4.1 Lyapunov Exponents Flashcards

1
Q

What is the general method used to quantify chaos?

A

Looking at the rate of divergence of the trajectories

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Describe the relationship between unstable fixed points and the local gradient

A
  • If the local gradient is bigger than 1, the fixed point is unstable and repulsive
  • If the local gradient is less than 1, it is stable and attractive
  • ie spiralling inwards or outwards and the stability just depends on the local gradient
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

What is M(x) on the tent map?

A

The line of x_N+1 = x_N

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

If we are looking at one iterate of the tent map, how can we determine the stability?

A

Look local to the fixed point

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

What is the relationship between the number of tent triangles and the number of fixed points?

A

One tent per two fixed points

- For p iterates, 2^p fixed points, 2^p-1 triangles

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

What is the nature of the stability of the 2^p iterates of the tent maps?

A

Always unstable as 2^p > 1 globally

- Globally chaotic

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

How can we turn a product into a sum?

A

Write f’(x_j) = exp [ ln ( f’(x_j) ) ]

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

What does the Lyapunov exponent measure?

A

The divergence of trajectories

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

State the Lyapunov exponent equation

A

λ = lim(N -> infinity) 1/N sum(j = 0 -> N-1) ln( | f’(x_j) | )

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Describe the behvaiour when the Lyapunov exponent is bigger than or less than 0

A

λ > 0: Chaos

λ < 0: Stable attractive points ie no chaos

How well did you know this?
1
Not at all
2
3
4
5
Perfectly