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)


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
$$ d({t_0}) = \,2\pi \delta \;\sin \frac{1}{\varepsilon }{t_0}\sec h\frac{1}{{2\varepsilon }}\pi +0({\delta_0}\delta \exp ( - \pi /2\varepsilon )), $$
and the angle of transversal intersection, ψ, in the t = 0 section satisfies
$$ \tan \frac{1}{2}\psi = \,\frac{\delta }{{2\varepsilon }}\pi \sec h\frac{1}{{2\varepsilon }}\pi +0({\delta_0}\frac{\delta }{\varepsilon }\exp ( - \pi /2\varepsilon )), $$


Periodic Solution Periodic Orbit Fourier Coefficient Unstable Manifold Homoclinic Orbit 
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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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