Sensitive Dependence on Initial Conditions

Sensitive Dependence on Initial Conditions(SDIC): “Butterfly Effect”, this is the idea that small changes in initial conditions can lead to vastly different outcomes. In chaotic systems, even a tiny perturbation can grow exponentially over time, making long-term prediction impossible.

Consider the map $f(x)=3x(mod1)$ on unit interval, Fixed point at $x=\frac{1}{2}$ and $0$. $x=1$ is identical to $x=0$

$$\begin{array}{rl}f\left(\frac{1}{4}\right)& =\frac{3}{4}\\ f\left(\frac{3}{4}\right)& =\frac{9}{4}(mod1=\frac{1}{4}\\ & \left\{\frac{1}{4},\frac{3}{4}\right\}\end{array}$$

This is a period 2 source, because $f^{\prime }=3$. $\frac{1}{3}$ is eventually periodic to source at 0.

$$(f^2{)}^1(p_{1}or{p}_2)$$

Desmos

All rational numbers will be eventually be periodic to a Repelling fixed point.

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Formal Definition of SDIC

Let $f$ be a map on $\mathbb{R}$. A point $x_0$ has SDIC if $\exists d>0$ such that any neighborhood of $x_0$ contains a point $x$ such that:

$$|f^K(x)-f^K(x_0)|\ge d$$

for some non negative integer $K$. I.e., a point $x$ will be sensitive if it has neighbors as close as desired that eventually move away.

In plain English, this means that no matter how close you start to $x_0$, there are points arbitrarily close that will eventually diverge from $x_0$ by at least a distance $d$ after some number of iterations.

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