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# 4 will give a multipart proof of the stable manifold

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Unformatted text preview: ian examples. Notice that the level set ! ! , ! = ! is preserved under the flow generated by the equations !" !" !" !" = !" !" + ! ! , ! − ! !! (! , !), !" = − !" + ! ! , ! − ! !! (! , !). 4 Ch5Lecs If this level set contains a homoclinic or heteroclinic orbit for the Hamiltonian system, the modified system will have the same orbit. ! Example. Let’s modify the shifted pendulum with ! ! , ! = ! ! ! + cos(! ). Notice ! 0,0 = 1. Consider the modified system ! = ! + (! ! , ! − 1)!" ! = sin ! + (! ! , ! − 1)!". The following phase portrait is obtained for ! = 0.1. The saddle connections are preserved, but otherwise the system is distorted. The equilibrium points within the separatrix cycles are now sinks. The flow is mostly toward the right in the upper half plane and toward the left in the lower half plane. This rule is broken only at large values of |!| and in a narrow strip near ! = 0. Outside the invariant sets, the flow is toward greater values of ! . The system is clearly no longer Hamiltonian. ■ 5.3 Stable Manifolds Section 5.4 will give a multi ­part proof of the stable manifold theorem for hyperbolic equilibrium points. In preparation, this section presents an analysis of a simplified system that that illustrates the properties of the stable manifold. Let ! , ! ∈ ℝ! and consider the skew product system 5 Ch5Lecs ! = −! , [1] ! = ! + !(! ). Assume ! ∈ ! ! and ! 0 = 0. Note that ! may contain a linear term. The origin is an equilibrium point. The Jacobian for this system evaluated at the origin is !" 0 = −1 !! 0 0 . 1 [2] Eigenvalues are ! = ±1, so !"(0) is hyperbolic. The unstable eigenvector is clearly !! = 0,...
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