An asymptotic symmetry of the rapidly forced pendulum

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The inhomogeneous differential equation (x{double prime} + sin x = {delta} sin (t + t{sub 0}){var epsilon}) describes the motion of a sinusoidally forced pendulum. The orbits that connect the two saddle points of the unforced ({delta} = 0) pendulum, (x = {pi}) and (x = {minus}{pi}), are called separatrices. If {var epsilon} {Omicron}(1), then one can use Melnikov's method to show that these separatrices can split for weak forcing ({delta} {much lt} 1), and that the perturbed motion is chaotic. If {var epsilon} 1, Melnikov's method fails because the perturbation term is not analytic in {var epsilon} at {var … continued below

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26 pages

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Chang, Yi-Hua & Segur, H. September 1990.

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  • Chang, Yi-Hua (Colorado Univ., Boulder, CO (USA). Applied Mathematics Program State Univ. of New York, Buffalo, NY (USA). Dept. of Mathematics)
  • Segur, H. (Colorado Univ., Boulder, CO (USA). Applied Mathematics Program)

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The inhomogeneous differential equation (x{double prime} + sin x = {delta} sin (t + t{sub 0}){var epsilon}) describes the motion of a sinusoidally forced pendulum. The orbits that connect the two saddle points of the unforced ({delta} = 0) pendulum, (x = {pi}) and (x = {minus}{pi}), are called separatrices. If {var epsilon} {Omicron}(1), then one can use Melnikov's method to show that these separatrices can split for weak forcing ({delta} {much lt} 1), and that the perturbed motion is chaotic. If {var epsilon} 1, Melnikov's method fails because the perturbation term is not analytic in {var epsilon} at {var epsilon} = 0. In this paper we show that for {delta} {much lt} 1 and {var epsilon} {much lt} 1, the solution of the perturbed problem exhibits a symmetry to all orders in an asymptotic expansion. From the asymptotic expansion it follows that the separatrices split by an amount that is at most transcendentally small. This proof differs from that of Holmes, Marsden and Scheurle. 16 refs.

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26 pages

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NTIS, PC A03/MF A01 - OSTI; GPO Dep.

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  • September 1990

Added to The UNT Digital Library

  • July 5, 2018, 11:11 p.m.

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  • Aug. 20, 2020, 7:28 p.m.

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Chang, Yi-Hua & Segur, H. An asymptotic symmetry of the rapidly forced pendulum, report, September 1990; Boulder, Colorado. (https://digital.library.unt.edu/ark:/67531/metadc1211797/: accessed May 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.

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