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Jupyter notebook version added
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‎ch05/Program_5.1_logisticmap.ipynb‎

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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"#\n",
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"# Program 5.1: Logistic map (logisticmap.ipynb)\n",
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"# J Wang, Computational modeling and visualization with Python\n",
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"#\n",
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"\n",
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"from __future__ import print_function # use print() as function\n",
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"import matplotlib.pyplot as plt # get matplotlib plot functions\n",
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"import sys\n",
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"%matplotlib notebook\n",
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"\n",
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"if (sys.version_info[0] < 3):\n",
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" x, r = input('enter x0, r: ')\n",
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"else:\n",
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" x, r = eval(input('enter x0, r: '))\n",
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"n, xn = 40, []\n",
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"for i in range(n):\n",
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" xn.append(x) # new line every 4 steps\n",
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" print('%8f' %x, end='\\n' if len(xn)%4==0 else ' ') # inline if \n",
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" x = 4*r*x*(1.0-x) # next iteration, logistic map\n",
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" \n",
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"plt.figure()\n",
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"plt.plot(range(n), xn, '--o') # plot dashed line with 'o' symbol\n",
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"plt.xlabel('$n$'), plt.ylabel('$x_n$')\n",
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"plt.ylim(0,1), plt.text(33, .1, 'r='+ repr(r)) # add text \n",
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"plt.show()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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],
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"display_name": "Python 3",
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"language": "python",
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"version": "3.6.1"
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"nbformat_minor": 2
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}

‎ch05/Program_5.2_perioddbl.ipynb‎

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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"#\n",
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"# Program 5.2: Period doubling (perioddbl.ipynb)\n",
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"# J Wang, Computational modeling and visualization with Python\n",
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"#\n",
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"\n",
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"import matplotlib.pyplot as plt # get matplotlib plot functions\n",
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"%matplotlib notebook\n",
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"\n",
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"ntrans, nsteps = 1000, 200 # num. of transients and to keep\n",
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"r, rend, dr, x, xa = 0.7, 1.0, 0.001, 0.5, [0.0]*nsteps\n",
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"plt.figure()\n",
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"while r <= rend:\n",
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" for i in range(ntrans): x = 4*r*x*(1-x) # discard transients\n",
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" for i in range(nsteps): xa[i], x = x, 4*r*x*(1-x) # keep rest\n",
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" plt.plot([r]*nsteps, xa, 'b,') # blue pixel markers\n",
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" r = r + dr\n",
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"plt.xlabel('r'), plt.ylabel('x'), plt.show()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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},
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],
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}

‎ch05/Program_5.3_logisdiff.ipynb‎

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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"#\n",
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"# Program 5.3: Initial value dependence of logistic map (logisdiff.ipynb)\n",
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"# J Wang, Computational modeling and visualization with Python\n",
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"#\n",
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"\n",
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"import matplotlib.pyplot as plt # get matplotlib plot functions\n",
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"import sys\n",
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"%matplotlib notebook\n",
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"\n",
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"if (sys.version_info[0] < 3):\n",
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" x1, r = input('enter x1, r; eg .4, .7 : ')\n",
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"else:\n",
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" x1, r = eval(input('enter x1, r; eg .4, .7 : '))\n",
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"x2 = x1 + 0.01 # initial difference\n",
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"n, xn1, xn2, diff = 20, [], [], [] # buffers\n",
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"for i in range(n):\n",
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" xn1.append(x1), xn2.append(x2), diff.append(abs(x1-x2))\n",
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" x1, x2 = 4*r*x1*(1.0-x1), 4*r*x2*(1.0-x2) # parallel iterates\n",
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" \n",
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"plt.figure() # plot the series\n",
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"plt.plot(range(n), xn1, 's-', range(n), xn2,'o-') # squares & circles\n",
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"plt.xlabel('$n$'), plt.ylabel('$x_n$')\n",
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"plt.text(3, .01+min(xn1),'$r=$' + repr(r))\n",
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"\n",
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"plt.figure() # plot the difference\n",
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"plt.plot(range(n), diff)\n",
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"plt.semilogy() # semilog scale\n",
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"plt.xlabel('$n$'), plt.ylabel('$\\Delta x$')\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": true
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},
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}
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],
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},
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}
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},
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}

‎ch05/Program_5.4_lyapunov.ipynb‎

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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"#\n",
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"# Program 5.4: Lyapunov exponent of logistic map (lyapunov.ipynb)\n",
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"# J Wang, Computational modeling and visualization with Python\n",
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"#\n",
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"\n",
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"import matplotlib.pyplot as plt # get matplotlib plot functions\n",
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"import math as ma # get math functions \n",
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"%matplotlib notebook\n",
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"\n",
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"def lyapunov(r): # compute Lyapunov exponent\n",
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" sum, x, ntrans, nsteps = 0.0, 0.5, 2000, 2000 # try diff num.\n",
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" for i in range(ntrans): # let transients pass\n",
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" x = 4*r*x*(1-x) \n",
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" for i in range(nsteps): # sum up next n steps\n",
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" x = 4*r*x*(1-x) \n",
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" dfdx = 4.0*r*(1.0-x-x) # \n",
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" sum += ma.log(abs(dfdx))\n",
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" return sum/float(nsteps) # lambda \n",
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" \n",
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"ra, lyap, r, dr = [], [], 0.7, 0.0001\n",
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"while r < 1.0: \n",
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" r = r + dr\n",
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" ra.append(r), lyap.append(lyapunov(r))\n",
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" \n",
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"plt.figure()\n",
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"plt.plot(ra, lyap, ',') # ','=pixels\n",
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"plt.axhline(0.), plt.ylim(-2, 1) # draw horiz. line, set y limits\n",
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"plt.xlabel('$r$'), plt.ylabel('$\\lambda$')\n",
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"plt.show()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {
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"collapsed": true
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},
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}
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],
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}

‎ch05/Program_5.5_nonlindro.ipynb‎

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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": [
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"#\n",
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"# Program 5.5: Nonlindro: Nonlinear driven oscillator (nonlindro.ipynb)\n",
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"# J Wang, Computational modeling and visualization with Python\n",
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"#\n",
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"\n",
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"import matplotlib.pyplot as plt # get matplotlib plot functions\n",
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"import ode, math as ma # get ODE, math functions \n",
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"%matplotlib notebook\n",
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"\n",
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"def pendulum(y, t): # y = [theta, omega], omega_0 = 1\n",
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" return [y[1], -ma.sin(y[0]) - b*y[1] + fd*ma.cos(omega_d*t)]\n",
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"\n",
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"def solution(n_periods): # find solutions for n_periods\n",
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" bins = 40 # number of points per period \n",
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" t, y, h = 0.0, [1.0, 0.0], 2*pi/(omega_d*bins) # init values \n",
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" ta, theta, omega = [], [], []\n",
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" for i in range(n_periods*bins):\n",
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" ta.append(t), theta.append(y[0]), omega.append(y[1])\n",
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" t, y = t+h, ode.RK4n(pendulum, y, t, h)\n",
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" return ta, theta, omega\n",
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"\n",
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"b, omega_d = 0.5, 0.6 # damping coeff., driving frequency\n",
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"subnum, pi = 1, ma.pi # subplot number, pi\n",
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"plt.figure()\n",
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"for fd in [0.7, 1.1]: \n",
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" ax1 = plt.subplot(2, 2, subnum) # 2x2 subplots \n",
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" ax2 = plt.subplot(2, 2, subnum+2)\n",
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" ta, theta, omega = solution(n_periods = 5)\n",
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" ax1.plot(ta, theta), ax2.plot(ta, omega)\n",
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" if (subnum == 1): # subplot specific label\n",
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" ax1.set_ylabel('$\\\\theta$ (rad)')\n",
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" ax2.set_ylabel('$\\\\omega$ (rad/s)')\n",
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" subnum, fdtxt = subnum + 1, '$F_d=$'+repr(fd)\n",
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" ax1.text(17, max(theta), fdtxt), ax2.text(17, max(omega), fdtxt)\n",
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" plt.xlabel('t (s)')\n",
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"plt.show()"
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]
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},
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