Lyapunov exponent -- Numerical calculations

[LagrangeEuler]

In computational physics is very often to calculate largest Lyapunov exponent. If largest Lyapunov exponent $LE$ is positive there is chaos in the system, if it is negative or zero there is no chaos in the system. But what can we say about some certain value of $LE$. For example $LE_1=-0.2$ and $LE_2=-0.4$. What is the difference between those particular values? Could we say something in small scales? Thanks for the answer.

 

[Dr. Courtney]

The more negative the LE is, the more quickly the trajectory returns to its unperturbed path in that dimension of phase space.

 

[LagrangeEuler]

Thank you for the answer. Is it necessary that if LE is negative motion is periodic? And when LE is zero motion is quasiperiodic?

 

[Dr. Courtney]

Too many variables in play for a simple answer. Conservative or not? Winding numbers? Dimensions?

A negative LE does not imply a periodic orbit.

 

[LagrangeEuler]

Ok thanks. But periodic orbits imply that LE is negative. Right?

 

[Dr. Courtney]

Ok thanks. But periodic orbits imply that LE is negative. Right?

Not always. There are cases of unstable periodic orbits where the LE is positive.

Try a google search for unstable periodic orbits. There are a lot of examples.