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the fixed points lose/gain stability. (iii) using a solid line for stable and dashed line for unstable. (i) Find the period-2 orbit(s) as a function of A. (ii) Plot these orbits as a function of A. [Hint: f2(x) = x is a polynomial of degree 9 but you can factor 3 roots from ² as a third period one and then cast the period-2 equation using the variable z = order polynomial on z. Note that z = A + 1 is a root of this last polynomial so you omlicitly solve 1

the fixed points lose/gain stability. (iii) using a solid line for stable and dashed line for unstable. (i) Find the period-2 orbit(s) as a function of A. (ii) Plot these orbits as a function of A. [Hint: f2(x) = x is a polynomial of degree 9 but you can factor 3 roots from ² as a third period one and then cast the period-2 equation using the variable z = order polynomial on z. Note that z = A + 1 is a root of this last polynomial so you omlicitly solve 1

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 91E
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1. Consider the cubic map In+1 = f(n) = In - I with -2≤ n ≤ 2 and 0 ≤ ≤ 3. (a) (i) Find all fixed points and study their stability. (ii) Find the values of λ at which the fixed points lose/gain stability. (iii) Plot these fixed points, as a function of X, using a solid line for stable and dashed line for unstable. (b) (i) Find the period-2 orbit(s) as a function of A. (ii) Plot these orbits as a function of A. [Hint: f2(x) = x is a polynomial of degree 9 but you can factor 3 roots from 1² as a third period one and then cast the period-2 equation using the variable z = order polynomial on z. Note that z = A + 1 is a root of this last polynomial so you can finally get a quadratic for z that you can explicitly solve.] 1.9 and one with o = 2.1. = (c) Let λ = 3. (i) Draw two cobweb orbits, one with co (ii) Explain the differences between the two orbits. (d) Draw a bifurcation diagram for the cubic map. Be sure to include positive and negative initial conditions. Compare the diagram with that of the logistic map.
Transcribed Image Text:1. Consider the cubic map In+1 = f(n) = In - I with -2≤ n ≤ 2 and 0 ≤ ≤ 3. (a) (i) Find all fixed points and study their stability. (ii) Find the values of λ at which the fixed points lose/gain stability. (iii) Plot these fixed points, as a function of X, using a solid line for stable and dashed line for unstable. (b) (i) Find the period-2 orbit(s) as a function of A. (ii) Plot these orbits as a function of A. [Hint: f2(x) = x is a polynomial of degree 9 but you can factor 3 roots from 1² as a third period one and then cast the period-2 equation using the variable z = order polynomial on z. Note that z = A + 1 is a root of this last polynomial so you can finally get a quadratic for z that you can explicitly solve.] 1.9 and one with o = 2.1. = (c) Let λ = 3. (i) Draw two cobweb orbits, one with co (ii) Explain the differences between the two orbits. (d) Draw a bifurcation diagram for the cubic map. Be sure to include positive and negative initial conditions. Compare the diagram with that of the logistic map.
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