(Chaos.) (16 marks) Consider the logistic map equation for one parameter family given by f(x)=Ax2 where A is the parameter. The objective of this question is to analyse the long term behavior of the system. (a) Find the fixed points for the logistic map equation. (b) Draw a bifurcation diagram when 41A2 and the initial condition is x0=0.9999. Submit the MATLAB code and the bifurcation diagram. (c) Find the range A for which this system has single-period cascading (system has only one cycle). Determine the value of A at which you first observed period-double cascading. (You may need to zoom in the bifurcation diagram). (d) What happens when A=45. Do you still get the similar bifurcation diagram as in (b)? If not, explain why? (e) Does this system have 3-cycles for any of the values of x0 and A as specified in part (b). (f) Is this system chaotic? That is: in class, we said that a system of difference equations is chaotic if arbitrarily tiny changes in the initial conditions can lead to wildly different outcomes: i.e. varying x0 by something as tiny as 0.0000001 could cause massive swings in the values of x100000. Does this system display chaotic behaviour like this, or not? Justify your answer. (g) Let us do some calculations by hand. Find the values of x for which this system is stable. For what values of x does it become unstable? Verify your answer using the bifurcation diagram you plotted in (b). (h) Plot the last 100 iterations on time course diagram ( x vs number of iterations) to show period-double cascading (2-cycles)