The logistic Function

Definition 1 (The logistic function)   The function $f_{\lambda}:\mathbb {R}\to\mathbb {R}$ with $f_{\lambda}(x)=\lambda x(1-x)$ is called the logistic function.

The logistic function is often used in relation with the fix-point iteration. For example the development of a population can be described by $x_{n+1}=f_{\lambda}(x_n)$ (see also section 4.1).

For $\lambda =3.2$ the logistic function is plotted in figure 1. As we can see for this parameter the logistic function has an stable two orbit attractor.

Figure 1: The logistic function $f_{\lambda }$ for $\lambda =3.2$

Theorem 1   The Fix-point of the logistic function $f_{\lambda }$ is at $x_1^*:=0$ and at $x_2^*:=\frac{\lambda-1}{\lambda}$. The second fix-point $x_2^*$ is stable for $-2<\lambda<3$ and instable for $\lambda>3$.

Proof. $x^*$ is a fix-point if and only if

$$f_{\lambda}(x^*)=x^* \quad \Leftrightarrow \quad x^*(\lambda-\lambda x^*)=x^*$$

$$\Updownarrow$$

$$x^*(\lambda-1-\lambda x^*)=0$$

$$\Updownarrow$$

$$x^*\in\left\{0,\frac{\lambda-1}{\lambda}\right\}$$

We know that the fix-point is stable if $\vert f_{\lambda}'(x_2^*)\vert<1$.

$$f'(x)=\lambda-2\lambda x \quad \Rightarrow \quad f'(x^*_2)=\lambda-2(\lambda-1)=2-\lambda$$

On the other hand the fix-point $x^*_2$ is instable if $\vert f_{\lambda}'(x^*_2)\vert>1$. $\qedsymbol$

Theorem 2   The logistic function $f_{\lambda }$ has an stable two orbit system if $3<\lambda<1+\sqrt{6}$ with the points

$$x^*\in\left\{ \frac{1+\lambda+\sqrt{\lambda^2-2\lambda-3}}{2\lambda}, \frac{1+\lambda-\sqrt{\lambda^2-2\lambda-3}}{2\lambda} \right\}$$

Proof. Obviously $f_\lambda$ has an stable two orbit system if and only if $f_\lambda \circ f_\lambda$ has there a stable fix-point. Therefor we have to examine $f_\lambda \circ f_\lambda$.

$$(f_\lambda\circ f_\lambda)(x)=f_\lambda(f_\lambda(x))= \lambda^2 x (1-x)(\lambda x^2+\lambda x -1)$$

A fix-point of $f_\lambda \circ f_\lambda$ is a value $x^*\in\mathbb {R}$ which satisfies the equation $f_\lambda(f_\lambda(x^*))=x^*$:

$$x(-\lambda^3 x^3+2\lambda^3 x^2-(\lambda^3+\lambda^2)x+\lambda^2-1)=0$$

This is the case if and only if

$$x^*\in\left\{0, \frac{\lambda-1}{\lambda}, \frac{1+\lambda+\sqrt{... ...3}}{2\lambda}, \frac{1+\lambda-\sqrt{\lambda^2-2\lambda-3}}{2\lambda} \right\}$$

The first derivative of $f_\lambda \circ f_\lambda$ in the last two points is exactly

$$-\lambda^2+2\lambda+4$$

of which the absolute value is smaller than one for all $\lambda\in (3,1+\sqrt{6})$ $\qedsymbol$

The function $f_\lambda \circ f_\lambda$ is represented for $\lambda =3.2$ in figure 2.

Figure 2: The logistic function $f_\lambda \circ f_\lambda$ for $\lambda =3.2$

For higher values $\lambda>1+\sqrt 6$ you can show that the orbit splits into 8 points then into 16, 32, 64 etc points. The limes is reached at about $\lambda\approx 3.570$. The orbit for this limes is a Cantor set.