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Bifurcation diagram rendered with 1-D Chaos Explorer
The simple logistic equation is a formula for approximating the evolution of an animal population over time. Many animal species are fertile only for a brief period during the year and the young are born in a particular season so that by the time they are ready to eat solid food it will be plentiful. For this reason, the system might be better described by a discrete difference equation than a continuous differential equation. Since not every existing animal will reproduce (a portion of them are male after all), not every female will be fertile, not every conception will be successful, and not every pregnancy will be successfully carried to term; the population increase will be some fraction of the present population. Therefore, if "An" is the number of animals this year and "An+1" is the number next year, then
An+1 = rAn
where "r" is the growth rate or fecundity, will approximate the evolution of the population. This model produces exponential growth without limit. Since every population is bound by the physical limitations of its surrounding, some allowance must be made to restrict this growth. If there is a carrying-capacity of the environment then the population may not exceed that capacity. If it does, the population would become extinct. This can be modeled by multiplying the population by a number that approaches zero as the population approaches its limit. If we normalize the "An" to this capacity then the multiplier (1 − An) will suffice and the resulting logistic equation becomes
An+1 = rAn(1 − An)
or in functional form
ƒ(x) = rx (1 − x).
The logistic equation is parabolic like the quadratic mapping with ƒ(0) = ƒ(1) = 0 and a maximum of ¼r at ½. Varying the parameter changes the height of the parabola but leaves the width unchanged. (This is different from the quadratic mapping which kept its overall shape and shifted up or down.) The behavior of the system is determined by following the orbit of the initial seed value. All initial conditions eventually settle into one of three different types of behavior.
The behavior of the logistic equation is more complex than that of the simple harmonic oscillator. The type of orbit depends on the growth rate parameter, but in a manner that does not lend itself to "less than", "greater than", "equal to" statements. The best way to visualize the behavior of the orbits as a function of the growth rate is with a bifurcation diagram. Pick a convenient seed value, generate a large number of iterations, discard the first few and plot the rest as a function of the growth factor. For parameter values where the orbit is fixed, the bifurcation diagram will reduce to a single line; for periodic values, a series of lines; and for chaotic values, a gray wash of dots.
Since the first two chapters of this work were filled will bifurcation diagrams and commentary on them, I won't go much into the structure of the diagram other than to locate the most prominent features. There are two fixed points for this function: 0 and 1 − 1/r, the former being stable on the interval (−1, +1) and the latter on (1, 3). A stable 2-cycle begins at r = 3 followed by a stable 4-cycle at r = 1 + √6. The period continues doubling over ever shorter intervals until around r = 3.5699457… where the chaotic regime takes over. Within the chaotic regime there are interspersed various windows with periods other than powers of 2, most notably a large 3-cycle window beginning at r = 1 + √8. When the growth rate exceeds 4, all orbits zoom to infinity and the modeling aspects of this function become useless.
|Bifurcation Diagram of the Logistic Equation|
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