The roadmap to chaos

The logistic function can behave strangely when numerically simulated. In this case, it is treated as a difference (or recursive) equation and called logistic map.

Since it does not show this behavior when calculated with its "exact" solution \(π‘₯_𝑑=\frac 1{(1+(\frac 1{π‘₯_0} βˆ’1) e^{βˆ’\gamma t}}\), the strange behavior is sometimes considered to be a modeling artefact.

However, as we noted before, it might be debatable whether β€žrealβ€œ continuity exists at all. What is more, there is evidence that the behavior of difference equations (or maps) can be found in natural dynamics as well. Some insect species in temperate climate zones for instance have relatively short seasons in which they are active, and a long period around winter during which they hibernate. This hibernation stage can be taken as undefined in respect to population size, and hence as a discrete time step in which the population size in the next year is recursively attained from the population size in the current year.

In order to see what might be called "the strange behavior of the logistic function", the following interactive model lets you increase the growth rate \(\gamma\) in the difference version of the Verhulst equation \(x_{t+1}= x_t + (\gamma*x_t*(1-\frac{x_t}{K}))*dt\).

Note that for rather small growth rates up to about 1.0 the dynamics show more or less the known behavior of the sigmoid-curve homing in on the carrying capacity \(K=0.7\). The ascent gets increasingly steep though.

However, for growth rates > 1.2 something strange happens. The dynamic overshoots the limit \(K\) only to fall back below it in the next step of time and subsequently overshoots it again several times till it finally homes in on \(K\).

If we increase \(\gamma\) further, the dynamics begin to oscillate periodically without settling down to around \(K\) anymore.

At \(\gamma\)-values > 2.4 the dynamics start oscillating with more than one period.

And finally at \(\gamma\)-values larger than \(\sim\)2.6 the dynamics do no longer show any discernible period. Mathematicians speak of deterministic chaos.

The following interactive model lets you test this with varying growth rates.

In the course of successively raising the growth rate, the periods of oscillations double repeatedly. This ongoing bifurcation of oscillation periods is called period doubling cascade.

This period doubling cascade is in fact rather un-chaotic in the sense that it is strictly deterministic and hence calculable. If you start the above calculations with exactly the same initial values you get the same results for sure. However, a characteristic feature of the logistic equation is that it shows high sensitivity to small changes in initial conditions. A small difference in the fifth or sixth decimal place can have dramatic impact on the results, just as the famous flap of a butterfly's wing in Brazil can cause a tornado in Texas (see Lorenz equations).

The add/remove-second-stock-button in the above model offers the option to add a second stock to the first one which differs in its initial value from the first stock by 0.000001. Running the model with a growth rate smaller than 2.6 will show just one blue curve as before, since both dynamics develop so closely together that differences are not discernible (one curve shields the other from view). Growth rates larger than 2.6 however will generate dynamics that visibly diverge. And as closer \(\gamma\) approaches 3 the divergence increases.

The two dynamics, although initially indiscernible close together, diverge to an extent that makes them seem to develop completely unpredictably.