or 6 18 ch5lecs

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

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online