# Lyapunov exponent -- Numerical calculations

1. Jun 30, 2015

### 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.

2. Jul 3, 2015

### Dr. Courtney

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

3. Jul 3, 2015

### 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?

4. Jul 3, 2015

### 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.

5. Jul 3, 2015

### LagrangeEuler

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

6. Jul 3, 2015

### Dr. Courtney

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.

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