M3A23/M4A23. Specimen Paper


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1 UNIVERSITY OF LONDON Course: M3A23/M4A23 Setter: J. Lamb Checker: S. Luzzatto Editor: Editor External: External Date: March 26, 2009 BSc and MSci EXAMINATIONS (MATHEMATICS) MayJune 2008 M3A23/M4A23 Specimen Paper Setter s signature Checker s signature Editor s signature
2 UNIVERSITY OF LONDON BSc and MSci EXAMINATIONS (MATHEMATICS) MayJune 2008 This paper is also taken for the relevant examination for the Associateship. M3A23/M4A23 Specimen Paper Date: examdate Time: examtime Credit will be given for all questions attempted but extra credit will be given for complete or nearly complete answers. Calculators may not be used.
3 1. Let X be a metric space with metric (distance function) d : X X R. List the properties that d must satisfy in order to be a metric. (b) Give the definition of a contraction F : X X, with respect to the metric d. (d) (e) For each of the combination of properties listed below of a contraction F on a metric space X, verify whether the situation can arise, and if so give one explicit example of F : X X that realizes the listed properties: (i) F has a unique fixed point and X is complete. (ii) F has no fixed point and X is complete. (iii) F has a unique fixed point and X is not complete. (iv) F has no fixed point and X is not complete. Prove the following variant of the Contraction Mapping Theorem: Theorem. Let F : X X be a Lipschitz map on a complete metric space X. Suppose that F is not a contraction, but F m is a contraction for some m > 1. Then F has a unique fixed point and all x X converge to this fixed point with exponential speed. Give an example of a map F that satisfies the conditions set out in the Theorem stated under (d). 2. Give a definition of topological mixing for a circle map F : S 1 S 1. (b) (d) (e) Show that invertible circle maps are never topologically mixing. [You may use results that were derived in the course without proof, but any such results should be stated carefully.] Show that all expanding circle maps are topologically mixing. Describe the condition on the parameters a, b and c that must be satisfied for F (x) = (ax + b cos 2 (2πx) + c) mod 1 to be an orientation preserving and expanding circle map. Show that if the circle map F of part (d) is orientation preserving and expanding then F k has a k 1 fixed points. Show that this implies that F has periodic orbits with period k for all k Z +.
4 3. Let F : S 1 S 1 defined by F (x) = (ax + b cos 2 (2πx) + c) mod 1 with x [0, 1) = S 1 diffeomorphism. and parameters a, b, c R be an orientation preserving circle (b) (d) (e) Determine what conditions on the parameters a, b, and c must be satisfied for F to be an orientation preserving circle diffeomorphism. Give a formula for the rotation number ρ(f ) of F. Show that although this formula refers to a specific point x S 1, ρ(f ) is independent of the choice of this point x. Prove the following proposition: Proposition. If ρ(f ) is irrational then for all x S 1 and any m, n N with m > n the following holds: for every y S 1 the forward orbit {F k (y) k N} intersects the interval I = [F m (x), F n (x)]. [You may use without proof results about properties of circle maps with rational rotation number, but any such result should be stated carefully.] Give the definition of the ωlimit set ω(x) for the map F and show that ω(x) is independent of x if ρ(f ) is irrational. [Hint: use the Proposition stated in part of this question.] Discuss the structure of the ωlimit set of F when ρ(f ) is irrational. [You may use results from the course without proof, but any such results should be stated carefully.]
5 4. Consider the cubic map with λ > 0. f λ : R R, x λx(1 x 2 ), For small λ > 0, x = 0 is an asymptotically stable fixed point. Determine the parameter value at which this fixed point loses stability. Determine the type of local bifurcation that occurs. Is it a generic (typical) local bifurcation of onedimensional maps with one parameter? [Motivate your answer.] (b) (i) Find the smallest value of λ > 0 at which a periodic doubling bifurcation occurs (from a fixed point). Note that at this parameter value two period doubling bifurcations occur at the same time. Verify that the branches of period two orbits have the explicit expression ±1 λ = x 1 x. 2 (ii) (iii) Show that in a typical period doubling bifurcation, the amplitude of the period doubled orbit grows as a the square root of the bifurcation parameter. Verify whether such growth is also observed in the period doubling bifurcation of f λ in part (i) of this question. Discuss how the map f λ in the neighbourhood of a perioddoubling bifurcation point can be considered as the Poincaré return map of a flow on a twodimensional surface. Describe (and sketch!) the local geometry of this suspension flow and illustrate the perioddoubling bifurcation on it (in pictures). Determine for which values of the parameter λ, the map f λ restricted to the set of initial conditions for which orbits do not escape from [0, 1] is topologically conjugate to a full shift on two symbols. [You do not need to prove the conjugacy, but need to show by what criteria you determine λ and how the conjugacy is defined.]
6 5. Let Σ 3 denote the set of infinite sequences {ω i } i N whose entries ω i are taken from a set of three symbols {1, 2, 3}. (Σ 3, d µ ) is a metric space with d µ (ω, ω ) := m N δ(ω m, ω m) µ m, where µ > 1 and δ(a, b) = 0 if a = b and δ(a, b) = 1 if a b. Determine for which values of µ R the cylinder C α0,...,α n 1 := {ω Σ 3 ω i = α i, 0 i < n}. is the open ball in Σ 3 around α = α 0... α n 1... with radius µ 1 n. (b) Consider the piecewise linear map F : [0, 1] [0, 1] given by 2x if x I 1 F (x) = 3 4x if x I 2 2x 3 if x I 2 3 where I 1 = [0, 1 2 ], I 2 = [ 1 2, 3 4 ], and I 3 = [ 3 4, 1]. (i) Show that the dynamics of this map is semiconjugate to a threestate Markov chain with a Markov graph as given in the figure that is depicted below, where points x [0, 1] are coded as ω Σ 3 with ω n = k if F n (x) I k. [Hint: Show that I ω0...ω n 1 ( 2) n, where I ω0...ω n 1 denotes the set of initial conditions in [0, 1] that are represented by a symbolic string in Σ 3 starting with ω 0... ω n 1.] (ii) Determine whether the following statements are correct the Markov chain in (i) is topologically transitive F is topologically transitive Motivate your answer. [You may use results from the course without proof, but any such results should be stated carefully.] (iii) Show that F 4 has 34 fixed points by a different method than writing out all admissible periodic words of length 4 for the Markov chain.
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