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Math 313-2 Test : May 7, 2002 C. RobinsonNo books, no notes. You may use hand calculators1. (20 Points) Consider the linear map92−14 −12 xyfrom the plane R2to itself. The eigenvalues are 3.5 and 0.5. Sketch the phase portrait,indicating the stable and unstable manifolds. Also, indicate the behavior of other typicalpoints.2. (20 Points) LetFxy=x + y + x22 x + 3 y.The fixed points are (0, 0) and (1, −1). Classify these fixed points are source, saddle, sink,or none of these.3. (20 Points) LetFxy=0.5 x2 y − x3,whose inverse isF−1xy=2 x0.5 y + 4 x3.(a) Show that the curve y =815x3is invariant by F..(b) Show that the stable manifold is given by y =815x3.(c) Show that the unstable manifold is given by x = 0.4. (20 Points) Take the H´enon map with a = 5 and b = −0.3,Fxy=5 − 0.3 y − x2x.Define the rectangles VL= [−3, −1] × [−3, 3] and VR= [1, 3] × [−3, 3].(a) Show that the net rotation of F on the boundary of the rectangles is nonzero for bothVLand VR, i.e., the map has nonzero index.(b) Show that { VL, VR} is a Markov partition, i.e., show that the images of VLand VRby F are correctly aligned with VLand VR.(Over next problem.)5. (20 Points) Let S = [0, 1] × [0, 1] be the unit square which contains four horizontal stripsH0, H1, H2, and H3of height 0.1, and four vertical strips V0, V1, V2, and V3of width1/12. See Figure below. Consider a map F from R2to itself, such that maps HjontoVjby stretching by 10 in the vertical direction and contracting by 1/12 in the horizontaldirection and with the appropriate translation. The strips H1and H3are also flipped over.Notice thatS ∩ F(S) = V1∪ V2∪ V3∪ V4,andS ∩ F−1(S) = H1∪ H2∪ H3∪ H4.(a) How many vertical strips does the intersectionS ∩ F(S) ∩ F2(S)contain and how wide are they?(b) How many horizontal strips does the intersectionS ∩ F−1(S) ∩ F−2(S)contain and how high are they?(c) Notice thatVj= {x : F−1(x) ∈ Hj} ≡ Sj..LetSs−2s−1.= { x : F−1(x) ∈ Hs−1and F−2(x) ∈ Hs−2}.What is the order of the vertical stripsS00., S10., . . . , S33.?(d) How many fixed points does the map F have?HHHHV V V V0


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NU MATH 313-2 - Math 313-2 Test

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