# data.gen.Lorenz: Lorenz system In synthesis: Generate Synthetic Data from Statistical Models

## Description

Generates a 3-dimensional time series using the Lorenz equations.

## Usage

 ```1 2 3 4 5 6 7 8``` ```data.gen.Lorenz( sigma = 10, beta = 8/3, rho = 28, start = c(-13, -14, 47), time = seq(0, 50, length.out = 1000), s ) ```

## Arguments

 `sigma` The sigma parameter. Default: 10. `beta` The beta parameter. Default: 8/3. `rho` The rho parameter. Default: 28. `start` A 3-dimensional numeric vector indicating the starting point for the time series. Default: c(-13, -14, 47). `time` The temporal interval at which the system will be generated. Default: time=seq(0,50,by = 0.01). `s` The level of noise, default 0.

## Details

The Lorenz system is a system of ordinary differential equations defined as:

dx/dt = sigma*( y - x )

dy/dt = rho*x - y - xz

dz/dt = -beta*z + xy

The default selection for the system parameters (sigma=10, rho=28, beta=8/3) is known to produce a deterministic chaotic time series.

## Value

A list with four vectors named time, x, y and z containing the time, the x-components, the y-components and the z-components of the Lorenz system, respectively.

## Note

Some initial values may lead to an unstable system that will tend to infinity.

## References

Constantino A. Garcia (2019). nonlinearTseries: Nonlinear Time Series Analysis. R package version 0.2.7. https://CRAN.R-project.org/package=nonlinearTseries

## Examples

 ```1 2 3 4 5``` ```###Synthetic example - Lorenz ts.l <- data.gen.Lorenz(sigma = 10, beta = 8/3, rho = 28, start = c(-13, -14, 47), time = seq(0, by=0.05, length.out = 2000)) ts.plot(cbind(ts.l\$x,ts.l\$y,ts.l\$z), col=c('black','red','blue')) ```

synthesis documentation built on Nov. 27, 2021, 5:07 p.m.