The logistic map is an archetypical example of how very complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was broadly known due to a seminal paper by the biologist Robert May in 1976. The logistic model was originally introduced as a demographicmodel by Pierre Fran%E7ois Verhulst. Later applied on surplus production of the population biomass of species in the presence of limiting factors such as food supply or disease, containing two causal loops:
due to reproduction the population will increase at a rate proportional to the current population
due to starvation, the population will increase at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population.
Mathematically this can be written as
xn is a number between zero and one, and represents the population at year n, and hence x0 represents the initial population (at year 0)
r is a positive number, and represents a combined rate for reproduction and starvation.
At r = 3.57 (approximately) is the onset of chaos. We can no longer see any oscillations. Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.
Most values beyond 3.57 exhibit chaotic behaviour, but there are still certain isolated values of r that appear to show non-chaotic behavior; for instance around 3.82 there is a range of parameters r which show oscillation between three values, and for slightly higher values of r oscillation between 6 values, then 12 etc. There are other ranges which yield oscillation between 5 values etc.; all oscillation periods do occur. These behaviours are again independent of the initial value.
Beyond r = 4, the values eventually leave the interval [0,1] and diverge for almost all initial values.
A bifurcation diagram summarizes this. The horizontal axis shows the values of the parameter r while the vertical axis shows the possible long-term values of x.
The bifurcation diagram is a fractal: if you zoom in on the above mentioned value r = 3.82 and focus on one arm of the three, say, the situation nearby looks just like a shrunk and slightly distorted version of the whole diagram. The same is true for all other non-chaotic points. This is an example of the deep and ubiquitous connection between chaos and fractals.
A GNU Octave script to generate bifurcation diagrams can be found in the description of the .
The following figure illustrates the streching and folding over a sequence of iterates of the map. Figure (a), left, gives a two-dimensional phase diagram of the logistic map for r=4, and clearly shows the quadratic curve of the difference equation (1). However, we can embed the same sequence in a three-dimensional phase space, in order to investigate the deeper structure of the map. Figure (b), right, demonstrates this, showing how initially nearby points begin to diverge, particularly in those regions of Xt corresponding to the steeper sections of the plot.
This stretching-and-folding does not just produce a gradual divergence of the sequences of iterates, but an exponential divergence (see Lyapunov exponents), evidenced also by the complexity and unpredictability of the chaotic logistic map. In fact, exponential divergence of sequences of iterates explains the connection between chaos and unpredictability: a small error in the supposed initial state of the sytem will tend to correspond to a large error later in its evolution. Hence, predictions about future states become progressively (indeed, exponentially) worse when there are even very small errors in our knowledge of the initial state.
It is often possible, however, to make precise and accurate statements about the likelihood of a future state in a chaotic system. If a (possibly chaotic) dynamical system has an attractor, then there exists a probability measure that gives the long-run proportion of time spent by the system in the various regions of the attractor. In the case of the logistic map with parameter r = 4 and an initial state in (0,1), the attractor is also the interval (0,1) and the probability measure corresponds to the beta distribution with parameters a = 0.5 and b = 0.5. Unpredictability is not randomness, but in some circumstances looks very much like it. Hence, and fortunately, even if we know very little about the initial state of the logistic map (or some other chaotic system), we can still say something about the distribution of states a long time into the future, and use this knowledge to inform decisionss based on the state of the system. All hope is not lost in a chaotic world.