Quadratic iteration, bifurcation, and chaos

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A simple iterative model of population growth (and decline) over time is

x maps to mu x (1 - x).

We choose a value for the parameter mu and a starting x value, then repeatedly substitute the results of each mapping into x on the right-hand side of the mapping; the list of partial results can be viewed as a time series of the population levels over a discrete sequence of times.

For example, if we let mu be 2, and let x start with 0.25, the first few iterates are

{0.25, 0.375, 0.46875, 0.498047, 0.499992, 0.5, 0.5, 0.5, ...}.

Similarly, if we let mu be 2.5, the first few iterates are

{0.25, 0.46875, 0.622559, 0.587448, 0.605882, 0.596973, 0.601491, 0.599249, 0.600374, 0.599813, 0.600094, 0.599953, 0.600023, 0.599988, 0.600006, 0.599997, 0.600001, 0.599999, 0.6, 0.6, 0.6, ...}.

For values of mu between 1 and about 3, the iteration settles down to a single value.

If we let mu be 3.2, the list of iterates is

{0.25, 0.6, 0.768, 0.570163, 0.784247, ..., 0.799455, 0.513045, 0.799455, 0.513045, ...};

the iteration eventually oscillates forever between the two values 0.799455 and 0.513045. The iteration has period two for values of mu between 3 and about 3.4494.

If we choose mu = 3.46, the iterates are

{0.25, 0.64875, 0.788442, 0.577132, ..., 0.838952, 0.467486, 0.861342, 0.413234, 0.838952, ...},

where the last four values repeat forever, or fall into a period-4 cycle. The phenomenon of period-doubling for greater values of mu is called bifurcation.

For greater values of mu, the list of iterates is chaotic; the iteration does not settle down to a fixed point or cycle, but rather apparently behaves randomly.

Bifurcation diagram:
bifurcation image

Here we visualize the iterations of the mapping for values of mu ranging from 2.4 to 3.8 in increments of 0.2. We see that for small values of mu the iteration settles down to a fixed point or a cycle, but that for larger values the behavior is chaotic. array of bifurcation images

Designed and rendered using Mathematica 3.0 for the Apple Macintosh, with more than a little help from Peitgen et al, Chaos and Fractals, Springer-Verlag, 1992.

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