Think of a number x, put it into a simple equation and feed the equation
to a computer. Put the answer back into the equation. Repeat the exercise
and watch chaos evolve before your eyes
Franco Vivaldi
If something is too large, make it smaller, if it is too small
make it larger. This is not a quotation from Mao's Little Red
Book, but rather a simple recipe for constructing a dynamical process
called feedback. It is a constant presence in our lives, and in many practical
situations, it is important to predict how large or small a variable quantity
associated with feedback will eventually become.
Common sense would
suggest that it will settle down somewhere in the middle, where it is
neither too large nor too small. But this answer looks suspiciously simple,
and in fact it can be terribly wrong.I want to show here how
feedback may turn into chaos.
The classic example comes from
population dynamics, where feedback
prevents populations of plants or animals from growing indefinitely. For
instance, imagine that our feedback variable is the number of fishes in an
ideal lake, free from pollution and fishermen. If there are few fishes, they
will thrive in the favourable environment and reproduce rapidly, and the
fish population will increase. But if there are
too many, they will compete for food and suffer from their own pollution
(an ideal lake, remember), and their number will decrease.
We can formulate this problem in an abstract setting using simple
mathematics. Let us call the variable x_{t }, where the subscript
t, a whole number, stands for the time, with the stipulation that
successive times correspond to successive measurements of the variable
x. Thus t does not necessarily represent the physical time,
but it is rather a convenient label that orders a succession of events. In
the population problem, t would label every new generation.
Assuming that t = 0 corresponds to the present, that is,
x_{0}, is the value measured at the beginning of the observations,
then predicting the future means computing x_{t }. The larger
the value of t, the more remote the future that we are probing. We
can even let t approach infinity, a privilege denied in real
forecasting.
Dynamics comes into play when we specify a rule for transforming
x, which sets the picture in motion. I assume that the result of the
measurement at time t + 1 is unambiguously determined by that
at time t. We can write this relation in a mathematical form :
x_{t + 1} =
f(x_{t})
The letter f denotes a "function" , which is the way
mathematicians indicate the precise relation between two quantities 
x_{t + l} depends on x_{t }, and only
x_{t }.
The detailed structure of the function f is of no concern at the moment,
but assuming that the link between the current and the successive values
of x is described by a function is no small deal. In one stroke, I
have removed any ambiguity in the determination of x_{t };
chance, unknown external factors, noise, are not allowed to play any role
herethis process is deterministic. In other words, the future of
deterministic systems follows from the present, without uncertainties.
"It is not like playing roulette",
it would be tempting to say. But do not say it, because we will be gambling
later on, using a simple deterministic feedback system.
All information about the system is stored within the function f.
Consider now a specific process, given by the following function
f:
(equation 1) which, if you remember from school mathematics,
represents a parabola. The function f depends on a certain parameter
l, which we have introduced
so as to incorporate in a single description of the behaviour of a whole
family of feedback systems. This parameter quantifies the strength of the
feedback, that is , the amount by which the feedback is correcting the value
of the variable x. Geometrically, by varying
l, we vary the shape
of f.

Figure1 A graph of Equation1 for different values of l 
It is useful to see what f looks like for various values
of l. To do this, we
can plot x_{t} against x_{t+1 }on a plane (see
Figure 1). All points lying above the diagonal given by x_{t+1
}= x_{t} have the property that x_{t+1
}is greater than x_{t} , while for those below it
we have x_{t+1 }less than x_{t}. The graph
of f will, therefore, lie above the diagonal for small x and
below it for large x. Then it must necessarily cross that line, at
least once. At the crossing point, we have x_{t+1 }=
x_{t}, that is x_{t} retains the same value
in two successive measurements.
Equation 1 is a famous model, called the
logistic map, originally proposed in
the context of population dynamics. The variable x is meant to be
restricted between 0 and 1, only apparently a limitation because there is
no harm in assuming that x has been suitably scaled so as to lie between
these limits. Does this simplelooking problem have an exact solution ? Can
we derive a formula expressing x_{t} explicitly as a function
of x_{0}, and
l? This would be the
formula for predicting the future. The answer is yes. There is a straightforward,
if naive, way of obtaining such a solution, as I shall indicate, but the
result will prove to be disappointing. If you find formulas hostile and
incomprehensible, bear with me for the next two paragraphs, and your worst
fears will be confirmed.
I begin by letting t=0 in equation 1. In this way,I obtain
the value of x_{1} as a function of x_{0},
and l, that is:
x_{1} =
x_{0}
(l  x_{0})
Now I can take equation 1 again, but with t = 1. This
gives me x_{2} =
lx_{1}(
l  x_{l} ), and I can substitute the equation for
x_{1},already obtained, to get the equation:
x_{2} =
l (lx_{0} (l  x_{0}) ( 1 
lx_{0} (l 
x_{0})))
One more time. From x_{3} =
lx_{2} (
l  x_{2} ), and using the expression for x_{2}
just found, 1 obtain x_{3} as a function of
x_{0} and
l, that is, the future
three steps ahead, as a function of the present, at the chosen value of the
parameter:
x_{3} =
l(l(lx_{0} (l 
x_{0})(1 
lx_{0}(l 
x_{0})))(1 
l(lx_{0}
(l  x_{0})(1 
lx_{0} (l 
x_{0})))))
You could scarcely get excited about this accomplishment. These
formulas rapidly become long and inscrutable ; the expression for
x_{3} is already unfriendly, I could not fit it on a
single line, x_{15} will fill a book and
x_{30} the British Library. This is because the formula
for x_{t} contains physically the formula for
x_{t1}, which in turn contains that for
x_{t2}, and so on. The formal solution to our problem is
clearly useless for predicting anything but the very near future, and should
make you ponder over the meaning of the word "solution", an attribute these
formulas do not deserve. We shall see that these equations cannot be simplified
plainly because what they describe is not simple. There is chaos in the system,
this problem has no solution.
What is actually happening to x_{t} as t
increases? We ought to perform an experiment. After all, this is what
people do in the physical sciences when theory fails. Rather than searching
for a result valid for all x_{0} and
l(the sequence of functions
above) , I shall choose specific values of x_{0} and
l, from them compute:
then:
and so forth, up to x_{t} . Manipulating numbers
instead of functions makes this process much more economical. At each step,
all we have to retain of the past is the previous value of x, which
is one number, rather than the entire past history of all possible processes.
The outcome of this mathematical experiment will be a sequence
of measurements x_{0},
x_{1},x_{2}....x_{t}, just like
in real experiments. We will not have a "formula" to pride ourselves with,
but we will gain precious information by means of iterative calculations.
Computers love
iteration  repeating the same task
over and over again. In our case, each individual task, the computation of
f(x) , is actually very simple, two multiplications and one subtraction
in all. With each arithmetical operation taking a tiny fraction of a second,
the prospect of computing x_{1000000 }becomes real. It can
be done overnight on your personal computer , and faster than a blink on
a Cray supercomputer. Moreover, in numerical experiments, the precision with
which the ingredients and the results can be measured is limited only by
the size and power of the computer.
To unveil the presence of chaos, 1 will choose a specific numerical experiment. Let x_{0} = 0.4, and compute x_{t} for t = 1, . ., 15, and for a few values of l. Those who are familiar with a programming language will find that the bulk of the program consists of a simple iterative loop,specified by a sequence of statements such as (here in Basic):
FOR I=1 TO 15 X=LAMB*X*(l  X) PRINT X NEXT 
Not a very intimidating program. The results may look like Figure
1.
For l = 2, the experiment
brings comforting news. We witness the predicted relaxation of x,
growing rapidly to a value that is neither too small, nor too large, x
= 0.5. This is the value at which the function f_{2}(0.5)
= 0.5 (see Figure 1), that is why once x reaches 0.5, it remains there.
The outcome of the second experiment is more puzzling. The value of
l of 1 +
Ö5 = 3.236 . .
. was not a random choice.
The sequence of numbers appears to approach a final regime
where two distinct values of x are alternating. Had we thought about
the feedback process more carefully we could have predicted this behaviour.
At this value of the parameter the feedback is strong enough to produce an
overcorrectiona value of x that is too small is followed by one that
is too large, and vice versa. This causes x to relax to a configuration
where the opposite overcorrections balance each other precisely and we get
regular oscillations between x = 0.50000 and x = 0.80902.
t  l= 2  l = 1 + Ö5  l = 4  t  l= 2  l= 1 + Ö5  l= 4  
0  0.40000  0.40000  0.40000  0  0.35000  0.35000  0.40001  
1  0.48000  0.77666  0.96000  1  0.45500  0.73621  0.96001  
2  0.49920  0.56133  0.15360  2  0.49595  0.62847  0.15357  
3  0.50000  0.79684  0.52003  3  0.49997  0.75561  0.51995  
4  0.50000  0.52387  0.99840  4  0.50000  0.59758  0.99841  
5  0.50000  0.80717  0.00641  5  0.50000  0.77820  0.00636  
6  0.50000  0.50368  0.02547  6  0.50000  0.55856  0.02526  
7  0.50000  0.80897  0.09928  7  0.50000  0.79792  0.09850  
8  0.50000  0.50009  0.35768  8  0.50000  0.52180  0.35518  
9  0.50000  0.80902  0.91898  9  0.50000  0.80748  0.91610  
10  0.50000  0.50000  0.29782  10  0.50000  0.50307  0.30743  
11  0.50000  0.80902  0.83650  11  0.50000  0.80899  0.85167  
12  0.50000  0.50000  0.54707  12  0.50000  0.50006  0.50531  
13  0.50000  0.80902  0.99114  13  0.50000  0.80902  0.99989  
14  0.50000  0.50000  0.03514  14  0.50000  0.50000  0.00045  
15  0.50000  0.80902  0.13561  15  0.50000  0.80902  0.00180  
Figure 2 The results of computing x_{t} in equation 1,starting from x_{0} = 0.4,for three different values of l.  Figure 3 The results of computing x_{t} starting from x_{0}_{ }= 0.35000.The numbers differing from those in Table 1 are in bold. 
The surprise comes from the sequence in the column on the far
right which you could have hardly guessed. It does not show any obvious pattern,
and you might think that there was a mistake in programming, but there is
no error. Successive values of x appear to be unrelated, but they
are, and by just two multiplications and one subtraction ; you can check.
This lack of a discernible structure is not a peculiarity of the first 15
data, it will persist as long as your computer can compute. This is chaos.
Not just a pretty pattern; this beautiful
and unusual computer generated picture shows what happens to equation 1 when
l is allowed to alternate
between two numbers A and B.The colours in the picture show the nature
of the motion, ranging from the orderly (the dark area) to completely
chaotic (lighter areas).A is the xaxis and B is the yaxis. 
It is now clear that we are in possession of a remarkable model,
whose simplicity contrasts with the variety of behaviour that it can produce.
So far, only the feedback strength
l has been changed,
but there is more than one reason for changing the initial state as well.
There was nothing special about x_{0}= 0.4, and we should
try other values. It is perhaps even more illuminating to vary the initial
state by small amounts, in order to simulate small uncertainties or errors
in the measurement of the initial datum, and assess their impact on the future
evolution of the system.
The second experiment is a replica of the first, but with different
values of x_{0}. This time we obtain the data in Figure 3.
For the first two values of
l, the initial state
was changed by 5 per cent. This would be a realistic uncertainty on the initial
data, if somewhat large. The discrepancy in the early values of
x_{t} fades away rapidly (more so in the first experiment)
, as the time evolution brings the system to the same final state. I invite
you to take the time to try other values of x_{0} in the unit
interval, to convince yourself that the resulting sequences are invariably
attracted to the same final regimes, a single point in the first experiment,
a pair of them in the second. No wonder these sets have been given the pictorial
name of
"attractors".
For
l= 4, the initial state
was changed by only one part in a 100 000, and we are entitled to expect
a virtually identical replica of the previous experiment. But this time the
error increases, and at a remarkable rate (show in bold in Figure 3). By
the 15th measurement, it has contaminated all available digits, making our
predictive power null. This is the signature of chaos. Had we doubled the
number of digits of accuracy, in other words made the accuracy 100 000 times
as great, we would just have postponed the problem for twice as long. There
is no way out; at some point the results of the experiments are going to
become meaningless.
We can make use of this property, and transform this process
into an honest mathematical toss of a coin, where the value of x smaller
or greater than 0.5 will mean "head" or "tail", respectively. Pick the value
of x_{0} of your choice, and then place your bet on
x_{20}.
We have arrived to the core of the issue, the realisation that there are
systems, even within mathematics, that are both deterministic and
unpredictable. We cannot blame this failure on the influence of unknown
factors, because there are none. It is rather the result of our own terminal
inability to measure or represent the present with infinite precision.
People have suspected for a long time that chaos exists in dynamical
systems, but it took computers to demonstrate it and assess the
implications. The history of the logistic map provides a marvellous example
of the interplay between theory and experiment within mathematics, something
that was once a prerogative of the physical sciences.
Period doubling and Feigenbaum numbers 

The first two columns of Figures 2 and 3 in the main text are examples of what are called periodic orbits. Chaologists like to use the geometrical language in describing changing values. So saying that an orbit is periodic just means that x eventually cycles back to its original value. In the case of the first column, once x settles down to a constant value, 0.5000. it returns to the same value after each successive step down the column. So the first column is said to have a period of 1. In general, we can say that if x returns exactly to its original value after T steps, the orbit has a period T and has only T distinct points. For the second column, then, the orbit settles down to period 2 as it alternates between 0.50000 and 0.80902. 
For any positive whole number T there are values of
l, with orbits of period
T, but their arrangement is extraordinarily complicated, as you will
find out if you experiment with a computer. 
That is the property of "renormalisation" When the periods are
sufficiently high, the magnified fine structure for one orbit, period 2048
for example, is indistinguishable from the structure for the previous period,
in this case period 1024, provided that you carry out the magnification to
a precise specification: the magnification in
l should be 4.66920166...
and the magnification in x should be 2.502908... Feigenbaum found
that the same numbers and the same structure appear for all sufficiently
smooth functions f(x), whose graph has only one maximum. So these
numbers, like p are
universal. Period doubling and Feigenbaum numbers appear not only on the
mathematician's computer screen but also in many kinds of natural chaos including
the dripping tap and the beating heart. 

Computers as a tool for mathematics
Early theoretical results had already explained some qualitative aspects
of the changing behaviour of the system as the parameter is varied (without
using my awkward formulas though). They encouraged mathematicians to run
extensive computations on their computers. These unveiled quantitative phenomena
crucial for the understanding of chaos. A new mathematical theory originated
from the computer experiments, and this process of symbiosis culminated in
a computer assisted proof of the main predictions regarding the transition
between order and chaos.
Common folklore relegates mathematics to the theoretical sciences,
where information and knowledge is reached
by logical steps within an abstract framework,
and not from experiments. In fact, mathematical discoveries are more likely
to spring from the patiently acquired experience of many specific computations.
The milestones of abstract thinking that have characterised the mathematics
of our century have left little room for public display of the usefulness,
and the charm, of mathematical experimentation.
Yet all great mathematicians of the past
were eager computers, and felt little obligation to hide their calculations
behind a polished façade of abstraction. Karl
Freidrich Gauss, one of the greatest mathematicians, once refrained
from disclosing a numerical table so as not to deprive the reader from
the pleasure of computing it. The tendency towards experimentation has been
particularly notable in number theory, where so many famous theorems have
been inferred from numerical data, and even used before proofs were available.
Computers have added a new dimension to the experimental side
of mathematics, and have made mathematical experimentation as fruitful and
tangible as that of the physical world. This is particularly true in dynamics,
because of the natural role of computers in iterative processes. Simple rules,
like the logistic map, can disclose unexpected treasures when applied over
and over again, but often the results materialise only after millions of
operations.
The marriage between chaos and computers has even more profound
roots. There is an intimate relationship between chaotic dynamics and the
structure of the number system itself, which iteration helps bring to the
surface, and which gives the science of chaos an appeal to fundamentals that
few other sciences have. Extreme sensitivity of the initial conditions
characterise a chaotic system. This mathematical experiment, brings the finest
details of the numbers representing the initial state
x_{0} into centre stage. In the final analysis, the key for
understanding the future is buried within the arithmetical properties of
those numbers.
I cannot help wondering what mathematics would be like if the
human brain could perform 10^{12 }arithmetical operations per second.
It would certainly be very different, and so would be our mathematical
description of the physical world. Machines of that speed are now being
conceived. Mathematics will change.
The Author
Franco Vivaldi is in the School of Mathematical Sciences
at Queen Mary and Westfield College, London. 
Further Reading
R. H. Abraham and C. D. Shaw. Dynamics the geometry
of behaviour. Volumes 14, Aerial Press, Santa Cruz, California. 
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