## Lyapunov spectrum

Purpose

Description

Macro Synopsis

Modules

Related Functions

References

### Purpose

Calculation of the Lyapunov exponents of a time series

### Description

The Lyapunov exponents are defined as follows :
For a discrete dynamical system

they are the logarithmic eigenvalues of the matrix

where J is the Jacobian matrix.
The exponents may be seen as the average exponential growth rates of
initially close points under the flow generated by f. By convention,
they are numbered from the largest to the smallest,

The set is the Lyapunov spectrum. It is calculated using the parameters

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

The results of "Lyapunov spectrum" including the Lyapunov exponents, the sum of
the Lyapunov exponents, and the Kaplan-Yorke-Dimension 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

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

signal x;

int m,M,pd;

### Modules

Nonlinear

### Related Functions

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

### References

Briggs [37]