In the non-linear case, this is the well-known logistic map: [+1]=[](1[])

a.Non-linear maps can produce strange behaviour known as period doubling, which you should be able to note in your previous plot for > 3. You can compute the values of for which this occurs if you apply the map to itself once, then find the fixed points. This is known as a period doubling bifurcation, found by solving [ + 2] = (([])) and setting [ + 2] = [ + 1] = [] = 2. Determine the period-2 fixed points and plot some trajectories. Determine the stability of the fixed points.

b.You could continue calculating further period doubling bifurcations by hand, but it is much easier to assess the logistic map numerically. One main way of producing such a plot is to plot the fixed points of the map against the bifurcation parameter (known as a bifurcation diagram). To do this, you will need to determine a method of numerically calculating fixed points. As a hint, consider you have computing power available, so do not try to continue calculating [ + ] = (( (([]) )) as this is not an efficient way to determine the fixed points.