Dynamical Systems - Logistic Difference Equation

The Logistic Map Function - Computing a Finite Difference Equation

• Defines a logistic map function to iteratively compute \(x_{t+1} = r x_t (1 - x_t)\).
  
• Simulates the map for four key values of \(r\), each corresponding to a distinct dynamical regime.
  
• Uses ggplot2 to create a faceted visualization, showing:
  • Stable equilibrium at \(r = 2.5\),
  • Periodic oscillation at \(r = 3.2\),
  • Period-doubling at \(r = 3.5\),
  • Chaos at \(r = 3.9\).

---

# Load required library
library(ggplot2)

# Logistic map function
logistic_map <- function(r, x0 = 0.5, n = 1000, discard = 500) {
  x <- numeric(n)
  x[1] <- x0
  for (t in 1:(n - 1)) {
    x[t + 1] <- r * x[t] * (1 - x[t])
  }
  # Discard initial transient behavior
  return(x[(discard + 1):n])
}

### Time Series Visualization for Selected r values ###
r_values <- c(2.5, 3.2, 3.5, 3.9)
simulations <- lapply(r_values, function(r) {
  data.frame(
    t = 1:100,
    x = logistic_map(r, n = 100, discard = 0),
    r = as.factor(r)
  )
})
df_traj <- do.call(rbind, simulations)

p1 <- ggplot(df_traj, aes(x = t, y = x, color = r)) +
  geom_line(size = 1) +
  facet_wrap(~r, ncol = 2, scales = "free_y") +
  theme_minimal(base_size = 14) +
  labs(
    title = "Logistic Map Dynamics for Selected r Values",
    subtitle = expression(paste(x[t+1] == r*x[t]*(1 - x[t]))),
    x = "Time step (t)",
    y = expression(x[t]),
    color = "Growth rate (r)"
  ) +
  theme(plot.title = element_text(face = "bold", color = "#003366"),
        strip.text = element_text(face = "bold", size = 13))

### Bifurcation Diagram ###
r_seq <- seq(2.5, 4, by = 0.001)  # range of r values
bifurcation_data <- data.frame()

for (r in r_seq) {
  x <- logistic_map(r, n = 1200, discard = 800)  # longer simulation for steady state
  bifurcation_data <- rbind(
    bifurcation_data,
    data.frame(r = rep(r, length(x)), x = x)
  )
}

p2 <- ggplot(bifurcation_data, aes(x = r, y = x)) +
  geom_point(color = "blue", alpha = 0.1, shape = 16, size = 0.4) +
  theme_minimal(base_size = 14) +
  labs(
    title = "Bifurcation Diagram of the Logistic Map",
    subtitle = expression(paste("Steady-state behavior of ", x[t+1] == r*x[t]*(1 - x[t]))),
    x = "Growth rate parameter (r)",
    y = "Steady-state values of x"
  ) +
  theme(plot.title = element_text(face = "bold", color = "#003366"))

# Display both plots
p1

p2