Transcendentally small transversality in the rapidly forced pendulum |
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Authors: | James A Ellison Martin Kummer A W Sáenz |
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Institution: | (1) Department of Mathematics, University of New Mexico, 87131 Albuquerque, New Mexico;(2) Department of Mathematics, University of Toledo, 43606 Toledo, Ohio;(3) Catholic University, 20064 Washington, D.C. |
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Abstract: | The rapidly forced pendulum equation with forcing sin((t/), where =<0p,p = 5, for
0, sufficiently small, is considered. We prove that stable and unstable manifolds split and that the splitting distanced(t) in the (
,t) plane satisfiesd(t) = sin(t/) sech(/2) +O(
0
exp(–/2)) (2.3a) and the angle of transversal intersection,, in thet = 0 section satisfies 2 tan/2 = 2S
s
= (/2) sech(/2) +O((
0
/) exp(–/2)) (2.3b) It follows that the Melnikov term correctly predicts the exponentially small splitting and angle of transversality. Our method improves a previous result of Holmes, Marsden, and Scheuerle. Our proof is elementary and self-contained, includes a stable manifold theorem, and emphasizes the phase space geometry. |
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Keywords: | Rapidly forced pendulum transversality separatrix splitting asymptotics beyond all orders exponentially small stable manifold theorem Hamiltonian |
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