Lyapunov Numbers Flashcards
(9 cards)
What is a lyapunov number?
The average per-step divergence rate of nearby points along the orbit.
What is the difference between the lyapunov exponent and the lyapunov number?
The exponent is the natural log of the lyapunov number.
How would you define a lyapunov exponent to someone?
A quantity that characterizes the rate of separation of infinitesimally close trajectories in a dynamical system.
What does a lyapunov number of 2 for orbit x mean about the distance between orbit x and the orbit of nearby point x?
That the distance doubles each iteration (on average).
What does a lyapunov number of 1/2 mean (as above)?
Distance would be halved each iteration (on average).
Why is there a spectrum of Lyapunov exponents?
Because the rate of separation can be different for different orientations of initial separation vector.
How many Lyapunov exponents are in its system’s spectrum?
The number is equal to the dimensionality of the phase space.
Describe what the global Maximal Lyapunov Exponent (MLE) is, and what information can be derived from it?
The MLE is the largest exponent in the spectrum, and it can determine how predictable the system is. A positive MLE usually indicates that the system is chaotic.
What is the difference between the global and local lyapunov exponents?
Global is used to determine the predictability of the whole system, whereas local is used to determine the predictability around a point in phase space.
Global lyapunov exponents are invariant under a nonlinear change in coordinates, local are not.
