Definitions

Logistic map

The logistic map is a polynomial mapping of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst. Mathematically, the logistic map is written
$\left(1\right)qquad x_\left\{n+1\right\} = r x_n \left(1-x_n\right)$
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.
This nonlinear difference equation is intended to capture two effects.

• reproduction where the population will increase at a rate proportional to the current population when the population size is small.
• starvation (density-dependent mortality) where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population.

However, as a demographic model the logistic map has the pathological problem that some initial conditions and parameter values lead to negative population sizes. This problem does not appear in the older Ricker model, which also exhibits chaotic dynamics.

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 the value

$frac\left\{r-1\right\}\left\{r\right\}$, independent of the initial population.

• With r between 2 and 3, the population will also eventually stabilize on the same value

$frac\left\{r-1\right\}\left\{r\right\}$, but first oscillates around that value for some time. The rate of convergence is linear, except for r=3, when it is dramatically slow, less than linear.

• With r between 3 and $1+sqrt\left\{6\right\}$ (approximately 3.45), the population may oscillate between two values forever. These two values are dependent on r.
• With r between 3.45 and 3.54 (approximately), the population may oscillate between four values forever.
• With r increasing beyond 3.54, the population will probably 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$dots$. This behavior is an example of a period-doubling cascade.
• At r approximately 3.57 is the onset of chaos, at the end of the period-doubling cascade. 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; these are sometimes called islands of stability. For instance, beginning at $1+sqrt\left\{8\right\}$ (approximately 3.83) 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.
• 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, the situation nearby looks 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 is available Logistic-burification.png.

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 a stretching-and-folding operation on the interval (0,1).

The following figure illustrates the stretching 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 system 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.

Since the map is confined to an interval on the real number line, its dimension is less than or equal to unity. Numerical estimates yield a correlation dimension of 0.500 ± 0.005 (Grassberger, 1983), a Hausdorff dimension of about 0.538 (Grassberger 1981), and an information dimension of 0.5170976... (Grassberger 1983) for r=3.5699456... (onset of chaos). Note: It can be shown that the correlation dimension is certainly between 0.4926 and 0.5024.

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 decisions based on the state of the system.

References

Textbooks

• Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 0-19-850840-9.
• Strogatz, Steven (2000). Nonlinear Dynamics and Chaos. Perseus Publishing. ISBN 0-7382-0453-6.
• Tufillaro, Nicholas; Tyler Abbott, Jeremiah Reilly (1992). An experimental approach to nonlinear dynamics and chaos. Addison-Wesley New York. ISBN 0-201-55441-0.