Logistic Map Numerics

In a fascinating and influential review article, Robert May (1976) emphasized that 

Suppose we fix r, choose some initial population x , and then use (1) to generate the subsequent x . What happens  

For small growth rate r 1, the population always goes extinct x 0 as n — oo. This gloomy result can be proven by cobwebbing (Exercise 10.2.2). 

Note that the successive bifurcations come faster and faster. Ultimately the converge to a limiting value r. The convergence is essentially geometric in the limit of large n, the distance between successive transitions shrinks by a constant factor 

We ll have a lot more to say about this number in Section 10.6. 

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Incidentally, this gives a remarkably accurate prediction of for the logistic map. Recall that p = 0 corresponds to the birth of period-2, which occurs at r = 3 for the logistic map. Thus p corresponds to = 3.56 whereas the actual numerical result is c = 3.57 ... 

Numerics of superstable cycles) Let denote the value of r at which the logistic map has a superstable cycle of period 2". 

Band merging and crisis) Show numerically that the period-doubling bifurcations of the 3-cycle for the logistic map accumulate near r = 3.8495..., to form three small chaotic bands. Show that these chaotic bands merge near r =3.857... to form a much larger attractor that nearly fills an interval. 

Fat fractals answer a fascinating question about the logistic map. Farmer (1985) has shown numerically that the set of parameter values for which chaos occurs is a fat fractal. In particular, if r is chosen at random between r and r = 4, there is about an 89% chance that the map will be chaotic. Farmer s analysis also suggests that the odds of making a mistake (calling an orbit chaotic when it s actually periodic) are about one in a million, if we use double precision arithmetic ... 

This idea, which was developed by Poincare around the turn of the century, is as follows. Instead of following up the whole path of trajectories in phase space we consider the crossing points of these trajectories with a plane (or a corresponding hyperplane). Quite often one finds from numerical studies or experimental 1y that the crossing points can be connected, at least to some approximation, by a line so that we can label the crossing points, xi,X2,X3,... In the next step one studies how the point x is connected with its previous point x. This is described by x = f(x ). A typical and by now wellknown example is the logistic map... 

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