Use Python

The Duffing map is defined as follows:

$x_{n+1}=y_n,y_{n+1}=-b*x_n+a*y_n-y_n^3,$

where a and $b$ are parameters. For this exercise we shall fix $a=2\mathrm{.}75$ and $b=0\mathrm{.}2$

throughout.

a$)$ Set up a function, which given $x_n,y_n,$a and $b$ returns $x_{n+1}$ and $y_{n+1}$ according to the Duffing map.

b$)$ Starting from the initial condition $x_0=0\mathrm{.}5$ and $y_0=0\mathrm{.}5,$ plot the first $400$ iterations of the Duffing map. You only need to plot the $x_n$ values.

c$)$ Find the fixed points of the Duffing map. You do not have to do this numerically.

d$)$ Represent the output of the first $3000$ iterations of the map on the $x-y$ plane, so

you can see the strange attractor. Remove transient effects by plotting only after $100$ iterations.

e$)$ Let's look at the fractal$-$like behaviour of the strange attractor. Run your map for many iterations $($say up to ${10}^8,$ but it will depend on your computer$).$ To do this, I recommend setting up a vector to hold all your iteration outputs, and then plotting at the end. This will be much faster than plotting on each time through the loop. Zoom in on a subset of the strange attractor, by plotting only the points that fall in the ranges $x$ in $[0\mathrm{.}85,1\mathrm{.}05]$ and $yin[1\mathrm{.}6,1\mathrm{.}75].$ If you run in interactive mode you will be able to zoom in$.$ Describe the strange attractor. Is it self$-$similar?