Variable time step option in Matlab (simulink) should have prevented calculation from quick divergence. Did you use fixed step ?

Yours sincerely 

J-F Antoine 

Integrate till abs(x) >1. There is no point to continue further as there is no returning force there. Basically you may change your potential after x > 1 to a small dip and a new linear returning force and collect your  ships there, in the dip. It would not affect the result.

Dear Dumitru:

Hsu, Cell to cell map , (Springer-Verlag,  "Cell to Cell Map, A Method of Global Analysis for Nonlinear Systems", 1987) is the best, -fast- method.(Or ceiling map).

There is abundant bibliography about, even for random systems. The main idea of the original methods is based in Markov Chain principles between cells, that evolves through time since the eighties of last century till  nowadays.

I believe you use this method when I saw your paper, through Matlab code.

Really,  I was thinking in e-mail you to ask , about your computing code efficiency.

I have copy of all  papers of Dr. Kan,(mainly in Japanese), et al (Mr. Saruta, Mr. Taguchi..),about, and he uses cell method together with analytical Melnikov to compare. (He used also time variant equations). (At that time Mr. Kan achieved his Dr.level)

Against the brute force of Runge-Kutta, there is a big time difference.  More than twenty  years ago in Vigo University,  I checked the cell code, against three Unix workstations, with Runge Kutta implemented, and my poor PC was much faster.

I worked with a code that implement  Cell to Cell map method, exactly, the code used for J.M.T. Thompson. It was written in Qbasic?, many years ago by Allan McRobie. I believe I have still the printed versión, But I do not found , at that year's, anybody who know the method, or could explain me the code in spite of  the book and bibliography I have. Also with many papers of the UCL group in London, about, that was then named Transient Basin.

There is some researcher, that implement ad hoc codes to solve soft duffing, with very nice colours., in function of the fraction of periods needs to capsize, with control variables , w and F.

In The excellent book "Nonlinear Vibrations" ,by  C. Schmidt and A.. Tondl, (CUP, 1986).you have a study of soft duffing equation with many graphics about. Even a description of classical methods to investigate stability in the large are described, as scan to scan TV, that was, I think so, the method that inspired my  first implementations, and my first results too..

It will be a big pleasure  if you contact me,  at  [email protected],  and we can interchange experiences with the plane of initial conditions, fractal erosion, and Dover Cliffs.(I start to work it in 1991 with a version of RK in Qbasic, it was very nice but I need one day to obtain a full diagram).

Dimitru, before I go further, can I ask you, what is "x" in your Duffing/capsizing equation?  You've never specified it. If, as I suspect, it is simply an angle between "gravitation vertical"  and "ship structure vertical" (or whatever they call it)  , i. e. similar to a pendulum problem in a  small-amplitude approximation (|x|

The most general form of Duffing equation d2x/dt2 + alpha*dx/dt ± beta*x - delta*x3 = gamma*cos(omega*t + phi) has no explicit symbolic solution.

Stability in special cases is understood by approximate methods. The nonlinear ode could be linearized about an operating point and consecutively an analyses of the eigenvalues can provide valuable insight.

The solution is bounded for unforced and undamped case with a catch; the coefficient of stiffness and nonlinearity must be both positive.

Regards.

http://www.scholarpedia.org/article/Duffing_oscillator

The method described in the article "A new reliable numerical method for computing chaotic solutions of dynamical systems: the Chen attractor case" can be transferred to this equation with modifications, because there is x^3, cos(omega*t) and the second derivative. Then the error of calculations is reduced to minimum.

Article A New Reliable Numerical Method for Computing Chaotic Soluti...