## Largest Lyapunov exponent (LLE)

Purpose

Description

Macro Synopsis

Modules

Related Functions

References

### Purpose

Calculate the largest Lyapunov exponent of a time series

### Description

Lyapunov exponents quantify the mean rate of divergence of neighboured
trajectories along various directions in phase space. For converging
trajectories, the corresponding Lyapunov exponents are negative.

The Lyapunov exponents of a discrete time-series () are
defined as the logarithms of the eigenvalues (denoted as *e.v.*) of the
tangent map

with embedding dimension m and n as the number of time steps.
(see Lyapunov spectrum.)

where J is the Jacobian matrix.

Time series of chaotic systems have a positive LLE.
Parameters of "Largest Lyapunov exponent (LLE)" are

- the embedding dimension
`m`,
- the number
`M` of neighbours taken into account in phase space,
- the degree
`pd` of the fitting polynome.

The result of "Largest Lyapunov exponent (LLE)" will appear in the message window.

**Warning:** *Phase space calculations on long signals are computationally very expensive, and can lead to large response times of the program. Try to use small radii and low embedding dimensions if possible.*

### Macro Synopsis

`LLE(x,m,M,pd);`

signal x;

int m,M,pd;

### Modules

Nonlinear

### Related Functions

Momentary largest Lyapunov exponent, Lyapunov spectrum,
Correlation dimension, Correlation integral,
Pointwise correlation dimension (PD2), Pointwise correlation integral.

### References

Briggs [37], Wolf et al. [38]