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GXWeb Chaos Explorer
Saltire Software, home of Geometry Expressions and GXWeb
Download a Geometry Expressions Model for Fractured Fractions
Symbolic computations on this page use Nerdamer Symbolic JavaScript to complement the in-built CAS of GXWeb
Take Continued Fractions to the Next Level
Consider a population, say of fish in a pond. If the pond is fixed in size and limited in the amount of food which it can provide, then the population of fish cannot grow unbounded. In fact, the size of the population itself will limit the growth - as the number of fish (\(x\)) gets large, it will act to slow down the rate of population growth (\(r\)). A simple model of this situation over time is given by the relationship: \[f(x)=r\cdot x\cdot (1-x)\]
In this simple model, the values of \(x\) may range between 0 and 1, and \(r\) is the growth rate of the population. This is an example of the logistic equation for population growth, a useful model for studying real-world populations. (Notice that as \(x\) approaches the upper value of 1, this will cause the population to decrease).
The points where the graph of this function meets the identity function (\(I(x) = x\)) are called equilibrium points and these indicate points at which there will be zero population growth in a year. If the initial population is \(x\), then the population after one year is \(f(x)\). After two years, \(f(f(x))\), and so on. If \(f(x) = x\), then \(f(f(x)) = x\) and so on. For example, when \(r\) = 4 we would solve the equation \(4 \cdot x \cdot (1-x)=x\) to give two equilibrium points, 0 and 0.75.
More interesting is to consider the behaviour (or orbit) of a point which is NOT an equilibrium point. For varying values of (r), the behaviour of such orbits is quite different. (The initial population actually has little effect in the medium to long term - however, the simple cobweb model below allows you to explore the immediate effects of both initial value and growth rate. NOTE that you can use the initial value slider to estimate the equilibrium value for different growth rates.)
The value of \(r\) has great impact upon the population growth. For small values of \(r\) (< 3) the population quickly settles to equilibrium. Between 3 and 4, the growth moves from stability to instability - in fact, to chaos!
The two graphs shown below capture this process in different ways: the first we have just encountered (called a cobweb graph) shows the orbit of a point over multiple iterations, and the second (a bifurcation map) beautifully illustrates the change known as period doubling.
Population Growth Rate r =
Start value =
Iterations (years) =
Bifurcation Map
© Jeremy Likness
One of the characteristics of chaotic behaviour in complex systems appears to be a feature known as period doubling. When \(r\) = 3, you should observe that the population oscillates between two stable values - this is an example of a period 2 cycle. By the time \(r\) = 3.5, however, there is clearly a change - in fact, we have moved to a period 4 cycle (hence the term period doubling).
It appears that for values of \(r\) between 3 and 3.43... the cycle has period 2.
For 3.43... < \(r\) < 3.53... the cycle is of period 4.
By around 3.83... all cycles of the form \(2^n\) appear to have been exhausted, and 3-cycles take over. These occur even more rapidly and by the time \(r\) = 4 the system has reached a chaotic state.
But what are these mysterious values which trigger such changes? Perhaps you might explore using the tools available here: continued fractions, CAS, graphs and tables of values? Follow a little further down this rabbit hole by investigating the Feigenbaum Constants, the first of which \(\delta \approx\) \(4.669201609102990671853203820466\) is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map - just like the one considered here!
Perhaps at this point you are wondering what chaos has to do with continued fractions? You have seen that, if we exclude rationals, quadratic irrationals and the odd transcendental number (like \(e\)) then all continued fractions are inherently chaotic in their patterns.
But it goes deeper than this. Think back to the process of building a continued fraction. After extracting the whole number part, the next two steps are to take the reciprocal of the tail, and subtract the floor of this reciprocal (the whole number part once again. This may be thought of as a function: \(\frac{1}{x}-floor(\frac{1}{x})\). When viewed graphically what we observe is an infinite number of random hyperbolic branches - a perfectly chaotic function lies at the very heart of every continued fraction!
These are the elements of chaos, a field at the cutting edge of mathematics and human knowledge. There is much yet to be learned about its properties and applications, and yet it is accessible through such simple models as those studied here. Be patient and systematic and you may discover something new and unexpected!
