4 min read

Chaos Butterfly


Package Odin has gone live on CRAN with version 1. It makes some stuff with deSolve a little easier: - Defining a set of ODEs looks a bit easier - it’s easier to change parameters - it can write its own C code for when you’ve got a massive set of ODEs and the efficiency helps.

Also, it’s a great punny name.

Lorentz’s Butterfly

Lorentz was trying to study a very simplified weather model with early computers. It’s easily described in 3 dimensions with respect to time - Link to his paper. At least skim it for the pictures, this is 1960’s fractal art! The 3 dimensions don’t seem to have simple interpretations once they’ve gone into this form.

It’s incredibly sensitive to initial conditions, he had problems trying to resume a simulation from a printout that only had 3 decimals, but the computer held more in memory.


lorentz <- odin({
  deriv(y1) <- sigma * (y2 - y1)
  deriv(y2) <- R * y1 - y2 - y1 * y3
  deriv(y3) <- -b * y3 + y1 * y2
  initial(y1) <- start_y_1
  initial(y2) <- start_y_2
  initial(y3) <- start_y_3
  start_y_1 <- user(10)
  start_y_2 <- user(1)
  start_y_3 <- user(1)
  ## These are the classical parameters:
  sigma <- 10
  R <- 28
  b <-  8 / 3
## gcc -std=gnu99 -I"/usr/share/R/include" -DNDEBUG      -fpic  -g -O2 -fdebug-prefix-map=/build/r-base-el7SHG/r-base-3.5.2=. -fstack-protector-strong -Wformat -Werror=format-security -Wdate-time -D_FORTIFY_SOURCE=2 -g  -c odin_a305fa58.c -o odin_a305fa58.o
## gcc -std=gnu99 -shared -L/usr/lib/R/lib -Wl,-Bsymbolic-functions -Wl,-z,relro -o odin_a305fa58.so odin_a305fa58.o -L/usr/lib/R/lib -lR

model <- lorentz()

tt <- seq(0, 100, by=1e-3)

model_result <- model$run(tt,method="lsode" ) %>%
  as_tibble() %>%

path <- function(data, mapping){
  ggplot(data=data, mapping=mapping) + 
    geom_path( aes(colour=t)) + 
    scale_color_viridis_c(option = "C") +

self <- function(data, mapping){
  ggplot(data=data, mapping=mapping) + geom_point(aes(y=t), size=0.1,alpha=0.1) + coord_flip()

model_result %>%
  ggpairs(columns =c("y1","y2","y3"),
          lower=list(continuous=path)) + ggthemes::theme_tufte()

It has 3 spatial dimensions, and 1 dimension of time, so let’s throw it at more stuff.

model_result %>%
  mutate(s=round(t,0)) %>%
  plot_ly(x=~y1,y=~y2,z=~y3, color=~t) %>%