Dynamic systems

About

This page provides some supplemental information for the discussion about dynamic systems.

It shows a simulation of the Lorenz system. The code is drawn from Wikipedia contributors (2025).

It also provides a link to an excellent video about convolution, a core operation in many areas of psychology and neuroscience.

Lorenz System

Equations

\[\dot{x} = \sigma(y-x) \] \[\dot{y} = \rho x - y - xz \] \[\dot{z} = -\beta z + xy \]

Simulation

Quietly load required libraries.

suppressPackageStartupMessages(library(deSolve))
suppressPackageStartupMessages(library(plotly))

Simulate n=10^{4} time points

# parameters
prm <- list(sigma = 10, rho = 28, beta = 8/3)
# initial values
varini <- c(
  X = 1,
  Y = 1, 
  Z = 1
)

Lorenz <- function (t, vars, prm) {
  with(as.list(vars), {
    dX <- prm$sigma*(Y - X)
    dY <- X*(prm$rho - Z) - Y
    dZ <- X*Y - prm$beta*Z
    return(list(c(dX, dY, dZ)))
   })
}

times <- seq(from = 0, to = 100, by = 0.01)
# call ode solver
out <- ode(y = varini, times = times, func = Lorenz,
           parms = prm)

Plot data.

# to assign color to points
gfill <- function (repArr, long) {
  rep(repArr, ceiling(long/length(repArr)))[1:long]
}

dout <- as.data.frame(out)

dout$color <- gfill(rainbow(10), nrow(dout))

# Graphics production with Plotly:
plot_ly(
  data=dout, x = ~X, y = ~Y, z = ~Z,
  type = 'scatter3d', mode = 'lines',
  opacity = 1, line = list(width = 6, color = ~color, reverscale = FALSE)
)

On convolution

Convolution is a fundamental operation in statistics, and it is a core operation found in Dynamic Field Theory and connectionist models. Here’s an excellent video to explain what convolution is about and how it works.

– 3Blue1Brown (2022)

References

3Blue1Brown. (2022). But what is a convolution? YouTube. Retrieved from https://www.youtube.com/watch?v=KuXjwB4LzSA

Wikipedia contributors. (2025, October 6). Lorenz system. Retrieved from https://en.wikipedia.org/wiki/Lorenz_system