data.gen.Rossler {synthesis}

Rössler system

Description

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

Usage

data.gen.Rossler(
  a = 0.2,
  b = 0.2,
  w = 5.7,
  start = c(-2, -10, 0.2),
  time = seq(0, by = 0.05, length.out = 1000),
  s
)

Arguments

a

The a parameter. Default: 0.2.

b

The b parameter. Default: 0.2.

w

The w parameter. Default: 5.7.

start

A 3-dimensional numeric vector indicating the starting point for the time series. Default: c(-2, -10, 0.2).

time

The temporal interval at which the system will be generated. Default: time=seq(0,50,by=0.01) or time = seq(0,by=0.01,length.out = 1000)

s

The level of noise, default 0.

Details

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

\dot{x} = -(y + z)

\dot{y} = x+a \cdot y

\dot{z} = b + z*(x-w)

The default selection for the system parameters (a = 0.2, b = 0.2, w = 5.7) is known to produce a deterministic chaotic time series. However, the values a = 0.1, b = 0.1, and c = 14 are more commonly used. These Rössler equations are simpler than those Lorenz used since only one nonlinear term appears (the product xz in the third equation).

Here, a = b = 0.1 and c changes. The bifurcation diagram reveals that low values of c are periodic, but quickly become chaotic as c increases. This pattern repeats itself as c increases — there are sections of periodicity interspersed with periods of chaos, and the trend is towards higher-period orbits as c increases. For example, the period one orbit only appears for values of c around 4 and is never found again in the bifurcation diagram. The same phenomenon is seen with period three; until c = 12, period three orbits can be found, but thereafter, they do not appear.

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 Rössler system, respectively.

Note

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

References

Rössler, O. E. 1976. An equation for continuous chaos. Physics Letters A, 57, 397-398.

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

wikipedia https://en.wikipedia.org/wiki/R

Examples

###synthetic example - Rössler

ts.r <- data.gen.Rossler(a = 0.1, b = 0.1, w = 8.7, start = c(-2, -10, 0.2),
                         time = seq(0, by=0.05, length.out = 10000))

oldpar <- par(no.readonly = TRUE)
par(mfrow=c(1,1), ps=12, cex.lab=1.5)
plot.ts(cbind(ts.r$x,ts.r$y,ts.r$z), col=c('black','red','blue'))

par(mfrow=c(1,2), ps=12, cex.lab=1.5)
plot(ts.r$x,ts.r$y, xlab='x',ylab = 'y', type = 'l')
plot(ts.r$x,ts.r$z, xlab='x',ylab = 'z', type = 'l')
par(oldpar)