Welcome to the new version of CaltechAUTHORS. Login is currently restricted to library staff. If you notice any issues, please email coda@library.caltech.edu
Published 1988 | Published
Book Section - Chapter Open

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

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)

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.

Attached Files

Published - HoMaSc1988.pdf

Files

HoMaSc1988.pdf
Files (1.1 MB)
Name Size Download all
md5:a5cd56c98c2802d30317843b0a0a372c
1.1 MB Preview Download

Additional details

Created:
August 19, 2023
Modified:
January 13, 2024