Chaos and Fractals

1.

Let \(f:\mathbb R\to \mathbb R\) be continuously differentiable. We say that \(x_0\) is a simple root of \(f\) if \(f(x_0)=0\) and \(f'(x_0)\neq0\text{.}\) Show that if \(x_0\) is a simple root of \(f\text{,}\) then \(x_0\) is a super-attracting fixed point of the Newton's method iteration function \(N\) for \(f\text{.}\)

2.

Find an example of a continuously differentiable function \(f:\mathbb R\to \mathbb R\) that attracts no critical point.

3.

Let \(f(x)=x^2-4x+5\text{.}\) Show that \(f\) has a super-attractive orbit of period 2.

4.

Let \(f(x)=3 x^2-6 x+3.415\text{.}\) Find all attractive orbits of \(f\text{.}\)

5.

Find a value of \(c\) such that \(f_c(x)=x^2+c\) is affinely conjugate to \(g(x)=(x-1)(x+2)\text{.}\) Show that both functions have neutral fixed points.

6.

Find an orbit of period 11 for the function \(g(x)=4x(1-x)\)

7.

We wish to find a number \(x_0 \in H=[0,1)\) whose orbit is dense in \(H\) under iteration of \(g(x)=4x(1-x)\text{.}\)

1. Outline a strategy for finding \(x_0\text{.}\)

2. Find a decimal approximation to \(x_0\) that is valid to 10 decimal places.