Journal of Marine Science and Technology

, Volume 15, Issue 1, pp 102–106 | Cite as

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

  • Atsuo MakiEmail author
  • Naoya Umeda
  • Tetsushi Ueta
Original Article


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.


Escape 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.


  1. 1.
    Devaney RL (2003) An introduction to chaotic dynamical systems, 2nd edn. Westview Press, BoulderzbMATHGoogle Scholar
  2. 2.
    Virgin LN (1987) The nonlinear rolling response of a vessel including chaotic motions leading to capsize in regular seas. Appl Ocean Res 9(2):89–95CrossRefGoogle Scholar
  3. 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
  4. 4.
    Thompson JMT (1997) Designing against capsize in beam seas: recent advance and new insights. Appl Mech Rev 50:307–325CrossRefGoogle Scholar
  5. 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. 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
  7. 7.
    Murashige S, Aihara K (1998) Experimental study on chaotic motion of a flooded ship in waves. Proc R Soc Lond Ser A 454:2537–2553zbMATHCrossRefGoogle Scholar
  8. 8.
    Murashige S, Yamada T, Aihara K (2000) Nonlinear analysis of roll motion of a flooded ship in waves. Philos Trans R Soc Lond Ser A 358:1793–1812zbMATHCrossRefGoogle Scholar
  9. 9.
    Holmes PJ (1980) Averaging and chaotic motions in forced oscillations. SIAM J Appl Math 38(1):65–80zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Salam FM (1987) The Melnikov technique for highly dissipative systems. SIAM J Appl Math 47(2):232–243zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Wu W, McCue L (2008) Application of the extended Melnikov’s method for single-degree-of-freedom vessel roll motion. Ocean Eng 35:1739–1746CrossRefGoogle Scholar
  12. 12.
    Kuznestov YA (2004) Elements of applied bifurcation theory, 3rd edn. Springer, HeidelbergGoogle Scholar
  13. 13.
    Grim O (1951) Das Schiff von Achtern Auflaufender See, Jahrbuch der Schiffbautechnischen Gesellschaft, 4 Band, pp 264–287Google Scholar
  14. 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
  15. 15.
    Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50(10):2061–2070CrossRefGoogle Scholar
  16. 16.
    McKean HP (1970) Nagumo’s equation. Adv Math 4:209–223zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Wan XY, Zhu ZS, Lu YK (1990) Solitary wave solutions of the generalized Burgers–Huxley equation. J Phys A Math Gen 23:271–274CrossRefGoogle Scholar
  18. 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

Copyright information

© JASNAOE 2009

Authors and Affiliations

  1. 1.Graduate School of Maritime SciencesKobe UniversityKobeJapan
  2. 2.Department of Naval Architecture and Ocean Engineering, Graduate School of EngineeringOsaka UniversitySuitaJapan
  3. 3.Centre for Advanced Information TechnologyTokushima UniversityTokushimaJapan

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