ÐÓ°ÉÔ­´´

IN FEBRUARY 1974, the Gaul, a trawler from Hull, disappeared in heavy
seas off the coast of Norway with the loss of 36 lives. The inquiry into
the disaster found that there was nothing wrong with the ship’s design:
the stability of the vessel against capsize conformed with all required
standards. One possible explanation put forward for the loss of the vessel
was that it capsized as the result of some transient phenomenon when hit
broadside by a short succession of abnormally large waves.

‘Transient phenomenon’ means that the vessel was moving in an irregular
fashion: rocking, swaying, pitching, heaving and rolling. A trawler in a
storm is obviously tossed around, so any capsize caused by waves hitting
a vessel is a transient phenomenon. Strangely, though, the methods that
engineers use to assess how stable a ship is assume that the ship is totally
motionless. The methods simply require that the forces on the stationary
vessel when tilted at various angles should be such as to try to right it.
Various naval engineers at the inquiry pointed out that these tried and
trusted methods of stability design might not always be safe. The motion
of a ship and the way that it rocks change its stability. En gineers suggested
that because of the dynamic nature of capsize a ship’s stability should
be evaluated when it is moving as well as when it is still.

The word ‘transient’, however, has a stronger meaning than just motion:
it is a mathematical term for unsteady motion – motion that has not yet
settled down to a steady and regular pattern. The mathematics of transient
motions gives equations that are more complicated than the equations of
steady rocking, which are often complicated enough. To many engineers, assessing
dynamic stability would mean analysing the steady rocking motions. Ships
capsize, however, as a result of transient motions so these should be considered
as well.

Everything moves, even buildings. Many people in Britain first realised
this in recent storms when office workers were evacuated from tower blocks
shaken about by high winds. Although the motion is usually imperceptible,
engineers spend a lot of effort analysing in detail the way a building will
move; the whole safety of the building and its occupants depends on an engineer
checking that the building will not wobble in a high wind so much as to
collapse.

At this moment, engineers around the world are busily analysing the
way nuclear power stations rock during earthquakes, how offshore oil platforms
behave when buffeted by waves, even how the next models of computer printers
vibrate when working.

Most systems that confront the engineer are examples of ‘nonlinear’
dynamical systems. Nonlinear means that the parameters of the system do
not vary proportionally. A graph representing this variation will not be
a straight line. To simplify things, engineers often assume that such systems
are linear. Often this is a good approximation. The force with which a material
resists being stretched, for example, is roughly proportional to the amount
that it is stretched. Stretch it too far, though, and the approximation
is not so good. Typically, the material will stretch more easily, and eventually
it may break. This is important: the linear approximation is usually worst
when things are about to fail.

Most engineering systems also experience random forces. The forces that
will act on the system are never known precisely beforehand and they will
fluctuate throughout the life of the system.

You need only envisage the motion of several large waves breaking across
the deck of a trawler to appreciate just how difficult the complete analysis
of a real nonlinear, random, dynamic situation can be. One offal chute accidentally
left open can easily invalidate an analysis. Faced with such complexity,
the engineer simplifies the calculations. Ship designers assume that the
vessel is still (the static approximation), but they retain the nonlinear
features of the acting forces in their analysis. Designing modern high-rise
buildings to withstand wind involves some comprehensive random dynamics
but assumes that material properties of the building behave proportionally
(the linear approximation).

Most engineering firms have computer packages that can take into account
nonlinear and random dynamic effects. The real skill in engineering, however,
is not in solving lots of complicated equations but in being able to identify
the important principles and parameters in the face of the overabundance
of detail and incompleteness of data. The major benefit of both the static
and the linear approximation is not that they are mathematically easier
to deal with but that such systems are well understood. Experienced engineers
have a ‘feel’ for the way they behave. It is in similar terms of understanding
that chaos theory has much to offer.

If we assume that the system is linear, we can represent the average
forces acting on it by a parabolic potential well. This means that the system
will obey laws of motion similar to those of a ball rolling in a trough
shaped like a parabola. One implication of this shape is that the ball can
never escape. The potential wells of real systems are not in general parabolic,
however (see Figure 1). Most wells allow the possibility of escape – which
corresponds to an engineering failure, be it a tower block collapsing or
a ship capsizing. Consider the ship in Figure 1. Small rocking motions are
like the ball rolling to and fro in the bottom of the well. If the oscillations
become too large, then just as the ball may roll out of the well and escape
to the left or the right, so the ship may capsize to starboard or port.

If you apply a periodic force to a linear system, the system will rock,
or oscillate, erratically for a while – these are the transients – before
settling down to a steady vibration with the same frequency as the applied
force. This final motion does not depend on the system’s initial conditions
– its velocity and position when the force was first applied. There is only
one long-term behaviour, called the steady-state solution.

If we include the nonlinearities of the system in the analysis, then
more than one steady-state solution is produced and the one on which the
system settles will depend on its initial conditions. Such a solution need
not have the same periodic time as the applied force, although it will usually
be some multiple of it. Occasionally, however, when the transients decay,
the system may not settle into any regular oscillation but may continue
to vibrate erratically, never quite repeating any earlier manoeuvre and
yet never escaping from the well. This is a chaotic solution, otherwise
called a strange attractor. It can be represented graphically, so as to
reveal its detailed form and inner structure, in a way that distinguishes
chaos from the fuzziness arising from ‘noisy’ external forces.

Starting from different initial conditions, similar systems will have
different transient behaviour but may eventually settle to the same chaotic
solution. They will probably not be in step with each other but the nature
of the long term irregularity will be identical. They will have the same
chaotic structure. Ian Stewart describes several examples of such chaotic
solutions (see New ÐÓ°ÉÔ­´´, ‘Portraits of chaos’, 4 November 1989). Chaotic
solutions are undoubtedly very interesting but, in terms of safety, the
solution that the engineer is usually most concerned about is the ‘attractor
at infinity’. This is where the system can escape to. The transients of
this attractor do not decay but grow without limit: the system escapes from
the potential well and fails, collapses or capsizes.

The computer graphics shown on the front cover of this issue and in
Figure 2 are examples of one way to represent how a nonlinear system responds
to a periodic force. They are based on a technique called cell-to-cell mapping
developed by Chieh Su Hsu at the University of California at Berkeley. The
method is particularly useful for personal computers which can store a lot
of information in the colours of the pixels forming the graphics screen.
The blue regions, of either shade, in the centre-left picture of Figure
2 correspond to those initial conditions from which the system escaped out
of the well; the yellow regions signify systems that converged to a steady-state
solution of the same period as the applied force; and the red regions converged
to a different solution at twice the period. Each steady-state solution
is referred to as an attractor and is represented by only as many points
as its period. The surrounding region that converges to that attractor is
called the basin of that attractor. The graphic on the cover of this week’s
issue also shows an attractor with period-eight and its basin is coloured
green. For the corresponding linear system, there would be one black pixel
representing the unique steady-state and the rest of the screen would be
yellow, because everything would converge to it. Not as pretty, and not
as realistic.

Most of the basin boundaries in the graphic on the front cover and in
Figure 2 are fractals. This means that no matter how much you magnify the
boundary regions a structure made up of infinitely finer layers is revealed.

By examining many such pictures for different parameters – frequency,
amplitude of forcing, for example – we can obtain an overview of how the
system behaves. Figure 3 shows the amplitudes of the main attractors plotted
against frequency for three similar systems. In the linear system, there
is only one response at any frequency and the sharp peak of the graph where
the amplitude of the response becomes large is known as resonance. The nonlinear
system, corresponding to a one-sided escape well similar to the one shown
in Figure 1, has a range of frequencies where two stable period-one solutions
coexist, one with a small and one with a large amplitude. The peak of the
graph has bent over to the left, and near the top of the curve, there are
period-doubling cascades similar to the Feigenbaum cascades encountered
in the simple iterative equations discussed in Franco Vivaldi’s article
‘An experiment with mathematics’ (New ÐÓ°ÉÔ­´´, 28 October 1989), earlier
in this series. At the end of each cascade, there is a narrow band of frequencies
where a chaotic solution exists. Adjust the frequency a little more and
the chaotic solution vanishes in an event called a ‘blue sky catastrophe’.

With gradual changes in parameter, attractors generally evolve smoothly,
but at certain critical points, called bifurcations, the attractor may split
into different attractors or may simply disappear. Such an event is a catastrophe
in the mathematical sense, introduced by the French mathematician Rene Thom,
at the Institute for Advanced Scientific Studies in Bures-sur-Yvette, France,
who used powerful theorems to show how such catastrophes could be classified
into a small number of elementary forms. It is a catastrophe in the everyday
sense only if it means that a system close to the attractor can suddenly
head for the attractor at infinity. Some catastrophes are relatively safe
in that they need not imply collapse, merely a jump from one stable attractor
to another.

Figure 4a shows a good example. It may be more dangerous to slow down
a rotating machine than to start it up, because start-up may pass through
a safe catastrophe, whereas wind-down might lead to an unsafe blue sky catastrophe.
In this case, these are not yet hard and fast rules, and studies of these
jumps are giving some interesting results. Even the first jump may not always
be safe, it seems. Figure 4b shows the response of a ship experiencing waves
of gradually increasing height. It reveals an early safe catastrophe before
the dangerous blue sky event where capsize is inevitable.

In engineering, the parameters rarely vary slowly enough and smoothly
enough for the system to adhere rigorously to the paths of attractors. The
applied force inevitably contains some noise. Any real system attempting
to follow the steady-state solution paths (as in Figure 4) may escape long
before reaching the period-doubling cascades. We can understand the reasons
for this by looking not just at the attractors, but also at the basins,
their boundaries and how they evolve.

Around any attractor there is an area that converges to that attractor.
This region may be so thin or wispy that small disturbances can knock the
system out of the basin to converge somewhere else. To be safe, any engineering
system must have its attractor at a prudent distance from its basin boundary
(see Figure 5). Structural engineers refer to failure as ‘exceedance of
the ultimate limit state’. Once a structure passes this condition, it will
collapse. In the dynamical case, the ultimate limit state is the outer basin
boundary, where the blue meets the yellow and red. Beyond that boundary,
the system must escape. The ultimate limit state is therefore often a fractal.
Not a lot of engineers are aware of this yet.

We have used various ‘integrity measures’ to quantify the margin of
safety between an attractor and the basin boundary (see Figure 5). In studies
of how ships capsize, we have observed these measures to decrease very dramatically
at a point on the solution path at which the realistic, noisy system is
liable to escape. We refer to the phenomenon as the ‘Dover cliff’ effect.
It occurs when the fractal ‘fingers’ of the basin boundary suddenly intrude
deep into the centre of the basin. The effect happens irrespective of the
exact equation describing the potential well. Provided the well looks rather
like the one-sided escape well of Figure 1 with one trough, one hump, the
effect appears to persist.

Most mathematical approaches to nonlinear dynamical systems focus on
finding steady-state solutions and resonant motions of large amplitude.
A trawler may capsize, however, as a result of a few abnormally large waves
during which a steady rocking motion has not had a chance to develop. In
terms of the ultimate safety, the emphasis of the analysis thus shifts.
The steady-state solutions are almost irrelevant; the transient behaviour
is where the action is. Did the ship capsize or not? Engineers can examine
the computer graphics, not to look for what or where the attractors are,
but to see what the transients did. How many transients led to failure?
Because many engineering failures happen within the first few transients,
particularly in ship stability, then methods even more straightforward than
cell-to-cell mapping can be used.

Using simple computer programs, engineers can thus readily determine
when any specific design is liable to fail. But safe engineering should
not just rely on a computer printout. It must involve understanding how
a design will behave and how and why it can fail. This is where chaos theory
and the study of the fractals in all their exquisite mathematical intricacy
is unmatched by earlier methods.

The fractal basin boundary is just one part of a geometrical structure
called a homoclinic tangle and the inner workings of these tangles can
explain a lot of the features of nonlinear dynamic behaviour.

The archetypal manifestation of chaos is the strange attractor and these,
it seems, do not occur very often in engineering. You can find strange attractors
in impacting systems where the smooth dynamics is interrupted by hitting
against some nearby body or stop. They happen where the forces arise internally
rather than from simple external application. Very often though, strange
attractors exist only over extremely narrow ranges of frequency and require
the sort of ideal conditions that few real engineering systems can provide.
Chaos, however, exerts a far stronger influence than this would suggest.
Not only does a strange attractor often herald a dangerous catastrophe,
but there exist other more important and more common chaotic solutions.
These solutions attract only in some directions and repel in others. They
occur throughout homoclinic tangles.

The most far-reaching benefit of applying chaos theory to engineering
is that it is helping us to understand nonlinear dynamic behaviour in real
situations. When nonlinearities are included in engineering dynamics, many
of the familiar landmarks of linear analysis are left behind. Good engineering
demands that engineers have an intuitive feel for how their designs behave,
rather than just relying on a set of numbers. The study of the shapes of
the underlying fractal structures and the rules that govern their evolution
can provide just such a deeper understanding.

* * *

Homoclinic tangles, horseshoe maps and holistic engineering

AROUND the turn of the century, the great French mathematician Henri
Poincare first glimpsed the structure of a homoclinic tangle. He was, apparently,
horrified. Homoclinic tangles occur in celestial mechanics, where Poincare
first met them on Saturn’s rings. They also occur in engineering. They are
formed by two fractal structures overlapping, the ‘inset’ and ‘outset’ of
an unstable solution called a saddle.

In a simple one-sided escape system, the inset corresponds to the global
basin boundary. All points ‘inside’ the inset (yellow) converge to some
attractor and all points ‘outside’ it (blue) escape to the attractor at
infinity. The fractal nature of the curves is such that under any magnification
a set of nested stripes-within-stripes can be observed. This striation is
strongly related to the Cantor set, a fractal that is simple to construct
but which has many unusual properties. The Cantor set was discovered in
1875 by Henry Smith and later used by the German mathematician, Georg Cantor.
Unlike the famous Mandelbrot set, which is an infinitely jagged fractal,
the inset and outset fractals are smooth curves. Sensitive dependence on
initial conditions results from this infinite layering.

Engineering systems dissipate energy. One result of this is that although
the outset and inset may appear to be somewhat similar in shape, the inset
has infinite area ‘inside’ while the outset has zero area inside. The outset,
therefore, has the peculiar property of being an infinitely long, smooth,
connected perimeter that bites out all of its own inside. It may not look
like that in the graphic opposite, but only the first few bites have been
drawn. On occasion, the inset can do the same thing. This is the phenomenon
of basin erosion. The fractal inset (the boundary) intrudes deeper and deeper
into its interior basin, until at the blue sky catastrophe (between chaos
and the zone of no stable solution), the boundary succeeds in biting out
all of its own inside, too. There is no yellow left and nothing is safe.

Before chaos theory, nonlinear dynamic problems were mostly tackled
by perturbation theory. It usually involved a lot of algebra, long equations
and lots of sines and cosines. Chaos theory uses a very different language,
the language of topology.

This is a language engineers do not usually learn. Using this very rigorous
and pure mathematics, you can learn things about dynamics that do not depend
on the exact equations but rather on the ‘shape’ of the graphs and the ‘form’
of the equations. This is the concept of ‘universality’. The power of chaos
theory is embodied in this concept. What chaos theory implies is: ‘If the
equations have these properties then the dynamic response will have these
features’. This is very important in engineering where you never know the
exact equations and values.

Engineers may ask: ‘How can something so mathematically precise and
detailed as chaos be of any help when we never know our equations exactly?’.
The answer lies in ‘universality’. While ignoring many of the details of
the system, chaos theory can make detailed predictions about how it can
and cannot behave. What is more, chaos theory applies to the cumbersome,
mathematically messy but realistic equations of engineering. Do not worry
about the fine details of your system, just look at the shapes.

One example of this approach is called ‘symbolic dynamics’. It relies
heavily on the work of the American mathematician Stephen Smale and his
construction called the ‘horseshoe map’. It refers to regions of the colourful
graphics, the homoclinic tangles, which some number of forcing cycles later
map back across themselves in the shape of a horseshoe. The fascinating
abstraction of Smale was to show that much of the important dynamics in
those regions is equivalent to all the possible ways of writing down infinite
sequences of two symbols (0 and 1, say) on either side of a decimal point,
for example: strings of the form:

. . . 001011101011010.00100111101001 . . .

At any point where the inset crosses the outset, there is a horseshoe,
and the symbol string above represents one state of the system. Move the
decimal point one place to the right and you get the state of the system
one forcing cycle later. This is called a Bernoulli shift. Dynamic analysis
becomes an exercise in simple binary logic, writing down sequences of 0s
and 1s and moving decimal points. This is very different to the pages of
sines and cosines that engineers are used to.

Using such holistic techniques, we are beginning to understand many
of the features of the response graphs such as Figure 3. Moreover, such
methods provide a way to analyse, quantify and understand the mechanism
of basin erosion, with all its implications for engineering safety.

J. M. T. Thompson is professor of civil engineering and Allan McRobie
is Wolfson research fellow, both at University College London.

Further reading Nonlinear Dynamics and Chaos – Geometrical Methods for
Engineers and ÐÓ°ÉÔ­´´s, J. M. T. Thompson and H. B. Stewart, Wiley, 1986.

On 15 June, there will be a one-day seminar called ‘Chaos in Science
and Engineering’ at University College London. It will include lectures
by many of the authors who have contributed to this New ÐÓ°ÉÔ­´´ series
on chaos. Phone Steve Bishop on 071 387 7050 ext 2728 for further details.