Cobweb diagram for the logistic equation

Cobweb diagram for the logistic equation with r=3.9

The discrete logistic equation is given by the difference equation .  It is easy to see that under ordinary assumptions, the logistic equation has a fixed point at .  However, depending on the initial condition and the value of , a number of different types of behaviors are possible.  The fixed point may be attractive (meaning that applying the logistic equation over and over pulls successive values closer to it) or repulsive (once the x value gets close to the fixed point, it is pushed away), or iterations of the logistic equations may orbit the fixed point retracing its steps in a periodic orbit.

Here is a link (http://http://sites.saintmarys.edu/~sbroad/example-logistic-cobweb.html) to a javascript implementation of a cobweb diagram for the logistic equation.  This diagram is generated based on the user’s choice of initial condition, parameter, and number of iterations.  The coordinate of the point of intersection between the parabola and the line is the fixed point of the logistic equation.  The diagram allows the user to see how much the behavior of the logistic equation is affected by the choice of the parameter.  It also shows the sensitive dependence on initial conditions for certain values of which are a hallmark of a chaotic dynamical system.

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