4.1 Lyapunov Exponents Flashcards
What is the general method used to quantify chaos?
Looking at the rate of divergence of the trajectories
Describe the relationship between unstable fixed points and the local gradient
- 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
What is M(x) on the tent map?
The line of x_N+1 = x_N
If we are looking at one iterate of the tent map, how can we determine the stability?
Look local to the fixed point
What is the relationship between the number of tent triangles and the number of fixed points?
One tent per two fixed points
- For p iterates, 2^p fixed points, 2^p-1 triangles
What is the nature of the stability of the 2^p iterates of the tent maps?
Always unstable as 2^p > 1 globally
- Globally chaotic
How can we turn a product into a sum?
Write f’(x_j) = exp [ ln ( f’(x_j) ) ]
What does the Lyapunov exponent measure?
The divergence of trajectories
State the Lyapunov exponent equation
λ = lim(N -> infinity) 1/N sum(j = 0 -> N-1) ln( | f’(x_j) | )
Describe the behvaiour when the Lyapunov exponent is bigger than or less than 0
λ > 0: Chaos
λ < 0: Stable attractive points ie no chaos