Measuring Chaos

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Bifurcation diagram rendered with 1-D Chaos Explorer


4.2 Logistic Equation

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.

  1. Fixed: The population approaches a stable value. It can do so by approaching asymptotically from one side in a manner something like an over damped harmonic oscillator or asymptotically from both sides like an under damped oscillator. Starting on a seed that is a fixed point is something like starting an SHO at equilibrium with a velocity of zero. The logistic equation differs from the SHO in the existence of eventually fixed points. It's impossible for an SHO to arrive at its equilibrium position in a finite amount of time (although it will get arbitrarily close to it).
  2. Periodic: The population alternates between two or more fixed values. Likewise, it can do so by approaching asymptotically in one direction or from opposite sides in an alternating manner. The nature of periodicity is richer in the logistic equation than the SHO. For one thing, periodic orbits can be either stable or unstable. An SHO would never settle in to a periodic state unless driven there. In the case of the damped oscillator, the system was leaving the periodic state for the comfort of equilibrium. Second, a periodic state with multiple maxima and/or minima can arise only from systems of coupled SHOs (connected or compound pendulums, for example, or vibrations in continuous media). Lastly, the periodicity is discrete; that is, there are no intermediate values.
  3. Chaotic: The population will eventually visit every neighborhood in a subinterval of (0, 1). Nested among the points it does visit, there is a countably infinite set of fixed points and periodic points of every period. The points are equivalent to a Cantor middle thirds set and are wildly unstable. It is highly likely that any real population would ever begin with one of these values. In addition, chaotic orbits exhibit sensitive dependence on initial conditions such that any two nearby points will eventually diverge in their orbits to any arbitrary separation one chooses.

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
bifurcation
 

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