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As mentioned above, the logistic map can be used as a model to consider the fluctuation of population size. In this case, the variable x of the logistic map is the number of individuals of an organism divided by the maximum population size, so the possible values of x are limited to 0 ≤ x ≤ 1.
An animated cobweb diagram of the logistic map = (), showing chaotic behaviour for most values of >. A cobweb plot , known also as Lémeray Diagram or Verhulst diagram is a visual tool used in the dynamical systems field of mathematics to investigate the qualitative behaviour of one-dimensional iterated functions , such as the logistic map .
The Feigenbaum scaling function provides a complete description of the attractor of the logistic map at the end of the period-doubling cascade. The attractor is a Cantor set , and just as the middle-third Cantor set, it can be covered by a finite set of segments, all bigger than a minimal size d n .
Feigenbaum originally related the first constant to the period-doubling bifurcations in the logistic map, but also showed it to hold for all one-dimensional maps with a single quadratic maximum. As a consequence of this generality, every chaotic system that corresponds to this description will bifurcate at the same rate.
Symmetry breaking in pitchfork bifurcation as the parameter ε is varied. ε = 0 is the case of symmetric pitchfork bifurcation.. In a dynamical system such as ¨ + (;) + =, which is structurally stable when , if a bifurcation diagram is plotted, treating as the bifurcation parameter, but for different values of , the case = is the symmetric pitchfork bifurcation.
The standard logistic function is the logistic function with parameters =, =, =, which yields = + = + = / / + /.In practice, due to the nature of the exponential function, it is often sufficient to compute the standard logistic function for over a small range of real numbers, such as a range contained in [−6, +6], as it quickly converges very close to its saturation values of 0 and 1.
English: A bifurcation diagram for the Logistic map: + = The horizontal axis is the r parameter, the vertical axis is the x variable. A starting value of x=0.25 was used, and the map was iterated 1000 times in order to stabilize the values of x. 1,000,000 x -values were then calculated for each value of r and for each x value, the corresponding (x,r) pixel in the image was incremented by one.
Archived 2022-06-28 at the Wayback Machine Dr. Ronald Joe Record's mathematical recreations software laboratory includes an X11 graphical client, lyap, for graphically exploring the Lyapunov exponents of a forced logistic map and other maps of the unit interval.