Authors

A. Tovbis

Comments

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Abbreviated Journal Title

SIAM J. Math. Anal.

Keywords

singular perturbations; periodic solutions; exponentially small; phenomena; EXPONENTIALLY SMALL REMAINDER; DIFFERENTIAL-EQUATIONS; HOMOCLINIC; CONNECTIONS; NORMAL FORMS; ASYMPTOTICS; ORBITS; ORDERS; Mathematics, Applied

Abstract

There are several recent developments in the well-known problem of breaking of homoclinic orbits (splitting of separatrices) of a system that undergoes a singular perturbation. First, survival of a homoclinic orbit is an exceptional situation that can be linked to triviality of the Stokes phenomenon of the underlying "truncated" equation. Second, homoclinic connections to exponentially small periodic orbits survive the perturbation in the generic case. In this paper we consider a different problem: we study deformations of "genuine" periodic orbits of the second order equation y '' = y + y(2) that undergoes the singular perturbation epsilon(2)y '''' + (1 - epsilon(2))y '' = y + y(2), where epsilon > 0 is a small parameter. We prove that if the period and the constant of motion do not change too rapidly (in epsilon), a genuine (nontrivial) periodic solution does not survive the perturbation.

Journal Title

Siam Journal on Mathematical Analysis

Volume

40

Issue/Number

4

Publication Date

1-1-2008

Document Type

Article

Language

English

First Page

1516

Last Page

1549

WOS Identifier

WOS:000263103500008

ISSN

0036-1410

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