Journal of Marine Science and Technology

, Volume 24, Issue 3, pp 846–854 | Cite as

On the loss of stability of periodic oscillations and its relevance to ship capsize

  • A. MakiEmail author
  • L. N. Virgin
  • N. Umeda
  • T. Ueta
  • Y. Miino
  • M. Sakai
  • H. Kawakami
Original article


This research revisits the analysis of roll motion and the possible capsize of floating vessels in beam seas. Many analytical investigations of this topic have adopted the softening Duffing equation, which is similar to the ship roll equation of motion. Here we focus on the loss of stability of periodic oscillations and its relevance to ship capsize. Previous researchers have found the thresholds of the saddle-node, flip, and heteroclinic bifurcations. They derived the flip condition from the negative stiffness condition in a Mathieu type variational equation. In our revisited analysis, we show that this threshold is identical to a pitchfork bifurcation. On the other hand, we simultaneously find that the generated asymmetry solution is unstable due to the limitation of the first order analysis.


Nonlinear dynamics Softening Duffing equation Capsizing Pitchfork bifurcation 



This work was supported by a Grant-in-Aid for Scientific Research from the Japan Society for Promotion of Science (JSPS KAKENHI Grant number 15H02327). The authors would like to thank Enago ( for the English language review.


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Copyright information

© JASNAOE 2018

Authors and Affiliations

  • A. Maki
    • 1
    Email author
  • L. N. Virgin
    • 2
  • N. Umeda
    • 1
  • T. Ueta
    • 3
  • Y. Miino
    • 4
  • M. Sakai
    • 1
  • H. Kawakami
    • 5
  1. 1.Department of Naval Architecture and Ocean Engineering, Graduate School of EngineeringOsaka UniversitySuitaJapan
  2. 2.School of EngineeringDuke UniversityDurhamUSA
  3. 3.Center for Administration of Information Technology Tokushima UniversityTokushimaJapan
  4. 4.Graduate School of Advanced Technology and Science, System Innovation EngineeringTokushima UniversityTokushimaJapan
  5. 5.Professor Emeritus of Tokushima UniversityTokushimaJapan

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