Practical unit: Logistic map, Chaos and Bifurcations

1.Step)

Write a function which solves the logistic difference equation  and returns the vector x(n).
Use an initial value x0 in [0,1], and a parameter value r in [1,4]
f<-function(r,N,x0)

2.) Investigate the sensitivity of solution on the parameter r (especially using r in [3,4])

3.) Now investigate the solutions dependent on r systematically:

4.) write a function which saves the local extrema of a vector (fixed points) and
returns them in a vector.

For each value of r, iterate the logistic difference equation 500 times, discard the first 200 times, and plot the fix-points / local extrema against r

What do you see ?

Zoom into the plot !

Special R-commands needed:

points(x,y)  #plots the points x,y    x,y can be vectors

before using points() you need to have a plot window open which defines the axes. Therefore call the plot command with ”n” which creates an empty plot.

e.g. plot(xlim=c(0,1),ylim=c(0,1),”n”)  define a 0,1 x 0,1  area

#logistic difference equation, with parameter r, N iterations, and initial value x0

f = function(r,N,x0)

{

  x <- vector()

  x[1]<-x0

  for (i in 2:N)  x[i]<-r*x[i-1]*(1-x[i-1])

  return(x)

}







#determine the values of the local extrema and give them back in a vector

local_extrema <- function(x)

{
result <- vector()

for (i in 1:(length(x)-2))
{

if ((x[i]<x[i+1]) && (x[i+1] > x[i+2])) result<-c(result,x[i+1]) #save the local maximum in the result vector

      if ((x[i]>x[i+1]) && (x[i+1] < x[i+2])) result<-c(result,x[i+1]) #savel the local minimum in the result vector

}

return(result)

}







#test the functions:


#Test the logistic difference equation function

plot(f(3.9,100,0.4),"l")  


hist(f(3.9,1000,0.4),1000)




#Test the local extrema function

temp<-sin((1:300)/30)

plot(temp)

local_extrema(temp)


#test histo


hist(f(4,2000,0.4)[1000:2000],breaks=50)


#main program

resolution<-400  #number of r-parameter values to be scanned

rlim<-c(1,4)     #minimum and maximum r-value

xlim<-c(0,1)     #minimum and maximum x-value


r<-rlim[1]+(1:resolution)*((rlim[2]-rlim[1])/resolution) #vector of all r-values we will scan

plot(xlim=rlim,ylim=xlim,1,"n",xlab="r",ylab="x",main="logistic map") #empty plot with axes and title



for (i in 1:resolution)

{

  temp<-f(r[i],300,0.5)[200:300]

  save<-local_extrema(temp)

  points(rep(r[i],length(save)),save,pch=".")

}