We present an extension of the Melnikov method which can be used for ascertaining the existence of homoclinic and heteroclinic orbits with many pulses in a class of near-integrable systems. The Melnikov function in this situation is the sum of the usual Melnikov functions evaluated with some appropriate phase delays. We show that a nonfolding condition which involves the logarithmic derivative of the Melnikov function must be satisfied in addition to the usual transversality conditions, in order for homoclinic orbits with more than one pulse to exist.
Unfortunately, the galley proofs for the published version of the paper were rather badly garbled-up, so some small but important parts may be incomprehensible. Click here to download a correct preprint of this paper.
This work was partly supported by the U.S. Department of Energy through grant DE-FG02-93ER25154, the National Science Foundation through grants DMS-9403750, DMS-9502142, and DMS-9510728, and the Alfred P. Sloan Foundation through a Sloan Research Fellowship.
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