# Generalized logistic map

xn+1 = 4rxn(1-xn) = 4 r ( xn -xn2)

The logistic series rule sets the next generation proportional to the existing one in its first term. This alone would lead to exponential growth for r > 1/4, and to exponential decline for r < 1/4). The second term introduces a diminution that depends on the square of the existing population (note that in the above formulation xn < 1 , hence xn2 < xn ).

The issue for a given growth rate r is: will the population approach a stable equilibrium (limit) value of linear growth and quadratic extinction − assuming an unlimited number of generations under identical conditions. If so, how does the equilibrium value depend on the growth rate r ?

Growth exists when xn+1 > xn, hence r > 1/(4(1-xn)). As 0 < x < 1 , for r < 0.25, all populations will iterate to zero, independent of the starting value. If for r > 0.25 an equilibrium value exists, a starting population greater than the limit should shrink to it, smaller ones should expand to it.

In the simulation r is increased in steps of 0.001 in the range 0 < r < 1 . The calculation for each step starts with a random value 0 < x1< 1 . In a calculation loop 2000 members of the series are calculated. The first ones differ largely in dependence on the random initial value. Therefore the first 1000 iterations are suppressed in the chart. For each step in r 1000 points on the ordinate could represent the iterations 1000 to 2000.

In the range 0.25 < r < 0.75 the iterations are so close together that they appear as one point only, resulting in a "limit curve" in dependence on r.

Then the curve splits in two (bifurcation), which means that the iteration now has two accumulation points for a certain r. The bifurcation repeats itself, until no accumulation points are visible any longer.

Quite surprisingly between the "filled" bands there are some quasi "empty" bands with only a few accumulation points.

The determining term is the product 4r; factors different from 4 just scale the abscissa differently.

It is not decisive for the bifurcation that the limiting term is exactly (1-xn). The crucial point is the nonlinearity of the conjunction xn -xn2.

To demonstrate this, a generalized series rule is used in this simulation, using a term (1-xnk), with k > 0 :

xn+1 = 4rxn(1-xnk)

When opening the simulation k = 1; Start produces the common logistic map.

After Stop k can be changed in the range 0.1 < k < 2 by a slider. The abscissa scaling is adjusted automatically.

The left chart displays the total range, the right one that of bifurcation with higher resolution. One can differentiate the calculation steps and the bifurcation structure in more detail if the window is expanded to full screen size.