# Melnikov integral formula for beam sea roll motion utilizing a non-Hamiltonian exact heteroclinic orbit

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## Abstract

The chaos that appears in the ship roll equation in beam seas known as the escape equation has been intensively investigated because it is closely related to capsizing incidents. In particular, many applications of the Melnikov integral formula have been reported in the literature; however, in all the analytical works concerning the escape equation, the Melnikov integral is formulated utilizing a separatrix for the Hamiltonian part or a numerically obtained heteroclinic orbit for the non-Hamiltonian part of the original escape equation. To overcome such limitations, this article attempts to utilise an analytical expression for the non-Hamiltonian part. As a result, an analytical procedure is provided that makes use of a heteroclinic orbit of the non-Hamiltonian part within the framework of the Melnikov integral formula.

## Keywords

Escape equation Chaos phenomenon Melnikov integral formula Analytical formulae Non-Hamiltonian heteroclinic orbit## Notes

### Acknowledgments

This research was partly supported by a Grant-in-Aid for Scientific Research of the Japan Society for Promotion of Science (no. 21360427) and the Ministry of Education, Culture, Sports, Science and Technology of Japan (MEXT) as an Advanced Research Grant.

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