# Compare nonlinear system to linearized system

# First, simulate the nonlinear system
######################################
# Define the time of simulation
maxTime <- 10
dt <- 0.01 # Precision of simulation

# Prepare variable for results of simulation
ts <- seq(0,maxTime,dt)
N <- length(ts) # Number of samples
x <- numeric( N ) # The results will go here

x[1] <- 0.01 # Initial condition
for (t in 2:N) {
  x[t] <- x[t-1] + dt*( sin(x[t-1] ) )
}

# Plot the nonlinear system
matplot( ts, x, xlab = "Time", ylab = "x", type = "l",col="red",add = FALSE )

# Now, simulate the linearized system
######################################
# This asks the user to click mouse on the curve
xy <- locator( n=1 )
# The first value is time, the second is x
t0 <- xy[[1]]
x0 <- xy[[2]]

# Time axis
ts <- seq(t0,maxTime,dt) # Starting from t0 !!!
# Analytic formula
xPred = pi + (x0-pi)*exp(-(ts-t0))

# Plot
matplot( ts, xPred, type = "l",col="green",add = TRUE )