Journal of Marine Science and Technology

, Volume 19, Issue 3, pp 257–264 | Cite as

Melnikov integral formula for beam sea roll motion utilizing a non-Hamiltonian exact heteroclinic orbit: analytic extension and numerical validation

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


In the research field of nonlinear dynamical system theory, it is well known that a homoclinic/heteroclinic point leads to unpredictable motions, such as chaos. Melnikov’s method enables us to judge whether the system has a homoclinic/heteroclinic orbit. Therefore, in order to assess a vessel's safety with respect to capsizing, Melnikov’s method has been applied for investigations of the chaos that appears in beam sea rolling. This is because chaos is closely related to capsizing incidents. In a previous paper (Maki et al. in J Mar Sci Technol 15:102–106, 2010), a formula to predict the capsizing boundary by applying Melnikov’s method to analytically obtain the non-Hamiltonian heteroclinic orbit was proposed. However, in that paper, only limited numerical investigation was carried out. Therefore, further comparative research between the analytical and numerical results is conducted, with the result being that the formula is validated.


Melnikov’s method Beam seas Roll motion Non-Hamiltonian exact heteroclinic orbit 



This research was partly supported by a Grant-in-Aid for Scientific Research of the Japan Society for Promotion of Science (No. 24360355). The authors express the sincere gratitude to the above organization. The authors are grateful to Mr. John Kecsmar from Ad Hoc Marine Designs, Ltd., for his comprehensive review of this paper as an expert in small craft technology and a native English speaker.


  1. 1.
    Maki A, Umeda N, Ueta T (2010) Melnikov integral formula for beam sea roll motion utilizing a non-Hamiltonian exact heteroclinic orbit. J Mar Sci Technol 15:102–106CrossRefGoogle 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, dynamics of marine vehicles and structures in wavesGoogle 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 Naval Archit Jpn 167:81–90CrossRefGoogle Scholar
  6. 6.
    Kan M, Taguchi H (1990) Capsizing of a ship in quartering seas (part 2 chaos and fractal in capsizing phenomenon) (in Japanese). J Soci Naval Archit 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 London Ser A 454:2537–2553CrossRefzbMATHGoogle 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 London Ser A 358:1793–1812CrossRefzbMATHGoogle Scholar
  9. 9.
    Falzarano JM, Shaw AW, Troesch AW (1992) Application of global method for analyzing dynamical systems to ship rolling motion and capsizing. Int J Bifurcation Chaos 2(1):101–115CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Bikdash M, Balachandran B, Nayfeh AH (1994) Melnikov analysis for a ship with a general roll-damping model. Nonlinear Dyn 6:101–124Google Scholar
  11. 11.
    Spyrou KJ, Cotton B, Gurd B (2002) Analytical expressions of capsizing boundary for a ship with roll bias in beam waves. J Ship Res 46(3):167–174Google Scholar
  12. 12.
    Holmes PJ (1980) Averaging and chaotic motions in forced oscillations. SIAM J Appl Math 38(1):65–80CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Springer, New-YorkCrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    Salam FM (1987) The Melnikov technique for highly dissipative systems. SIAM J Appl Math 47(2):232–243CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Endo T, Chua O (1993) Piecewise-linear analysis of high-damping chaotic phase-locked loops using Melnikov’s method. IEEE Trans Circuits Syst I, Fundam Theory Appl 40:801–807CrossRefzbMATHGoogle Scholar
  17. 17.
    Kudryashov A (2005) Exact solitary waves of the fisher equation. Phys Lett A342:99–106CrossRefMathSciNetGoogle Scholar
  18. 18.
    Kawakami H, Yoshinaga T, Ueta T (1997) Methods of computer simulation on dynamical systems. Bull Jpn Soc Ind Appl Math 7(4):49–57 in JapaneseGoogle Scholar
  19. 19.
    Yagasaki K (1996) The Melnikov theory for subharmonics and their bifurcation in forced oscillations. SIAM J Appl Math 56(6):1720–1765CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Maki A, Umeda N, Ueta T, Kobayashi E (2010) Theoretical analysis for nonlinear beam sea roll using heteroclinic orbit. Faculty of Maritime Sciences, Kobe University 7:27–38 (in Japanese)Google Scholar

Copyright information

© JASNAOE 2014

Authors and Affiliations

  1. 1.Graduate School of Maritime SciencesKobe UniversityKobeJapan
  2. 2.Naval System Research CentreTechnical Research & Development Institute, Ministry of DefenceTokyoJapan
  3. 3.Department of Naval Architecture and Ocean Engineering, Graduate School of EngineeringOsaka UniversitySuitaJapan
  4. 4.Centre for Advanced Information TechnologyTokushima UniversityTokushimaJapan

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