R Language Solving ODEs in R The Lorenz model
Example
The Lorenz model describes the dynamics of three state variables, X, Y and Z. The model equations are:
The initial conditions are:
and a, b and c are three parameters with
library(deSolve)
## -----------------------------------------------------------------------------
## Define R-function
## ----------------------------------------------------------------------------
Lorenz <- function (t, y, parms) {
with(as.list(c(y, parms)), {
dX <- a * X + Y * Z
dY <- b * (Y - Z)
dZ <- -X * Y + c * Y - Z
return(list(c(dX, dY, dZ)))
})
}
## -----------------------------------------------------------------------------
## Define parameters and variables
## -----------------------------------------------------------------------------
parms <- c(a = -8/3, b = -10, c = 28)
yini <- c(X = 1, Y = 1, Z = 1)
times <- seq(from = 0, to = 100, by = 0.01)
## -----------------------------------------------------------------------------
## Solve the ODEs
## -----------------------------------------------------------------------------
out <- ode(y = yini, times = times, func = Lorenz, parms = parms)
## -----------------------------------------------------------------------------
## Plot the results
## -----------------------------------------------------------------------------
plot(out, lwd = 2)
plot(out[,"X"], out[,"Y"],
type = "l", xlab = "X",
ylab = "Y", main = "butterfly")
