Lorenz Attractor in R.

Lorenz Attractor

The Lorenz System is one of the most famous system of equations in the realm of chaotic systems first studied by Edward Lorenz. The double lob remembering a butterfly wing is on the imagination of any complex systems enthusiast.

Lorenz Attractor Equations

The three equations that govern its behavior are:

$\frac{dx}{dt}=\sigma(y-x)$

$\frac{dy}{dt}=x(\rho-z)-y$

$\frac{dz}{dt}=xy-\beta z$

where usually $\sigma=10,\rho=28,\beta=8/3$

So let's define the parameters, the initial state of the system and the equations in the form of a function called Lorenz

parameters <- c(s = 10, r = 28, b = 8/3)
state <- c(X = 0, Y = 1, Z = 1)
 
Lorenz <- function(t, state, parameters) {
    with(as.list(c(state, parameters)), {
        dX <- s * (Y - X)
        dY <- X * (r - Z) - Y
        dZ <- X * Y - b * Z
        list(c(dX, dY, dZ))
    })
}

We can now start processing this and plotting it.

times <- seq(0, 50, by = 0.01)
library(deSolve)
out <- ode(y = state, times = times, func = Lorenz, parms = parameters)
 
par(oma = c(0, 0, 3, 0))
plot(out, xlab = "time", ylab = "-")
plot(out[, "Y"], out[, "Z"], pch = ".", type = "l")
mtext(outer = TRUE, side = 3, "Lorenz model", cex = 1.5)

The above example is mainly copied from the the deSolve package documentation that uses the same example. What I'd like to point out in this is how simple it is to solve and plot a system of differential equations in R. The language is simple and clear and it is much more practical than implementing everything from scratch.

If you want to play with a couple of Lorenz attractors synchronizing please visit this Java implementation.

Up next: Animating the Lorenz Attractor in R.