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.
Note
Some initial values may lead to an unstable system that will tend to infinity.
Examples
###synthetic example - Rösslerts.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)
References
Rössler, O. E. 1976. An equation for continuous chaos. Physics Letters A, 57, 397-398.