Consider the decimal shift map on the unit interval given by x n+1 =10x n (mod1) . a) The graph of the map is to be drawn. b) Find all fixed points. c) Prove that the map has periodic points of all periods, but all of them are unstable. d) Prove that the map has infinitely many aperiodic orbits. e) Show that the map has sensitive dependance on initial conditions by considering the rate of separation of two nearby orbits. Concept Introduction: ➢ The logistic map is a second-degree equivalent mapping. It has mention how its complex points and the chaotic logistic map is used to arise into a very straightforward non-linear dynamic equation. ➢ The logistic map function is x n+1 = f(x n ) .

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Page of 2 ZOOM + 1) Answer the following questions by circling TRUE or FALSE (No explanation or work required). i) If A = [1 -2 1] 0 1 6, rank(A) = 3. (TRUE FALSE) LO 0 0] ii) If S = {1,x,x², x³} is a basis for P3, dim(P3) = 4 with the standard operations. (TRUE FALSE) iii) Let u = (1,1) and v = (1,-1) be two vectors in R². They are orthogonal according to the following inner product on R²: (u, v) = U₁V₁ + 2U2V2. ( TRUE FALSE) iv) A set S of vectors in an inner product space V is orthogonal when every pair of vectors in S is orthogonal. (TRUE FALSE) v) Dot product of two perpendicular vectors is zero. (TRUE FALSE) vi) Cross product of two perpendicular vectors is zero. (TRUE FALSE) 2) a) i) Determine which function(s) are solutions of the following linear differential equation. - y (4) — 16y= 0 • 3 cos x • 3 cos 2x -2x • e • 3e2x-4 sin 2x ii) Find the Wronskian for the set of functions that you found from i) as the solution of the differential equation above. iii) What does the result…

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Question

Chapter 10.3, Problem 7E

Interpretation Introduction

Interpretation:

Consider the decimal shift map on the unit interval given by xn+1=10xn(mod1).

a) The graph of the map is to be drawn.

b) Find all fixed points.

c) Prove that the map has periodic points of all periods, but all of them are unstable.

d) Prove that the map has infinitely many aperiodic orbits.

e) Show that the map has sensitive dependance on initial conditions by considering the rate of separation of two nearby orbits.

Concept Introduction:

➢ The logistic map is a second-degree equivalent mapping. It has mention how its complex points and the chaotic logistic map is used to arise into a very straightforward non-linear dynamic equation.

➢ The logistic map function is xn+1= f(xn).

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Page of 2 ZOOM + 1) Answer the following questions by circling TRUE or FALSE (No explanation or work required). i) If A = [1 -2 1] 0 1 6, rank(A) = 3. (TRUE FALSE) LO 0 0] ii) If S = {1,x,x², x³} is a basis for P3, dim(P3) = 4 with the standard operations. (TRUE FALSE) iii) Let u = (1,1) and v = (1,-1) be two vectors in R². They are orthogonal according to the following inner product on R²: (u, v) = U₁V₁ + 2U2V2. ( TRUE FALSE) iv) A set S of vectors in an inner product space V is orthogonal when every pair of vectors in S is orthogonal. (TRUE FALSE) v) Dot product of two perpendicular vectors is zero. (TRUE FALSE) vi) Cross product of two perpendicular vectors is zero. (TRUE FALSE) 2) a) i) Determine which function(s) are solutions of the following linear differential equation. - y (4) — 16y= 0 • 3 cos x • 3 cos 2x -2x • e • 3e2x-4 sin 2x ii) Find the Wronskian for the set of functions that you found from i) as the solution of the differential equation above. iii) What does the result…

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1. Give a subset that satisfies all the following properties simultaneously: Subspace Convex set Affine set Balanced set Symmetric set Hyperspace Hyperplane 2. Give a subset that satisfies some of the conditions mentioned in (1) but not all, with examples. 3. Provide a mathematical example (not just an explanation) of the union of two balanced sets that is not balanced. 4. What is the precise mathematical condition for the union of two hyperspaces to also be a hyperspace? Provide a proof. edited 9:11