Logistic map

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 popularized by the biologist Robert May in 1976. It was originally made as a very simple model for the population numbers of species in the presence of limiting factors such as food supply or disease, containing two causal loops: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

(1)     ,

where:

x_n 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.

Table of contents

1 Behaviour dependent on r 2 Chaos and the logistic map 3 External links

Behaviour dependent on r

By varying the parameter r, the following behaviour is observed:

• With r between 0 and 1, the population will eventually die, independent of the initial population.
• With r between 1 and 2, the population will quickly stabilize on a single value; this value depends on r but does not depend on the initial population.
• With r between 2 and 3, the population will also eventually stabilize on a single value, but first oscillates around that value for some time. Again, the final value does not depend on the initial population
• With r between 3 and 1+√6 (approximately 3.45), the population will oscillate between two values forever. These two values are dependent on r but independent of the initial population.
• With r between 3.45 and 3.54 (approximately), the population will oscillate between four values forever; again, this behavior does not depend on the initial population.
• With r slightly bigger than 3.54, the population will oscillate between 8 values, then 16, 32, etc. The lengths of the parameter intervals which yield the same number of oscillations decrease rapidly; the ratio between the lengths of two successive such bifurcation intervals approaches the Feigenbaum constant δ = 4.669. All of these behaviors do not depend on the initial population.
• 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 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 above image.

Chaos and the logistic map

The relative simplicity of the logistic map makes it an excellent point of entry into a consideration of the concept of chaos. A rough description of chaos is that chaotic systems exhibit a great sensitivity to initial conditions -- a property of the logistic map for most values of r between about 3.57 and 4 (as noted above). A common source of such sensitivity to initial conditions is that the map represents a repeated folding and stretching of the space on which it is defined. In the case of the logistic map, the quadratic difference equation (1) describing it may be thought of as stretching-and-folding operation on the interval (0,1).

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 X_t 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.

Note, however that this unpredictability is not the same as randomness: if we did have perfect (that is, error-free) knowledge of the initial state and the system, we could (in principle, given also access to perfect computation) be able to make error-free predictions about any future state. The system is unpredictable because knowledge and computation are (in practice) always subject to some degree of error, and any error in measurement or computation, no matter how small, will grow exponentially in our predictions. Greater accuracy and precision will improve predicitions of future states, but in chaotic systems of the type described above, precise and accurate predictions about the state of the system in the arbitrarily distant future can never be made. To contrast, predictions about a random process always contain some error, even given knowledge of the initial state and computation that are without error.

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.

For more information see the article Chaos theory.

External links

• Dan Marthaler: The Logistic Map, http://mathpost.la.asu.edu/~daniel/logistic.html . Contains an interactive computer simulation of the logistic map and also allows to zoom in on the bifurcation diagram.
• Another interactive simulation: http://www.geocities.com/CapeCanaveral/Hangar/7959/logisticmap.html
• Logistic model. An unusual, although straightforward, Java simulation.
• Emergence of Chaos. The famous bifurcation diagram is linked interactively to iterations on a parabola.