! ≥ 1, with a fixed point at the origin. Let the eigenspaces of !" 0 = ! be written ! ! ⨁! ! ⨁! ! . Then there is a neighborhood of the origin in which there exist ! ! invariant manifolds: the local stable !
!
manifold !!"# , tangent to ! ! , on which ! ! → 0 as ! → ∞, the local unstable manifold !!"# , tangent to ! ! , on which ! ! → 0 as ! → −∞, and a local center manifold ! ! tangent to ! ! . No proof is given. Example. ! = ! ! , ! = −!. Jacobian !" = 2!
0 0
00
and !" 0 =
−1
0 −1 Center and stable spaces !
: ! ∈ ℝ and ! ! =
0 !! = 0
: ! ∈ ℝ ! Solve the equations. Start with the chain rule !"
!" = !"
!"
!"
!" ! = − ! ! Separate variables ! !! !" = −! !! !" Integrate ln ! = ! !! + ! Exponentiate !! !(! ) = ! ! ! Notice that ! As ! → ∞, ! → 0 and ! → ! . As ! → 0! , ! → ∞ [1] 15 Ch5Lecs As ! → 0! , ! → 0 Graph 〈… 0 … !
,… 0 … !
, flow on the ! and ! axes, horizontal lines ! = ±! , trajectories〉 The stable manifold is the !
axis. What is the center manifold? How smoothly does ! → 0 as ! → 0! ? From [1] ! ! = !(−! !! ) ! !! = ! ! !! + 2! !! !! ! (!) is a linear combination of terms of form ! !! ! ! where ! is a positive integer. Consider lim!→!! ! !! ! ! !! = lim!→!! ! ! ! ! = lim!→! −1 ! ! ! ! !! !! = −1 ! lim!→! ! ! Use L’Hospital’s rule ! times = −1 ! !! lim!→! ! ! = 0 As ! → 0! , ! (!) → 0 for ! = 0, 1, 2, … Notice that any trajectory in the left half plane has the center manifold properties mentioned in theorem 5.8: •
• {! ! : ! ∈ ℝ} is an invariant manifold ! (!) is tangent to ! ! at the origin There is an infinite family of center manifolds of form anytrajectoryinthelefthalfplane ∪ theorigin ∪ {thepositive!
axis} However, there is a...
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 Fall '14
 MarcEvans
 Sets, rod, the00

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