Bifurcation

Given a starting population with a certain reproduction rate and a death rate, we can predict (calculate) the population for specific generations.  If we plot the population by generation with a given birth rate, the trajectory will be linear in a straight line.

If we add an additional rule to confine the population by a carrying capacity, the trajectory changes to a point until it become impossible to predict.  In other words, a small variation in birth or death rate, the population for a specific generation will not be close together but may attain a very diverse value

The above can be modeled bu a logistic map formula.  We combine the birth rate with the death rate into a variable called R.  The carrying capacity is replaced by a variable called fraction of carrying capacity x. Logistic map is represented by the following formula: Xi+1    = R*Xi *(1-Xi )

(1) For a small value or R, e.g. 2, x will converge to a single value after a few generations.
(2) When R increases to a value of 3.1, the trajectory evolves to an oscillation of 2 values.  \
(3) As R increases further, the oscillation changes to among 4, 8 values etc.
(4) The spread of oscillation increases rapider and rapider as R picks up gradually.
(5) When R reaches about 3.569946, the trajectory becomes chaotic.  Even a small difference in the value of x (e.g. at the 10 decimal point) will produce a completely different trajectory.  In other words, the trend becomes unpredictable.

Because of this property, the logistic map is used as an algorithm to generate pseudo-random number.

Among the chaos, there are 2 universal characteristics observed:
(1) The trajectories follow a period-doubling (bifurcation) route to chaos. From a single value to 2, to 4, to 8 etc.
(2) As R increases, their bifurcation points move closer and closer. The rate of convergence is found to be a constant value of 4.6692016. This is called the Feigenbaum constant.