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Exponentially Small Phenomena in the Rapidly Forced Pendulum

  • Martin Kummer
  • James A. Ellison
  • A. W. Sáenz
Part of the NATO ASI Series book series (NSSB, volume 284)

Abstract

The rapidly forced pendulum equation with forcing δ sin t/ε, where δ = δ0 εp, p = 5, for δ0, ε sufficiently small, is considered. We sketch our proof that stable and unstable manifolds split and that the splitting distance d(t0) in the ẋ-t plane satisfies
$$ d({t_0}) = \,2\pi \delta \;\sin \frac{1}{\varepsilon }{t_0}\sec h\frac{1}{{2\varepsilon }}\pi +0({\delta_0}\delta \exp ( - \pi /2\varepsilon )), $$
(1)
and the angle of transversal intersection, ψ, in the t = 0 section satisfies
$$ \tan \frac{1}{2}\psi = \,\frac{\delta }{{2\varepsilon }}\pi \sec h\frac{1}{{2\varepsilon }}\pi +0({\delta_0}\frac{\delta }{\varepsilon }\exp ( - \pi /2\varepsilon )), $$
(1)
.

Keywords

Periodic Solution Periodic Orbit Fourier Coefficient Unstable Manifold Homoclinic Orbit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M. Kummer, J.A. Ellison and A.W. Saenz, to be submitted.Google Scholar
  2. 2.
    P. Holmes, J. Marsden and J. Scheuerle, “Exponentially Small Splittings of Separatrices with Applications to KAM Theory and Degenerate Bifurcations”, Contemporary Mathematics 81 (1988) 213.CrossRefGoogle Scholar
  3. 3.
    J. Scheuerle, “Chaos in A Rapidly Forced Pendulum Equation”, Contemporary Mathematics 97 (1989) 411.CrossRefGoogle Scholar
  4. 4.
    J. Marsden, this proceedings.Google Scholar
  5. 5.
    H.S. Dumas and J.A Ellison in “Local and Global Methods of Nonlinear Dynamics”, pp. 200–230 (A. W. Saenz, W.W. Zachary and R. Cawley, Eds.) Springer Verlag, New York, 1986.CrossRefGoogle Scholar
  6. 6.
    J. Scheuerle, this proceedings.Google Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Martin Kummer
    • 1
  • James A. Ellison
    • 2
  • A. W. Sáenz
    • 3
  1. 1.Department of MathematicsUniversity of ToledoOhioUSA
  2. 2.Department of MathematicsUniversity of New MexicoAlbuquerqueUSA
  3. 3.Naval Research Laboratory and Department of PhysicsCatholic UniversityWashington, D. C.USA

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