This applet shows stability properties of orbits of order 1 (fixed points) and 2 of the logistic map, explaining why the Feigenbaum diagram bifurcates even if the fixed points do not disappear. As we have seen in the previous activity, if [math]x_0[/math] is a fixed point of a map [math]f(x)[/math] and [math]s[/math] is the slope of the tangent line to the graph of [math]f[/math] at [math](x_0,x_0)[/math], then writing [math]y=f(x)=f(x_0)+\Delta y=x_0+\Delta y[/math], [math]x=x_0+\Delta x[/math], and approximating [math]f[/math] with the tangent line [math]y=s(x-x_0)+x_0[/math] for values of [math]x[/math] "sufficiently near" [math]x_0[/math], the iteration reduces to [math]\Delta y=s\cdot\Delta x[/math], giving the iterates [math]\Delta x_{n+1}=s\cdot\Delta x_n[/math]: this is the equation of the Malthus' model with [math]s[/math] as value of the parameter [math]p[/math]. When [math]\vert s\vert<1[/math] dynamics correspond to extinction, so [math]\Delta x_n\rightarrow 0[/math], i.e. [math]x_n\rightarrow x_0[/math]: the fixed point is an [i]attractor[/i] in the sense that neighboring points are transformed, by iteration, in points even closer. When [math]\vert s\vert>1[/math] dynamics correspond to demographic explosion , so [math]\Delta x_n\rightarrow\infty[/math]: the fixed point is a [i]repulsor[/i] in the sense that neighboring points are transformed, by iteration, in points further away. It turns out that fixed points possess basins of attraction or repulsion depending on their type, and a point that is in a basin of repulsion is gradually moved away by the iteration, until ending up in a basin of attraction and thus be "captured" by its attractor. The same applies to orbits of order greather than 1, since these can be described as fixed points of suitable maps.
To show fixed points or orbits of order 2 click to the corresponding Check Box. Near each fixed point of the corresponding map little axes [math]\Delta x[/math] and [math]\Delta y[/math] are shown. To change the position of fixed points move the red one.