ISSN:
1572-9222
Keywords:
Rapidly forced pendulum
;
transversality
;
separatrix splitting
;
asymptotics beyond all orders
;
exponentially small
;
stable manifold theorem
;
Hamiltonian
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract The rapidly forced pendulum equation with forcing δ sin((t/ε), where δ=〈δ0εp,p = 5, forδ 0,ε sufficiently small, is considered. We prove that stable and unstable manifolds split and that the splitting distanced(t) in the ( $$\dot x$$ ,t) plane satisfiesd(t) = πδ sin(t/ε) sech(π/2ε) +O(δ 0 δ exp(−π/2ε)) (2.3a) and the angle of transversal intersection,ψ, in thet = 0 section satisfiesψ ∼ 2 tanψ/2 = 2S s = (πδ/2ε) sech(π/2ε) +O((δ 0 δ/ε) exp(−π/2ε)) (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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01053162
