Math 53 Chaos!: Homework 3due Fri Oct 19 . . . but best if do relevant questions after each lectureAlthough it looks like lots of questions this week, most of them are pretty fast (I believe. . . )2.3 (easy). Are the fixed points hyperbolic?A. Find the slight subtlety in the proof that AB and BA always have the same eigenvalues, which underliesthe lovely fact tha t stability is the same whichever point in a periodic orbit you pick (Remark 2.14).Specifically: write the relation stating λ is an eigenvalue o f AB. Left-multiply by B and interpret thisas an eigenvalue relation for BA. Are the corresponding eigenvectors the same? This argument failsfor one case of λ: explain why, then use the characteristic equation to prove it in this case.T2.7 a,b only.2.8Compu. Expt. 2.2: Here you can take the guts of the explormap2d.m code and wrap it with something to do a bifurcationdiagram as requested. This is not hard but will be good programming experience building on whatyou already know. Print out your x-coordinate diag ram for b = −0.3 and 0 ≤ a ≤ 2 .2.T2.8 (easy)T2.10 Give two ve c tors parallel to the axes. Explain the surprising result that even though one of theeigenvalues of AATexceeds 1 in absolute value, the ellipses AnN shrink to the origin.2.9 Show a sketch as in Fig. 2.29 showing the action of the inverse cat map.Challenge 2: Glancing at Fig. 2.31 you see this linear map has complex behavior which makes it fun to investigate.Make sure you’re happy up to Step 5. Do Step 7 too on your own (darn Fibonacci again!). Then writeup:Step 6 (easy)Step 8: plotting the solutions in the unit squar e will help you count them.Step 9. (I found a simpler formula than theirs—can you?)Step 11. Write out table only to k = 6 (yo u don’t need Step 10), and treat the proof that all periodsexist only as an optional BONUS.Compu. Expt. 3.1: You can combine bits of code from HW1 and fr om Compu. Expt. 2.2 above, to make this Lyapunov-exp onent-vs-a plot. Use fine steps in a, e.g. 10−3, to se e the jagge d quality. Only once you’re happywith your plot, c ompare to p. 237.Hints: look at the hw1_iter_sol.m code I provided on the HW page. You notice it plots the differenceof two nearby orbits on a log scale. If you take the ln o f this difference, the slope of the resulting graphis literally h, the Lyapunov exp onent (as explained p. 107). So you could mea sure the slope of thisgraph using eg 25 its (but not too many otherwise it stops gr owing). Since h can be negative, I sug gestyou don’t start at 10−15difference (since it could get smaller but you’d not be able to see this due toround-off error). Instead, why not choose a number somewhere between this and 1 (‘between’ in whatsense?) so that you can detect + or − exponents.1A better alternative is to use only one orbit xkbut to keep track of the product g′(x1)g′(x2) . . . forthat orbit. For each a, use Defn 3.1 to estimate the exponent. This method allows you to go for morethan 25 its (why?)Postponed to HW4:T3.2 (easy and has some review of Ch. 1. Be sure to find Lyapunov exponent not