# NU MATH 313-2 - Math 313-2 Test (2 pages)

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## Math 313-2 Test

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## Math 313-2 Test

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Pages:
2
School:
Northwestern University
Course:
Math 313-2 - R. Clark Robinson

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Math 313 2 Test May 7 2002 C Robinson No books no notes You may use hand calculators 1 20 Points Consider the linear map 9 1 4 12 2 x y from the plane R2 to 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 typical points 2 20 Points Let x y x2 x F y 2x 3y The fixed points are 0 0 and 1 1 Classify these fixed points are source saddle sink or none of these 3 20 Points Let x 0 5 x F y 2 y x3 whose inverse is 2x 1 x F y 0 5 y 4 x3 8 3 a Show that the curve y x is invariant by F 15 8 3 b Show that the stable manifold is given by y x 15 c Show that the unstable manifold is given by x 0 4 20 Points Take the He non map with a 5 and b 0 3 5 0 3 y x2 x F y x 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 both VL and VR i e the map has nonzero index b Show that VL VR is a Markov partition i e show that the images of VL and VR by F are correctly aligned with VL and VR Over next problem 5 20 Points Let S 0 1 0 1 be the unit square which contains four horizontal strips H0 H1 H2 and H3 of height 0 1 and four vertical strips V0 V1 V2 and V3 of width 1 12 See Figure below Consider a map F from R2 to itself such that maps Hj onto Vj by stretching by 10 in the vertical direction and contracting by 1 12 in the horizontal direction and with the appropriate translation The strips H1 and H3 are also flipped over Notice that S F S V1 V2 V3 V4 and S F 1 S H1 H2 H3 H4 a How many vertical strips does the intersection S F S F2 S contain and how wide are they b How many horizontal strips does the intersection S F 1 S F 2 S contain and how high are they c Notice that Vj x F 1 x Hj Sj Let Ss 2 s 1 x F 1 x Hs 1 and F 2 x Hs 2 What is the order of the vertical strips S00 S10 S33 d How many fixed points does the map F have F S S H H 3 2 H1 V 0 V 1 V2 V 3 H0

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