Ass5 - Assignmetn 5 PDF

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School of Mathematics and Statistics Victoria University of Wellington

MATH462 Assignment 5 - last one DUE: 10 October 2018 This is the assignment for Chapter Five in Alligood et al. 1. Find a general formula for area contracting skinny baker maps (as in Example 5.3), where the strips have width w rather than 1/3 — find the area contraction factor per iteration, and the Lyapunov exponents.

that f (sL ) and f (sR ) surround sL and sR (in terms of x-coordinates), and that f (sT ) and f (sB ) surround sT and sB (in terms of y coordinates). Show that f has a fixed point in S .

2. Find the Lyapunov dimension of the generalised skinny baker maps in the previous question.

5. Find the Lyapunov dimension of the cat map.

3. Prove the Brouwer Fixed-point Theorem in two dimensions: If f is is a continuous map, S is a closed square, and f (S) is contained in S, then there is a fixed point in S . 4. Let f be a continuous map on IR2 , and S a rectangle with vertical sides sL and sR , and horizontal sides sT and sB . Assume

6. Find the Lyapunov dimension of the Ikeda map. 7. Find the (x1 , x2 ) coordinates of the period two orbits of the skinny baker map in the strips .LRL and .RLR of the figure below, which is Fig. 5.12 in Alligood et al. Are there any other periodic orbits in those two strips?

8. Show that the invariant set of the horseshoe map contains a dense chaotic orbit. Mark McGuinness...