My Math Forum Sensitive Dependence on Initial Condtion

 Complex Analysis Complex Analysis Math Forum

 December 2nd, 2016, 07:32 AM #1 Newbie   Joined: Dec 2016 From: Montreal Posts: 4 Thanks: 0 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.
 December 2nd, 2016, 01:00 PM #2 Senior Member     Joined: Sep 2015 From: USA Posts: 1,982 Thanks: 1027 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
 December 2nd, 2016, 01:40 PM #3 Newbie   Joined: Dec 2016 From: Montreal Posts: 4 Thanks: 0 Wow thanks! That makes a lot of sense.

 Tags condtion, dependence, initial, sensitive

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post sivela Physics 4 April 2nd, 2011 08:49 AM sivela Physics 0 March 30th, 2011 06:55 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top