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Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations

Holmes, Philip and Marsden, Jerrold and Scheurle, Jürgen (1988) Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations. In: Hamiltonian dynamical systems : proceedings of the AMS-IMS-SIAM joint summer research conference. Contemporary mathematics . No.81. American Mathematical Society , pp. 213-244. ISBN 0821850865 http://resolver.caltech.edu/CaltechAUTHORS:20100924-133113111

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Abstract

Both upper and lower estimates are establishedfor the separatrix splitting of rapidly forced systems with a homoclinic orbit. The general theory is applied to the equation φ + sin φ =S sin(^t-_Є) for illustration. There are two types of results. First,fix η > 0 and let 0 < E ≤ 1 and 0 ≤ S ≤ S_0 where S_0 is sufficiently small. If the separatrices split, they do so by an amount that is no more than C S exp(-^1-є(^π-2-η)) where C = C(S_0) is a constant depending on S_0 but is uniform in E and S . Second, if we replace S by єP S, p ≥ 8, then we have the sharper estimate C_2 є^p S e ^(-π/2є) ≤ splitting distance ≤ C_1є^P S є -^(π/2є) for positive constants C_1 and C_2 depending on So alone. In particular, in this second case, the Melnikov criterion correctly predicts exponentially small splitting and transversal intersection of the separatrices. After developing this theory we discuss some of its applications, concentrating on a 2:I resonance that occurs in a KAM (Kolmogorov, Arnold, and Moser) situation and in the/orced saddle node bifurcation described by X + µX + x^2 + x^3 = Sf(t)


Item Type:Book Section
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http://www.cds.caltech.edu/~marsden/bib/1988/06-HoMaSc1988/HoMaSc1988.pdfAuthorUNSPECIFIED
Additional Information:© 1988, American Mathematical Society. We thank B. Birnir, J. Carr, B. Chirikov, W. Eckhaus, M. Golubitsky, J. Guckenheimer. K. Kirchgassner. M. Kummer. R. Meyer, and H. Segur for useful comments. The research of P. Holmes was partially supported by NSF grant MSM 84-02069 and AFOSR contract 84-0051. and that of J. Marsden and J. Scheurle was partially supported by NSF grant OMS 87-02502 and DOE contract OE-A T03-85ERI2097.
Funders:
Funding AgencyGrant Number
NSFMSM 84-02069
Air Force Office of Scientific Research (AFOSR)84-0051
NSFDMS 87-02502
Department of EnergyDE-AT03-85ERI2097
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Deposited By: Ruth Sustaita
Deposited On:24 Sep 2010 21:09
Last Modified:26 Dec 2012 12:27

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