Dynamical Systems — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

Namè of the Paper Dynamical Systems 1. - a) LetQ(x) = x? - 2. Apply Newton's method with the initial guess xo = 1. Compute x1, 2z and xy. Compare these values with the exact root V2. c) Define 9(0) = 8 + Esin(20) for 0 < € <} i) Find the periodic points of the function g. ii) Sketch the phase portraits of g. 2. a) Letp be a hyperbolic fixed point with If'(p)| > 1.Then there is an open. interval U of p such that if x E U, ‡ p,then there exists k>0 such that f*(x) $U. b) Find all the periodic points of f(x) = x- x'. Classify them as attracting, repelling with the phase portraits. c) For the function F¿(x) = 2x (1 - x). Prove that F2 (x) → 1/2 as n → 3. a) Let stE Ez and suppose si = ti for is n, then prove that dis, t) ≤ En where dis, t] = I stil Conversely, if dis, t) < in then si = t, for i 5n. ii) Compute dis, t] for s = (1000 1000 1000...) and t = b) Define o: Ez → Izas a(So545...) = (45253...) Prove that the cardinality of Per, (0) = 27. Let Qc(x) = x? + c. Prove that if c < 1/4, there is a unique / > 1 such that Oc is topologically conjugate to F, (x) = ux (1 - x) via a map of the form h(x) = ax + B. 4. a) Define Periodic point. Find a collection F of blocks over (0, 1} so that Let Xbe the full A-shift. Explain statements that no longer hold if we merely assume that X is a shift space. c) Define the higher block shift. Prove that the higher block shifts of a shift space are also shift spaces. The image of a shift space under a sliding block code is a shift space. b) Define a shift of firite type with example. Prove that if X is a shift of finite type, then there is an M ≥ O such that X is M-step. c) Suppose that H is a subgraph of G. What is the relationship between the adjacency matrices An and Aç? 6. a) Let X be a shift space. If X* is shift of finite type, must X be a shift of finite iype? b) Define Pseudo orbit tracing property (Shadowing property). c) Let fbe a linear map of the Euclidean space R" and d be the Euclidean metric for R". Then f has Pseudo orbit tracing property (POTP) under - d if and only if f is hyperbolic, i.e. it has no eigenvalues of modulus 1.