3.3 Discrete Systems: Tent Maps I

Question 1

Why is the condition that trajectories not being able to cross in phase space not evidence for chaos?

Answer 1

Due to the smooth dynamics of the system

- Chaos also neds at least 3D

Question 2

What happens to the trajectories at a saddle?

Answer 2

They diverge

Question 3

State the equations which describe the tent map

Answer 3

x_N+1 = 2x_n when x < 1/2

= 2(1-x_N) when x > 1/2

Question 4

State the properties of the tent map

Answer 4

It is 1D, non invertible, piecewise, linear, bounded between [0,1]

Question 5

What does the regular tent map look like on a graph, and state the x and y axes

Answer 5

Y axis is x_N+1, x axis is x_N

Increases linearly until (1/2, 1) then decreases back to 0

Question 6

What does the tent map being non invertible mean?

Answer 6

That 2 values of x_N e.g. x_k and x’_k give one value of n_N+1 e.g. x_k+1
Think of the x_N+1 = x_N line and two x values give 1 y value

Question 7

What is the fixed point equal to for the LHS of the tent map?

Answer 7

LHS: x_N+1 = 2x_N

Fixed point is at x bar = 0

Question 8

What is the fixed point equal to for the RHS of the tent map?

Answer 8

RHS: x_N+1 = 2(1 - x_N)

x bar = 2( 1 - x bar ), x bar = 2/3

Question 9

How do we examine the stability of the fixed points on the tent map?

Answer 9

Linearize
x_N = x bar + 𝛿x_N
x_N+1 - x bar + 𝛿x_N+1

Question 10

Describe the global behvaviour of the tent map with the cobweb diagrams

Answer 10

Fixed points are unstable and repulsive

• Lines come in towards the f.p. then get repelled away
• Order shuffled = global chaos