4.4 Logistic Map 2

Question 1

State the logistic map equation and its fixed points

Answer 1

M(x) = λx(1-x)

Fixed points at x = 0, λ - 1 / λ

Question 2

How do we obtain the fixed points of the logistic map?

Answer 2

Differentiate M(x) and set it to equal 0

Question 3

Describe the stability and logistic map shapes for 1 < λ < 3 and 3 < λ < 4

Answer 3

Two logistic maps peaking at λ/4 on the y axis, but the 3 < x < 4 is a much taller shape

• Attractor for 1 -> 3 as the local gradient is less than 1
• Repeller for 3 -> 4
• Gradient scales with λ

Question 4

Describe how we can find the fixed points of M^2(x)

Answer 4

We know two at x = 0 and x = λ-1/λ as fixed points of M(x) are fixed points of M^2(x)

Question 5

Describe the two different quartic tent map shapes for λ < 3 and λ > 3

Answer 5

λ < 3: Tall quartic with only one intercept for M^2(x)
λ > 3: More exaggerated quartic with three total intercepts with M^2(x). 2 new unknown fixed points which are not fixed points of M(x)

Question 6

What is the relationship between fixed points in M(x) and M^2(x)?

Answer 6

All fixed points in M(x) are fixed points in M^2(x) but the reverse isnt always true

Question 7

What is a period 2 of M(x)?

Answer 7

When one of the two new unknown fixed points when 3 < λ < 4 are attractive
- Get a sort of quadrilateral orbit with the iterations

Question 8

Describe the bifurcation behaviour on an x-λ graph

Answer 8

First bifurcation at λ = 3, then successive doublings until global chaos at λ = 4