Asymptotics beyond All Orders pp 187-195 | Cite as

# Exponentially Small Estimates for Separatrix Splittings

## Abstract

This paper reviews our previous estimates and gives an example exhibiting a new phenomenon. In problems involving asymptotics beyond all orders in a perturbation parameter *ε*, it is a common *assumption* that the quantity being studied (such as a separatrix splitting distance or angle, a solitary wave mismatch, etc.) can be “estimated” by an expression of the form *aε* ^{ b } *e* ^{ -c/ε } as *ε* → 0. Here, *a, b* and *c* are constants (where *b* can be negative and *c* is “sharp”, often the distance from the real axis to a pole in the complex plane). The main purpose of our example is to show that this assumption can be wrong. The example, which concerns the splitting of separatrices in a rapidly forced system with a heteroclinic orbit shows that the even the estimate from above (using the sharp value of c) is incorrect. We argue that this situation is not isolated or particular, but happens rather generally. We especially note that in situations involving asymptotics beyond all orders, when an estimate of the form *aε* ^{ b } *e* ^{ -c/ε } is assumed, it needs to be justified.

## Keywords

Homoclinic Orbit Heteroclinic Orbit Essential Singularity Melnikov Function Heteroclinic Connection## Preview

Unable to display preview. Download preview PDF.

## References

- T. Bountis, V. Papageorgiou, and M. Bier [1987] On the singularity analysis of intersecting separatrices in near-integrable dynamical systems
*Physica D***24**, 292–304MathSciNetADSMATHCrossRefGoogle Scholar - P. Holmes, J. Marsden and J. Scheurle [1988] Exponentially small splittings of separatrices with applications to KAM theory and degenerate bifurcations,
*Cont. Math., AMS***81**, 213–244MathSciNetCrossRefGoogle Scholar - E. Fontich and C. Simo [1990] The splitting of separatrices for analytic diffeomor-phisms,
*Ergodic Th. and Dyn. Sys.***10**, 295–318MathSciNetMATHGoogle Scholar - A. I. Nieshtadt [1984] The separation of motions in systems with rapidly rotating phase,
*PMM USSR***48**, 133–139Google Scholar - J. Scheurle [1989] Chaos in a rapidly forced pendulum equation,
*Cont. Math. AMS***97**, 411–419MathSciNetCrossRefGoogle Scholar