data.gen.Lorenz {synthesis}

Lorenz system

Description

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

Usage

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:

\dot{x} = \sigma(y-x)

\dot{y} = \rho x-y-xz

\dot{z} = -\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

###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'))