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Unformatted text preview: d pendulum. It will now be convenient to measure the angle ! from the ! upwards vertical. 〈Modified sketch〉 Now ! ! , ! = ! ! ! + cos(! ). Then ! = !, ! = sin(! ). !!
The linearization of this system about the origin is simply ! = ! and ! = ! , or !" ! =
01!
01
. If ! =
then ! = !" ! = 0 and ! = det ! = −1. The origin is a saddle. 10!
10
The energy at the saddle is ! 0,0 = ! = 1. This corresponds to the curves !± = ± 2 1 − cos ! . 〈Sketch curves between (0,0), (!, 0) and (2!, 0) and indicate qualitative flows on either side. 〉 〈Sketch ! ! = cos ! vs. ! for 0 ≤ ! ≤ 2! and horizontal lines ! < 1, ! = 1, ! > 1〉 The top curve is a branch of the unstable manifold of the eq pt at the origin. The bottom curve is part of the stable manifold. Both of these curves are separatrices. Together they form a separatrix cycle. ■ A separatrix is a trajectory that separates qualitatively different parts of a 2D flow. A separatrix cycle is a collection of separatrices and equilibrium points that divides the plane into two regions. Example (continued) . Find the stable and unstable eigenspaces of the linearized system. The characteristic equation !! − !" + ! = !! − 1. The eigenvalues are !± = ±1. The associated eigenvectors are !! = 1,1 ! and !! = 1, −1 ! . Then ! ! = span!! and ! ! = span!! . A Taylor series expansion shows that !± = ±! + ⋯. The stable and unstable manifolds are tangent to the stable and unstable eigenspaces at the origin. ■ We will see that the stable manifold is the unique invariant set emanating from the origin that is tangent to ! ! , and the unstable manifold is the unique invariant set...
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This document was uploaded on 02/24/2014 for the course MATH 512 at Washington State University .
 Fall '14
 MarcEvans
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