#!/usr/bin/env python3
# -*- coding: utf-8 -*-
"""
Cobweb plot
Created on Mon Aug 24 08:43:32 2020
@author: sal
"""

# scientific libraries
import numpy as np
import matplotlib.pyplot as plt

# parameters and initial conditions
x0 = 0.123 # initial state
n = 7 # number of iterations 

# transformation
l = 3.333
def map(x,l):
	return l * x * (1-x)  # logistic map l=lambda

# cobweb arrays 
X, Y = [x0] , [x0]
xn = x0 
for i in range(0,n):
    yn = map(xn,l) # f(xn) 
    X.append(xn) , Y.append(yn) # (xn, f(xn)) 
    X.append(yn) , Y.append(yn) # (f(xn), f(xn))
    xn = yn 
    
# picture
plt.figure()
plt.title('Cobweb plot of the map $f(x)=$' + str(round(l,3)) +'$\cdot x (1-x)$')
#plt.xlabel('x')
#plt.ylabel('f(x)')
xmin, xmax = 0.0 , 1.0
ymin, ymax = 0.0 , 1.0
plt.axis([xmin,xmax,ymin,ymax])
plt.axis('equal')
x = np.arange(xmin, xmax, 0.01)  
plt.scatter(x0, x0, c='lightblue', s=40, label='$x(0)=$' + str(x0)) # initial state
plt.plot(x,x, 'pink') # diagonal
plt.plot(x,map(x,l), 'r') # graph of the tranformation
plt.plot(X,Y,'lightblue') # cobweb plot 
plt.scatter(Y[2*n], Y[2*n], c='C0', s=40, label='$x($' 
            + str(n) + '$) \simeq $' + str(round(xn,3)) ) # final state
plt.legend()