Chaos in Newtons Method

Question

Im trying to prove that Newtons method applied to ${\rm f}\left(\, x\,\right) =x^{2} + c$, is chaotic for $c > 0$.

I know I need to prove:
(a) The periodic points of ${\rm f}$ are dense in $X$,
(b) ${\rm f}$ is topologically transitive and
(c) ${\rm f}$ exhibits sensitive dependence on initial conditions.

and have the following definitons for (a), (b) and (c).

(a) Let $G \subset X$ where $X$ is a metric space with metric ${\rm d}$. Then $G$ is said to be dense in $X$ if, for any given $x \in X$ and $\epsilon > 0,\exists\ y \in g$ such that ${\rm d}\left(\, x,y\,\right) < \epsilon$. Equivalently, $G$ is dense in $X$, if for any given $x \in X,\exists\ {x_{n}} \subset G$ such that $x_{n} \to x$ in $X$ as $n \to\ \infty$.

(b) The function ${\rm f}:X \to X$ is topologically transitive if for all open sets $U$ and $V$ in $X$, $\exists\ x \in U$ and a natural number $n$ such that ${\rm f}^{n}\left(\, x\,\right)$ is in $V$.

(c) Let $X$ be a metric space with metric ${\rm d}$. The function ${\rm f}:X \to X$ exhibits sensitive dependence on initial conditions if $\exists\ \delta > 0$ such that $\forall\ x \in X$ and $\forall\ \epsilon > 0$, there is a $y \in X$ and a natural number $n$ such that ${\rm d}\left(\, x,y\,\right) < \epsilon$ and ${\rm d}\left(\,{\rm f}^{n}\left(\, x\,\right),{\rm f}^{n}\left(\, y\,\right)\right) > \delta$.

Struggling to get started as I can't find any examples of anyone doing this sort of thing anywhere. Most books have this sort of thing as an exercise but with no solutions.

Any help would be brilliant !.

Answer 1

I recommend that you use the notion of dynamically conjugacy - an extremely important tool in dynamics. The function $f$ is said to be dynamically conjugate to $g$ if there is a function $\varphi$ such that $f\circ\varphi = \varphi\circ g$. It's not hard to show that this implies $f^n\circ\varphi = \varphi\circ g^n$. As a result, $\varphi$ maps an orbit of $g$ starting at $z_0$ to an orbit of $f$ starting at $\varphi(z_0)$ so that the dynamics of $f$ and $g$ will be closely related. Typically, the conjugacy $\varphi$ is assumed to have some nice properties. The nicer $\varphi$, the closer the relationship between $f$ and $g$.

For example, let $f_c(z)=z^2+c$ so that the Newton's method iteration function is $$n_c(z)=\frac{z}{2}-\frac{c}{2\,z}.$$ It turns out that $n_c(z)$ is conjugate to $n_1$ via $\varphi(z)=\sqrt{c}\,z$. We can show this with a couple simple computations:

\begin{align} n_c(\varphi(z)) &= \frac{\sqrt{c}\,z}{2} - \frac{c}{2\sqrt{c}\,z} = \frac{\sqrt{c}\,z}{2} - \frac{\sqrt{c}}{2\,z} \\ \varphi(n_1(z)) &= \sqrt{c}\left(\frac{z}{2}-\frac{1}{2\,z}\right) = \frac{\sqrt{c}\,z}{2} - \frac{\sqrt{c}}{2\,z}. \end{align}

Since these are equal, we have the conjugacy you were looking for. Note that this already helps your cause - if you can show that $n_1$ is a chaotic map on the line, then so is $n_c$ for any choice of $c$. Thus, you can focus on just the one choice of $c$.

Now to show that $n_1$ is a chaotic map, you can show that it is dynamically conjugate to some function that is known to be chaotic. I'll let you fill out the details, but here are some possibilities.

• The complex squaring function $g(z)=z^2$ is known to be chaotic on the unit circle. Your function $n_1$ is conjugate to $g$ via the appropriate Mobius transformation.
• The squaring function is conjugate to $f_{-2}(z)=z^2-2$ via the conjugacy $\varphi(z)=z+1/z$. Furthermore, $\varphi$ maps the unit circle to the interval $[-2,2]$. Thus, $f_{-2}$ is chaotic on that interval. If you happen to know already that $f_{-2}$ is chaotic on the interval (perhaps by studying it's relationship to the logistic function), then you can compose this conjugacy with the Mobius transformation above to obtain a conjugacy between $n_1$ and $f_{-2}$ thus showing that $n_1$ is chaotic. The advantage is that your final argument can avoid complex variables altogether.
• The doubling function $d(x) = 2x \mod 1$ is chaotic on the interval $[0,1)$. There's a fairly obvious conjugacy between the squaring function on the circle and the doubling function on the interval arising from the fact that the squaring function doubles the complex argument. By composition, $n_1$ is conjugate to the doubling function. Again, this would allow you to bypass the use of complex arithmetic, if desired. The advantage here is that the doubling function is very well known to be chaotic.
• The Bernoulli map is a chaotic map in symbolic dynamics. While it requires some advanced notions from metric topology, it is super simple to see that it is a chaotic map, once those ideas are understood. It's not at all hard to show that the doubling function is conjugate to the Bernoulli map via it's itinerary map. Again via composition, you could set up a conjugacy between $n_1$ and the Bernoulli map.