* GIF images produced by "Chaos for Java", animation by "Adobe Photshop Elements".

Just as Lorenz studied a simple non-linear system to gain profound new understanding in relation to the weather, the equations

dx/dt = y

dy/dt = - x(1 - x) - cy + k sin (ft)

have been studied to gain insight into ship stability. The variables x and y repesent the angle of roll, and the rate of roll, respectively.

The right hand side of the second equation is the sum of three terms

1. A restoring force, treating the effect of buoyancy as if the ship were a simple harmonic oscillator, but with a non-linear bias to one side, crudely modelling wind forces, which can lead to capsize.
2. A linear damping force proportional to the rate of roll.
3. A periodic driving force representing wave action on the ship, of amplitude k and frequency f.

The unit of roll is that x = 1 represents the point at which the restoring force on the ship is exactly zero, any further roll leads to capsize with no wave action. The unit of frequency is the natural frequency at which the ship would rock, with no wave loading, after a small disturbance.

Starting from any particular initial state, we expect one of two potential long-term behaviours:

1. The ship could lock into regular oscillation in sympathy with the driving frequency.
2. The ship could capsize, represented in this crude model by unbounded increase of the angle of roll.

The pictures below show the long-term fate of the system for many different initial states (one per pixel). The horizontal direction is the initial angle of roll, the vertical direction is the initial rate of roll. All are for a driving frequency of 0.85.

The pictures may be read as follows:

1. Initial states in the light green or light blue areas are stable, and settle to the dark cross (+) in that area. These are basins of attraction.
2. The light crosses (x) are unstable periodic orbits, which, although they are perfectly good solutions of the equations, can never be observed in practice.
3. Initial states in the black regions lead to capsize; this is the basin of attraction for capsize.

On an (arbitrary) scale of 1 to 10, the wave loading (value of k) is 3 for the top picture, 5 for the middle one, and 8 for the last.

For low wave loading, the picture shows the desirable behaviour: the ship simply rolls through a relatively small angle in sympathy with the waves.

At the middle loading, a typical non-linear behaviour is seen. There are two possible stable responses, one of much larger amplitude than the other, and the boundary between the two basins is governed by an unstable orbit.

The last picture is the worry! By now there is only one stable response - the large amplitude one, and its basin has suffered fractal erosion. Since the wind and wave loadings are never constant (only in the model) this last situation leads inexorably to capsize, perhaps after a prolonged period of apparently normal rolling.

These pictures are for an initial time of zero, but they would be the same for any time which is at the same phase in the cycle of the periodic disturbance. After all, the meaning of the term "periodic" for the sine function is that it is the same function after shifting the origin by any multiple of 360 degrees.

This is the simplest example of a Poincaré section. The dynamics can be described by samples, taken just once per cycle, at the same phase of the periodic force. As we change the phase, the global picture changes continuously, but it repeats itself each time a complete 360 increment has been made. Of course, if the motion of the ship does not have the same period as the driving force, it will be in a different position at the end of each cycle. Nevertheless, its long-term fate is completely determined by the Poincaré sections.

The animated picture at the head of this page shows 24 Poincaré sections, taken at equal intervals over one cycle, for the fractal case. It shows that the factal basin is formed by the action of "sucking in" the basin of capsize, and winding it round endlessly. This is controlled by properties of the unstable orbit on the boundary - it generates a homoclinic tangle, an infinitely complex object first discovered by Poincaré.

These pages are hosted by Mathematical Sciences Institute, ANU, and maintained by Brian Davies .

e-mail to Brian.Davies@anu.edu.au
Department of Mathematics
Mathematical Sciences Institute
Australian National University
Canberra ACT 0200
Australia
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