Maths Examples of Chaos
Bifurcation diagram
Simple Chaos
Chaos does not require complex equations. It can be produced by very simple ones. Take a look at the simple equation cx2-1 where c is a real valued constant.
Now, by iterating this equation on x, varying results can be obtained, some displaying chaos.
By 'iterating on x' I mean take some starting value of x = x0, then keep putting the value of the equation back into itself, such that: xn+1 = cxn2-1
What we get is an infinite sequence of values, x0, x1, x2,... which can then be analysised.
- The sequence might eventually settle down to some value, the sequence is said to converge to this value.
- The sequence might settle into a cycle of a finite number of values, in which case the sequence is said to be a cycle.
- The sequence may just head off towards +/- infinity. This is called divergence.
- However the sequence may not display any sort of pattern, and in fact will continue not to exhibit any sort of pattern infinitely. It is these sequences which are said to exhibit chaos.
Referring to the above equation, values of c of about 1.5 and above appear to cause chaos, with the larger c, the more chaotic. However this is not entirely true. At c=1.74 the sequence does not ever repeat, and is chaotic. At c=1.75 the same appears to be true at first, but after many iterations (>100) the sequence settles down to a cycle of length 3. Here we have an example where a small change in the conditions changes the output dramatically. The chaos, and the simple pattern of length 3, are produced by similar conditions, but in the long run are completely different. This is the hallmark of chaos.
The Logistic Mapping
The logistic mapping is an important mapping. It is a basic mapping from which many other chaotic mappings can be studied. First though, what is a mapping? You may already be familiar with the concept, and the word mapping here has the same meaning as it always does in maths. A mapping is another word for a function, that is a rule to get from an input x to an output f(x). For example, above the mapping given is: x -> cx2-1
The logistic mapping is similar in form, and is given by: x -> kx(1 - x) where k is a real valued constant between 0 and 4, and x0 has initial value between 0 and 1
Iterating, as above, we get: xt+1 = kxt(1 - xt)
Geometrically, the Logistic mapping represents stretching/compressing a line segment in a non-uniform way, and then folding it in half. This stretching/folding combination is a classic recipe for chaos. We shall now examine the logistic mapping, and the way it behaves.
Steady State Regimes
For values of k between 0 and 3 iterating on any x0 between 0 and 1 will produce a sequence that converges. For example, take k = 2, and x0 = 0.9. The values for the sequence {xt} are as follows:
| value of t | xt |
| 0 | 0.9 |
| 1 | 0.18 |
| 2 | 0.2952 |
| 3 | 0.41611392 |
| 4 | 0.485926251164 |
| 5 | 0.499603859187 |
| 6 | 0.499999686145 |
| 7 | 0.5 |
| 8 | 0.5 |
| 9 | 0.5 |
As can be seen, the sequence has converged to 0.5. 0.5 is called a point attractor for the system, a steady stable state. In the long term, such a system remains at rest, it does nothing. Now this is not the most interesting bit of maths, but it leads onwards...
Period-doubling Cascades
For k = 3, we have a kind of intermediate stage. The sequence will converge at k = 3, but it converges infinitely slowly. As k is increased, the sequence becomes unstable, but then settles down into a period 2 cycle - there are 2 point attractors. But at about k = 3.5 this 2-cycle becomes unstable, and as k is increased we get a period 4 cycle. The rate at which this doubling occurs increases, and by k = 3.56 there is a period 8 cycle, by k = 3.567 there is a 16-cycle. This continues infinitely, but happens so quickly that by k = 3.58 it has finished. At this point the mapping becomes chaotic. There are more period doubling cascades to come though. At k = 3.835 there is a period 3 cycle, doubling to 6, 12, 24 etc as k is increased very slightly. There is a period 5 attractor at k = 3.739, which again forms a period doubling cascade (5, 10, 20, 40,...). The simple logistic mapping produces this extremely complex mixture of chaos and order.
Bifurcation Diagrams
The behaviour of the logistic mapping can be described graphically by a bifurcation diagram. This is a graph which plots the value of k across, against the attractors of the sequence vertically:
Bifurcation diagram - click to enlarge
The single branch at the beginning represents the steady state, which branches into two. This is the period 2 cycle. Now we can graphically see the period doubling cascade, as the branches each divide into two, until there is chaos. Amidst the chaos, new branches spring up, and then double themselves - these are the new period doubling cascades. (To see the proper detail, you will need to examine the larger version of the diagram.) I wrote a simple qbasic program to produce the bifurcation diagram, click here to see the code.
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Will Bolam 2001
