Melnikov integral formula for beam sea roll motion utilizing a non-Hamiltonian exact heteroclinic orbit
- 179 Downloads
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.
KeywordsEscape equation Chaos phenomenon Melnikov integral formula Analytical formulae Non-Hamiltonian heteroclinic orbit
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.
- 3.Thompson JMT (1991) Transient basins: a new tool for designing ships against capsize. In: Dynamics of marine vehicles and structures in waves. Elsevier, AmsterdamGoogle Scholar
- 5.Kan M, Taguchi H (1990) Capsizing of a ship in quartering seas (part 1 model experiments on mechanism of capsizing) (in Japanese). J Soc Nav Arch Jpn 167:81–90Google Scholar
- 6.Kan M, Taguchi H (1990) Capsizing of a ship in quartering seas (part 2 chaos and fractals in capsizing phenomenon) (in Japanese). J Soc Nav Arch Jpn 168:213–222Google Scholar
- 12.Kuznestov YA (2004) Elements of applied bifurcation theory, 3rd edn. Springer, HeidelbergGoogle Scholar
- 13.Grim O (1951) Das Schiff von Achtern Auflaufender See, Jahrbuch der Schiffbautechnischen Gesellschaft, 4 Band, pp 264–287Google Scholar
- 14.Maki A (2008) On broaching-to phenomenon in following and quartering seas, based on non-linear dynamical system theory and optimal control theory (in Japanese). Ph.D. thesis, Osaka UniversityGoogle Scholar
- 18.Hodgkin AL, Huxley AF (1952) A qualitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544Google Scholar