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December 2nd, 2016, 08:32 AM   #1
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Sensitive Dependence on Initial Condtion

I would like to prove that a function f(x) has sensitive dependence on initial conditions.

The theory, as I understand it is:

We let J be an interval, and f : J → J.
Then f has sensitive dependence on initial conditions at x if
there is an ε > 0 such that for each d > 0 there is y ∈ J and a
positive integer n such that

|x − y| ≤ d and |(f^n)(x) − (f^n)(y)| ≥ ε

How do I show that f(x) = x^3 has sensitive dependence at -1, and no sensitive dependence at 0.
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December 2nd, 2016, 02:00 PM   #2
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well looking at this what's happening is that the $\delta$ interval about $-1$, i.e.

$(-1-\delta, -1+\delta),~\delta>0$ will necessarily contain some values

$x:~|x+1|<\delta \wedge |x|>1$

$\displaystyle{\lim_{n\to\infty}}\left |\left(x^3\right)^n\right| =

\begin{cases}
0 &|x|<1 \\
1 &|x| = 1 \\
\infty &|x|>1
\end{cases}$

So for these values with magnitude greater than 1 in the delta interval you can make $\left|\left(x^3\right)^n\right|$ as large as you like no matter how small $\delta$ is.

On the other hand the $\delta$ interval about $x=0$ is such that

$|x| < \delta < 1 \Rightarrow \displaystyle{\lim_{n\to\infty}}\left |\left(x^3\right)^n\right| = 0$
Thanks from zactops
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December 2nd, 2016, 02:40 PM   #3
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Wow thanks! That makes a lot of sense.
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