Ph 106b Midterm ExamDue: Thursday 14 February 2008, 4pm• This exam is to be taken in one continuous time interval not to exceed 3hours, beginning when you first open the exam. (You may take one 15minute break during the exam, which does not count as part of the 3hours.)• You may consult the textbooks Chaos in Dynamical Systems by Ott andClassical Mechanics by Goldstein, the distributed lecture notes, your ownlecture notes, and the problem sets and solutions. If you wish, you may usea calculator, computer, or integral table for doing calculations. However,this probably won’t be necessary. No other materials or persons areto be consulted.• There are three problems, each with multiple parts, and 100 possible points;the value of each problem is indicated. You are to work all of the problems.• The completed exam is to be deposited in the box outside 448 Lauritsen, nolater than 4:00 pm on Thursday 14 February 2008. No late exams willbe accepted.• Good luck!1. A one-dimensional map (30 total points)Consider a one-dimensional map on the circle (i.e., the periodically identifiedunit interval) defined byM (x)=ax (mo d 1) ,where a is a positive integer.a) (10 points) What is the Lyapunov exp onent of the map M ?b) (10 points) For this map M , how many orbits are there of length p, wherep is a prime number?c) (10 points) Are the periodic points of M dense in the unit circle? Explainyour answer.2. A two-dimensional map (30 total p oints)Consider the two-dimensional mapM :w → w0= w + J (mod 1) ,J → J0= J,where w is a periodic variable with period 1 and J is a real number. We mayinterpret this map as describing a rotor with constant angular velocity; wetake a “snapshot” of the position of the rotor in phase space at regular timeintervals. The angular position of the rotor (measured counterclockwise relativeto the vertical) is 2πw, and J is proportional to Lδt, where L is the angularmomentum of the rotor, and δt is the interval between successive snapshots.a) (10 points) Find all of the periodic orbits of the map M . Are the periodicpoints dense in the (w, J ) phase space?b) (10 points) Suppose that (w0,J0)isnot a periodic point of M . Then theorbit containing this point densely fills an invariant curve in the (w, J)phase space. Describe this curve.c) (10 points) Suppose that (w0,J0) is not a periodic point of M . Evaluate thetime average of J2sin2(πw)hJ2sin2(πw)itime≡ limN →∞1NN −1Xn=0J2nsin2(πwn) ,where (wn,Jn)=Mn(w0,J0).3. Another two-dimensional map (40 total points)Now consider a new mapM :w → w0= w + J (mod 1) ,J → J0= J +F2πsin(2π (w + J )) ,which reduces for F = 0 to the map considered in Problem 2. Thismap also describes a rotor, but now the rotor receives a vertical impulseproportional to F an instant before each snapshot is taken.a) (5 points) Find the fixed p oints of the new map, for F 6=0.b) (10 points) The effect of the map on the separation between two nearbypoints in phase space is described by a linearized map. That is, if the points(w, J)and(w+δw,J +δJ ) are mapped to (w0,J0)and(w0+δw0,J0+δJ0),then, to linear order,δw0δJ0= AδwδJ,where A is a 2 × 2 matrix. Find this matrix A, expressed as a function ofw0and J0.c) (5 points) Show that the linearized map preserves the phase space areaelement dw ∧ dJ.d) (10 points) Consider the fixed points found in (a), with −12<J<12.For each such fixed point, find the range of values of F for which thefixed point is elliptic, and the range of values for which the fixed point ishyperbolic. Assume F>0.e) (10 points) Consider the period-two orbits of the F = 0 map that you foundin part (a) of Problem 2, with 0 <J<1. Describe qualitatively whathappens to these orbits for 0 <F <<1. What do the new invariant curveslook like? Are there chaotic regions in phase space? Do not calculateanything, but draw a picture of the orbits in the (w, J)