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

## Description

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

## Usage

 ```1 2 3 4 5 6 7 8``` ```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:

dx/dt = -(y + z)

dy/dt = x + a*y

dz/dt = 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

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33``` ```###synthetic example - RÃ¶ssler 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) # Application to testing variance transformation method in: # Jiang, Z., Sharma, A., & Johnson, F. (2020) data <- list(x = ts.r\$z, dp = cbind(ts.r\$x, ts.r\$y)) dwt <- WASP::dwt.vt(data, wf="d4", J=7, method="dwt", pad="zero", boundary="periodic") par(mfrow = c(ncol(dwt\$dp), 1), mar = c(0, 2.5, 2, 1), oma = c(2, 1, 0, 0), # move plot to the right and up mgp = c(1.5, 0.5, 0), # move axis labels closer to axis pty = "m", bg = "transparent", ps = 12) # plot(dwt\$x, type="l", xlab=NA, ylab="SPI12", col="red") # plot(dwt\$x, type="l", xlab=NA, ylab="Rain", col="red") for (i in 1:ncol(dwt\$dp)) { ts.plot(cbind(dwt\$dp[, i], dwt\$dp.n[, i]), xlab = NA, ylab = NA, col = c("black", "blue"), lwd = c(1, 2)) } ```

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