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<h1 id="non-linear-systems-logistic-maps-and-chaos">Non-Linear Systems: Logistic Maps and Chaos</h1>
<p><em><strong>Suggested Method of Viewing the report</strong></em>
Since this is a big report, for your convinience please visit this website to see the report clearly
<a href="https://logistics-map-group10.netlify.com/">non-linear-systems-group10</a></p>
<p>The logistic map is a model of population growth that exhibits many different types of behavior, dependingon the value of a few constants.
The equation then, for somepopulation $X_n+1$ after an arbitrary time step, starting with population $X_n$ is:</p>
<p>$X_n+1$ = r*$X_n$*(1 - $X_n$)</p>
<h3 id="experiment-report-by-group-10a">Experiment Report by Group 10A</h3>
<ul>
<li>Aarushi Goenka IMT2018001</li>
<li>Abhigna Banda IMT2018002</li>
<li>Agam Kashyap IMT2018004</li>
<li>Mohd Nazar Kashif IMT2018042</li>
</ul>
<h3 id="references-used">References Used</h3>
<ul>
<li>Explanation of Logistic maps by <a href="https://www.youtube.com/watch?v=ovJcsL7vyrk">Veritasium</a></li>
<li>The Feigenbaum Constant (4.669) - <a href="https://www.youtube.com/watch?v=ETrYE4MdoLQ">Numberphile</a></li>
</ul>
<h1 id="time-series-plots">Time Series Plots</h1>
<p>Code used to generate the time series plots:</p>
<pre class="hljs"><code><div>import numpy as np
import matplotlib.pyplot as plt
def fun(r,x):
return r*x*(1-x)
r_value = 2.85
init_value = 0.5
max_n = 600
xn_values = []
xn_values.append(init_value)
for i in range(1,max_n):
xn_values.append(fun(r_value,xn_values[i-1]))
plt.xlabel('n', fontsize=20)
plt.ylabel(r'$X_n$', fontsize=20)
plt.title(r'$X_n$ vs n time series at r = %f'%(r_value))
plt.plot(range(0,max_n),xn_values)
plt.savefig('timeseries%f.png'%(r_value))
plt.show()
</div></code></pre>
<p>Corresponding graphs with r values ranging from 1 to 4</p>
<p><em><strong>Graph1 (Done by Abhigna)</strong></em></p>
<p><img src="0.5.png" alt=""></p>
<p><em><strong>Graph2 (Done by Abhigna)</strong></em></p>
<p><img src="1.0.png" alt=""></p>
<p><em><strong>Graph3 (Done by Abhigna)</strong></em></p>
<p><img src="1.3.png" alt=""></p>
<p><em><strong>Graph4 (Done by Abhigna)</strong></em></p>
<p><img src="1.5.png" alt=""></p>
<p><em><strong>Graph5 (Done by Agam)</strong></em></p>
<p><img src="./timeseries2.000.png" alt=""></p>
<p><em><strong>Graph6 (Done by Agam)</strong></em></p>
<p><img src="timeseries2.50000.png" alt=""></p>
<p><em><strong>Graph7 (Done by Agam)</strong></em></p>
<p><img src="./timeseries2.85000.png" alt=""></p>
<p><em><strong>Graph8 (Done by Agam)</strong></em></p>
<p><img src="timeseries2.789000.png" alt=""></p>
<p><em><strong>Graph9 (Done by Kashif)</strong></em></p>
<p><img src="timeseries3.35.png" alt=""></p>
<p><em><strong>Graph10 (Done by Kashif)</strong></em></p>
<p><img src="timeseries3.35.png" alt=""></p>
<p><em><strong>Graph11 (Done by Kashif)</strong></em></p>
<p><img src="timeseries3.37.png" alt=""></p>
<p><em><strong>Graph12 (Done by Aarushi)</strong></em></p>
<p><img src="3.5.png" alt=""></p>
<p><em><strong>Graph13 (Done by Aarushi)</strong></em></p>
<p><img src="3.545.png" alt=""></p>
<p><em><strong>Graph14 (Done by Aarushi)</strong></em></p>
<p><img src="3.57.png" alt=""></p>
<p><em><strong>Graph15 (Done by Aarushi)</strong></em></p>
<p><img src="3.64.png" alt=""></p>
<p><em><strong>Graph16 (Done by Aarushi)</strong></em></p>
<p><img src="3.82843.png" alt=""></p>
<p><em><strong>Graph17 (Done by Aarushi)</strong></em></p>
<p><img src="3.9.png" alt=""></p>
<h2 id="conclusions-drawn-from-this">Conclusions Drawn from this</h2>
<p><em><strong>By Aarushi</strong></em></p>
<table>
<thead>
<tr>
<th style="text-align:center">Values</th>
<th style="text-align:center">Observation</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center"><= 1</td>
<td style="text-align:center">Population Will Die i.e. $X_n$ goes to 0</td>
</tr>
<tr>
<td style="text-align:center">1.5</td>
<td style="text-align:center">Gives a Constant Value</td>
</tr>
<tr>
<td style="text-align:center">2</td>
<td style="text-align:center">Gives a Constant Value</td>
</tr>
<tr>
<td style="text-align:center">2.5</td>
<td style="text-align:center">Becomes constant after fluctuating for a while</td>
</tr>
<tr>
<td style="text-align:center">2.789</td>
<td style="text-align:center">Becomes constant after fluctuating for a while</td>
</tr>
<tr>
<td style="text-align:center">2.85</td>
<td style="text-align:center">Becomes constant after fluctuating for a while</td>
</tr>
<tr>
<td style="text-align:center">3</td>
<td style="text-align:center">The rate of convergence is less than linear</td>
</tr>
<tr>
<td style="text-align:center">3.34560</td>
<td style="text-align:center">Oscillates between 2 values : Period 2</td>
</tr>
<tr>
<td style="text-align:center">3.5</td>
<td style="text-align:center">Oscillates between 4 values : Period doubles</td>
</tr>
<tr>
<td style="text-align:center">3.545</td>
<td style="text-align:center">Oscillates between 8 values : Again Period has doubles from before</td>
</tr>
<tr>
<td style="text-align:center">3.57</td>
<td style="text-align:center">It is chaotic, Can't find repeating values periodically</td>
</tr>
<tr>
<td style="text-align:center">3.64</td>
<td style="text-align:center">It is Chaotic</td>
</tr>
<tr>
<td style="text-align:center">3.82843</td>
<td style="text-align:center">should've Oscillated between 3 values but didn't happen</td>
</tr>
<tr>
<td style="text-align:center">3.9</td>
<td style="text-align:center">Chaotic</td>
</tr>
<tr>
<td style="text-align:center">4</td>
<td style="text-align:center">Chaotic</td>
</tr>
</tbody>
</table>
<blockquote>
<p>One fine point that we noticed is,
after 3.57, a smal change in the value
of r can make the system change suddenly from
chaotic to stable and vice versa.</p>
</blockquote>
<h3 id="time-series-graphs-made-from-value-of-r-using-r--3847xxx">Time-Series Graphs made from Value of r using r = 3.847XXX</h3>
<p><em><strong>Code used by Aarushi</strong></em></p>
<pre class="hljs"><code><div>import numpy as np
from matplotlib import pyplot as plt
r = 3.847001
x = np.arange(0,151)
y = [0.5]
for i in range(150):
g = y[-1]
y.append(r*g*(1-g))
plt.xlabel('n', fontsize=10)
plt.ylabel('x_n', fontsize=10)
plt.title(r'$X_n$ vs n logistics map for r=3.847001')
plt.plot(x,y)
plt.show()
</div></code></pre>
<p><em><strong>Graph by Aarushi Roll number IMT2018001</strong></em></p>
<p><img src="imt2018001_3.847001.png" alt=""></p>
<p><em><strong>Code used by Abhigna</strong></em></p>
<pre class="hljs"><code><div>import numpy as np
import matplotlib.pyplot as plt
x0=0.5
r=3.847002
X=[]
Y=[]
X.append(0)
Y.append(x0)
count=0
for n in range (1,600):
X.append(n)
Y.append(r*Y[count]*(1-Y[count]))
count=count+1
plt.xlabel('n',fontsize=20)
plt.ylabel('$X_n$',fontsize=20)
plt.title('$X_n$ vs n')
plt.plot(X, Y)
plt.show()
</div></code></pre>
<p><em><strong>Graph by Abhigna Roll number IMT2018002</strong></em></p>
<p><img src="imt2018002.png" alt=""></p>
<p><em><strong>Code by Agam</strong></em></p>
<pre class="hljs"><code><div>import numpy as np
import matplotlib.pyplot as plt
def fun(r,x):
return r*x*(1-x)
r_value = 2.85
init_value = 0.5
max_n = 600
xn_values = []
xn_values.append(init_value)
for i in range(1,max_n):
xn_values.append(fun(r_value,xn_values[i-1]))
plt.xlabel('n', fontsize=20)
plt.ylabel(r'$X_n$', fontsize=20)
plt.title(r'$X_n$ vs n time series at r = %f'%(r_value))
plt.plot(range(0,max_n),xn_values)
plt.savefig('timeseries%f.png'%(r_value))
plt.show()
</div></code></pre>
<p><em><strong>Graph by Agam Roll number IMT2018004</strong></em></p>
<p><img src="timeseries004.png" alt=""></p>
<p><em><strong>Code by Kashif</strong></em></p>
<pre class="hljs"><code><div>import numpy as np
import matplotlib.pyplot as plt
r = 3.847042
X = []
X.append(0.5)
for i in range(1,800):
X.append(r*X[i-1]*(1-X[i-1]))
plt.xlabel('n')
plt.ylabel(r'$X_n$')
plt.show()
</div></code></pre>
<p><em><strong>Graph by Kashif Roll number IMT2018042</strong></em></p>
<p><img src="imt2018042.png" alt=""></p>
<h1 id="bifurcation-diagrams">Bifurcation Diagrams</h1>
<p>A bifucation is a period-doubling, a change from an N-point attractor to a 2N-point attractor, which occurs when the control parameter is changed.
A Bifurcation Diagram is a visual summary of the succession of period-doubling produced as r increases. The next figure shows the bifurcation diagram of the logistic map, r along the x-axis. For each value of r the system is first allowed to settle down and then the successive values of x are plotted for a few hundred iterations.</p>
<p><strong>Code 1 used (Written by Agam Kashyap)</strong></p>
<pre class="hljs"><code><div>import numpy as np
import matplotlib.pyplot as plt
def fun(r,x):
return r*x*(1-x)
def calc(n1,n2,nr,init_value):
r_values = np.linspace(nr,4,1000)
for r in r_values:
alt = []
xn_values = []
alt.append(init_value)
for i in range(1,n1):
alt.append(fun(r,alt[i-1]))
xn_values.append(alt[n1-1])
for i in range(1, n2-n1):
xn_values.append(fun(r,xn_values[i-1]))
xn_values = np.array(xn_values)
r_arr = xn_values*0.0 + r
plt.plot(r_arr,xn_values,'ko',markersize=1)
r_min = 1
plt.xlabel('r', fontsize=20)
plt.axis([r_min,4.0,0,1.0])
plt.title(r'$X_n$ vs r bifurcation graph')
calc(100,200,r_min,0.05)
plt.show()
</div></code></pre>
<p><strong>Bifurcation Graph from Code 1</strong></p>
<p><img src="bifurcation_agam1.png" alt=""></p>
<p><em><strong>Code 2 by Aarushi</strong></em></p>
<pre class="hljs"><code><div>import matplotlib.pyplot as plt
import numpy as np
P=np.linspace(0.7,4,50000)
m=0.7
X = []
Y = []
for u in P:
X.append(u)
m = np.random.random()
for n in range(1500):
m=(u*m)*(1-m)
Y.append(m)
plt.xlabel('r', fontsize=10)
plt.ylabel('x_n', fontsize=10)
plt.title(r'$X_n$ vs r bifurcation graph')
plt.plot(X, Y, ",", color='k')
plt.show()
</div></code></pre>
<p><strong>Bifurcation Graph from Code2</strong></p>
<p><img src="bifurcatiob_diagram.png" alt=""></p>
<h1 id="feigenbaum-constants">Feigenbaum Constants</h1>
<p>The first Feigenbaum constant is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map</p>
<p>$x_i+1$ = f( $x_i$ )</p>
<p>where f(x) is a function parameterized by the bifurcation parameter a.</p>
<p>It is given by the limit</p>
<p>$\delta$ = $\displaystyle \lim_{n \to \infty} g(x)$ = 4.669201609....</p>
<p>where g(x) = $\frac{a_{n-1} - a_{n-2}}{a_n - a_{n-1}}$</p>
<p>where $a_n$ are discrete values of a at the n-th period doubling.</p>
<p><em><strong>Observations (Done by Abhigna and Kashif)</strong></em></p>
<table>
<thead>
<tr>
<th style="text-align:center">r1</th>
<th style="text-align:center">r2</th>
<th style="text-align:center">r3</th>
<th style="text-align:center">g(x)</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">3.002</td>
<td style="text-align:center">3.44912</td>
<td style="text-align:center">3.53708</td>
<td style="text-align:center">5.083219645</td>
</tr>
<tr>
<td style="text-align:center">3.002</td>
<td style="text-align:center">3.44912</td>
<td style="text-align:center">3.54197</td>
<td style="text-align:center">4.815508885</td>
</tr>
<tr>
<td style="text-align:center">3.002</td>
<td style="text-align:center">3.44668</td>
<td style="text-align:center">3.54197</td>
<td style="text-align:center">4.666596705</td>
</tr>
<tr>
<td style="text-align:center">3.002</td>
<td style="text-align:center">3.44668</td>
<td style="text-align:center">3.53708</td>
<td style="text-align:center">4.919026549</td>
</tr>
</tbody>
</table>
<p>Average value of Feigenbaum constant = 4.871087945999999</p>
<p><em><strong>Report Made by Agam</strong></em></p>
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