Fixed Point Classification

Resuming work on Linear Systems regarding the Logistic Map using a Cobweb Plot:

A fixed point is a point $p$ such that $f(p)=p$. Usually a sink/source, but due to periodic points, can be a rotation set of points.

The epsilon-neighborhood $N_{ϵ}(p)$ of a point $p$ is the internal $\{x\in \mathbb{R}:|x-p|<ϵ\}$.

If $\exists ϵ>0:\forall x\in N_{ϵ}(p)$, ${\lim }_{k\to \infty }{f}^k(x)=p$ then we call $p$ a sink or Attracting fixed point

In plain English, if there is some neighborhood around point $p$ such that all points in that neighborhood converge to $p$ under iteration of $f$, then $p$ is a sink.

If $\exists ϵ>0:$ all $x\in N_{ϵ}(p)$ except $p$ eventually map outside of $N_{ϵ}(p)$, then we call $p$ a source or a Repelling fixed point.

In plain English, if there is some neighborhood around point $p$ such that all points in that neighborhood (except for $p$ itself) eventually leave that neighborhood under iteration of $f$, then $p$ is a source.

For cases in this class, $ϵ$ represents a small positive real number parameter.

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Cubic Example

Looking at the function:

$$f(x)=\frac{3x-x^3}{2}$$

Desmos

The fixed point sinks are: $x=\pm 1$

The fixed point Sources are: $x=0$

Key observations:

• Sinks at $x=\pm 1$ are super-attracting: $f^{\prime }(\pm 1)=0$ (critical points = fixed points)
• Source at $x=0$ has $f^{\prime }(0)=\frac{3}{2}>1$ (linearly unstable)
• Odd symmetry: $f(-x)=-f(x)$ creates symmetric basins of attraction
• Topologically similar to Logistic Map at $r=3$ (pre-chaotic regime)

Theorem

Let $f$ be a smooth map (derivatives continousous), and assume $p$ is fixed point of $f$.

If $|f^{\prime }(p)|<1\Rightarrow p$ is a sink.

If $|f^{\prime }(p)|>1\Rightarrow p$ is a Source.

If $|f^{\prime }(p)|=1$, we need more information.

Origin is a “neutral” fixed point, all other $x\in \mathbb{R}$ are period 2. In essence, some starting position will take a minute to read the period two orbit, whereas others like $p_1$ will already be on the period two orbit.

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Periodic Sinks and Source

Let $f$ be a map and assume $p$ is a period $k$ point. The period k orbit of $p$ is a periodic sink (source) if $p$ is a sink( source ) for the map $f^k$

In plain English, if $p$ is a period $k$ point, then we can look at the $k$th iterate of $f$ and classify $p$ as a sink or source based on that.