library(deSolve)
library(ggplot2)

# Define the ODE system function
simple_ode <- function(time, state, parameters) {
  x <- state[1]  # State variable
  k <- parameters[1]  # Parameter
  
  # Condition where x is below 0.5 and dx/dt is negative
  if (x < 0.5 && k * x < 0) {
    dx <- 1  # Set dx to 1
  } else {
    dx <- -k * x  # Exponential decay
  }
  
  # Condition where x is below 0.9
  if (x < 0.9) {
    dx <- 1  # Set dx to 1
  }
  
  return(list(dx))
}

# Initial state and parameter
initial_state <- c(x = 1)  # Initial value of x
parameters <- c(k = 0.1)  # Decay rate

# Time range
time <- seq(0, 100, by = 0.1)  # Time range from 0 to 10 with a step size of 0.1

# Solve the ODE system
solution <- ode(y = initial_state, times = time, func = simple_ode, parms = parameters)

# Create a data frame from the solution
df <- as.data.frame(solution)

# Plot the state variable over time
plot <- ggplot(df, aes(x = time, y = x)) +
  geom_line() +
  xlab("Time") +
  ylab("x") +
  ylim(0,5)+
  ggtitle("Simple ODE System") +
  theme_minimal()

# Save the plot as a PDF file
ggsave("simple_ode_plot.pdf", plot, width = 8, height = 6)