A fundamental feature of chaos in dynamical systems is as follows: the evolution of physical quantities in time, although specified by definite rules, shows an extreme sensitivity to initial conditions. (The word "extreme" has in fact a precise quantitative meaning here, which we shall explain subsequently.) For the purpose of illustration, let us count time in discrete steps. The rule of time evolution is then called a "map". We will consider three maps as examples: (i) the so-called CAT MAP, (ii) a map related to the cat map which we shall call the SQRT MAP (because it is like the "square root" of the cat map), and (iii) the so-called BAKER MAP. These whimsical names have interesting origins. The cat map was so named because its action was first illustrated by Arnold with the help of a cartoon of a cat's face that got progressively distorted as the map was iterated. The baker map or baker transformation gets its name from the fact that its action is similar to the way a baker kneads dough for making bread.
Each of the maps above is a map that takes each point of the unit square in the x-y plane to another point in the square according to a definite rule. Any point in the unit square has coordinates (x,y), where x and y take values between 0 and 1. As mentioned earlier, time is measured in discrete steps, and denoted by the index n, where n = 0,1,2,.... The rules of evolution are specified as follows. Let x n and y n denote the values of x and y at time n. The values of these variables at time n + 1 are then specified recursively in terms of x n and y n by the following rules:
CAT MAP
x n+1 = [x n + y n ] mod 1
yn+1 = [xn + 2 yn] mod 1
SQRT MAP
x n+1 = y n mod 1
y n+1 = [x n + y n + 0.5] mod 1
BAKER MAP
When x n < = 0.5,
x n+1 = 2 x n
y n+1 = 0.5 y n
When x n > 0.5,
x n+1 = 2 x n - 1
y n+1 = 0.5 y n + 0.5
In the above, "mod 1" means that we omit the integer part of the expression to the left and retain just the fractional part that lies between 0 and 1. This is equivalent to "folding back" the variables so as to fit them into the interval between 0 and 1; it is a crucial step in obtaining chaotic behaviour because it makes the maps nonlinear . Otherwise the examples chosen would be relatively simple "linear" maps, with rather simple dynamical behaviour.
The following applet illustrates the extreme sensitivity to initial conditions of three maps above. You can choose any initial point and see how it evolves under the map. Further, you can also see the evolution of a second point which only differs from the first initial point by a very small amount. (This can be regarded as the "error" or level of accuracy in specifying the initial point.) The extreme sensitivity to initial conditions is demonstrated by the dramatically divergent evolution of the two initially close points within a few time steps.
In the above applet, you have a choice of three maps (cat map, sqrt map and the baker map) and of three parameters in each case: the initial position (x,y) in the square and the size of the error. Choose the map and the values of these three numbers, and press the start button. The red lines indicate the evolution of the point (x,y) under the map -- two consecutive points are connected by a line segment to make the evolution easier to follow by the eye. The blue lines indicate the evolution of the second point which is separated from your initial choice of (x,y) by two random numbers whose magnitude is less than the error. As you will see, even for a small choice of error such as 0.001, the two curves deviate quite rapidly, indicating extreme sensitivity to initial conditions.
We can now state what the precise meaning of the phrase "extreme sensitivity" is, in this context. It means that the divergence or separation of two initially very close points actually occurs exponentially rapidly as a function of time. Such an exponential separation is in stark contrast to the cases in which such a separation does not occur at all, or in which the separation grows like some positive power of the time. In the latter cases the evolution is computable . On the other hand, an exponentially growing initial error or imprecision makes the system variables uncomputable.