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<p class="admonition-title">Note</p>
<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-algorithms-plot-iterated-dynamical-systems-py"><span class="std std-ref">here</span></a>
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<section class="sphx-glr-example-title" id="iterated-dynamical-systems">
<span id="sphx-glr-auto-examples-algorithms-plot-iterated-dynamical-systems-py"></span><h1>Iterated Dynamical Systems<a class="headerlink" href="#iterated-dynamical-systems" title="Permalink to this heading">#</a></h1>
<p>Digraphs from Integer-valued Iterated Functions</p>
<section id="sums-of-cubes-on-3n">
<h2>Sums of cubes on 3N<a class="headerlink" href="#sums-of-cubes-on-3n" title="Permalink to this heading">#</a></h2>
<p>The number 153 has a curious property.</p>
<p>Let 3N={3,6,9,12,…} be the set of positive multiples of 3. Define an
iterative process f:3N->3N as follows: for a given n, take each digit
of n (in base 10), cube it and then sum the cubes to obtain f(n).</p>
<p>When this process is repeated, the resulting series n, f(n), f(f(n)),…
terminate in 153 after a finite number of iterations (the process ends
because 153 = 1**3 + 5**3 + 3**3).</p>
<p>In the language of discrete dynamical systems, 153 is the global
attractor for the iterated map f restricted to the set 3N.</p>
<p>For example: take the number 108</p>
<p>f(108) = 1**3 + 0**3 + 8**3 = 513</p>
<p>and</p>
<p>f(513) = 5**3 + 1**3 + 3**3 = 153</p>
<p>So, starting at 108 we reach 153 in two iterations,
represented as:</p>
<p>108->513->153</p>
<p>Computing all orbits of 3N up to 10**5 reveals that the attractor
153 is reached in a maximum of 14 iterations. In this code we
show that 13 cycles is the maximum required for all integers (in 3N)
less than 10,000.</p>
<p>The smallest number that requires 13 iterations to reach 153, is 177, i.e.,</p>
<p>177->687->1071->345->216->225->141->66->432->99->1458->702->351->153</p>
<p>The resulting large digraphs are useful for testing network software.</p>
</section>
<section id="the-general-problem">
<h2>The general problem<a class="headerlink" href="#the-general-problem" title="Permalink to this heading">#</a></h2>
<p>Given numbers n, a power p and base b, define F(n; p, b) as the sum of
the digits of n (in base b) raised to the power p. The above example
corresponds to f(n)=F(n; 3,10), and below F(n; p, b) is implemented as
the function powersum(n,p,b). The iterative dynamical system defined by
the mapping n:->f(n) above (over 3N) converges to a single fixed point;
153. Applying the map to all positive integers N, leads to a discrete
dynamical process with 5 fixed points: 1, 153, 370, 371, 407. Modulo 3
those numbers are 1, 0, 1, 2, 2. The function f above has the added
property that it maps a multiple of 3 to another multiple of 3; i.e. it
is invariant on the subset 3N.</p>
<p>The squaring of digits (in base 10) result in cycles and the
single fixed point 1. I.e., from a certain point on, the process
starts repeating itself.</p>
<p>keywords: “Recurring Digital Invariant”, “Narcissistic Number”,
“Happy Number”</p>
</section>
<section id="the-3n-1-problem">
<h2>The 3n+1 problem<a class="headerlink" href="#the-3n-1-problem" title="Permalink to this heading">#</a></h2>
<p>There is a rich history of mathematical recreations
associated with discrete dynamical systems. The most famous
is the Collatz 3n+1 problem. See the function
collatz_problem_digraph below. The Collatz conjecture
— that every orbit returns to the fixed point 1 in finite time
— is still unproven. Even the great Paul Erdos said “Mathematics
is not yet ready for such problems”, and offered $500
for its solution.</p>
<p>keywords: “3n+1”, “3x+1”, “Collatz problem”, “Thwaite’s conjecture”</p>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>Building cubing_153_digraph(10000)
Resulting digraph has 10000 nodes and 10000 edges
Shortest path from 177 to 153 is:
[177, 687, 1071, 345, 216, 225, 141, 66, 432, 99, 1458, 702, 351, 153]
fixed points are []
</pre></div>
</div>
<div class="line-block">
<div class="line"><br /></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">networkx</span> <span class="k">as</span> <span class="nn">nx</span>
<a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">nmax</span></a> <span class="o">=</span> <span class="mi">10000</span>
<a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">p</span></a> <span class="o">=</span> <span class="mi">3</span>
<span class="k">def</span> <span class="nf">digitsrep</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">10</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Return list of digits comprising n represented in base b.</span>
<span class="sd"> n must be a nonnegative integer"""</span>
<span class="k">if</span> <span class="n">n</span> <span class="o"><=</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">dlist</span> <span class="o">=</span> <span class="p">[]</span>
<span class="k">while</span> <span class="n">n</span> <span class="o">></span> <span class="mi">0</span><span class="p">:</span>
<span class="c1"># Prepend next least-significant digit</span>
<span class="n">dlist</span> <span class="o">=</span> <span class="p">[</span><span class="n">n</span> <span class="o">%</span> <span class="n">b</span><span class="p">]</span> <span class="o">+</span> <span class="n">dlist</span>
<span class="c1"># Floor-division</span>
<span class="n">n</span> <span class="o">=</span> <span class="n">n</span> <span class="o">//</span> <span class="n">b</span>
<span class="k">return</span> <span class="n">dlist</span>
<span class="k">def</span> <span class="nf">powersum</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">p</span></a><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">10</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Return sum of digits of n (in base b) raised to the power p."""</span>
<span class="n">dlist</span> <span class="o">=</span> <span class="n">digitsrep</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="nb">sum</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="n">dlist</span><span class="p">:</span>
<span class="nb">sum</span> <span class="o">+=</span> <span class="n">k</span><span class="o">**</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">p</span></a>
<span class="k">return</span> <span class="nb">sum</span>
<span class="k">def</span> <span class="nf">attractor153_graph</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">p</span></a><span class="p">,</span> <span class="n">multiple</span><span class="o">=</span><span class="mi">3</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">10</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Return digraph of iterations of powersum(n,3,10)."""</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">DiGraph</span><span class="p">()</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<span class="k">if</span> <span class="n">k</span> <span class="o">%</span> <span class="n">multiple</span> <span class="o">==</span> <span class="mi">0</span> <span class="ow">and</span> <span class="n">k</span> <span class="ow">not</span> <span class="ow">in</span> <span class="n">G</span><span class="p">:</span>
<span class="n">k1</span> <span class="o">=</span> <span class="n">k</span>
<span class="n">knext</span> <span class="o">=</span> <span class="n">powersum</span><span class="p">(</span><span class="n">k1</span><span class="p">,</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">p</span></a><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="k">while</span> <span class="n">k1</span> <span class="o">!=</span> <span class="n">knext</span><span class="p">:</span>
<span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">k1</span><span class="p">,</span> <span class="n">knext</span><span class="p">)</span>
<span class="n">k1</span> <span class="o">=</span> <span class="n">knext</span>
<span class="n">knext</span> <span class="o">=</span> <span class="n">powersum</span><span class="p">(</span><span class="n">k1</span><span class="p">,</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">p</span></a><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="k">return</span> <span class="n">G</span>
<span class="k">def</span> <span class="nf">squaring_cycle_graph_old</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">10</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Return digraph of iterations of powersum(n,2,10)."""</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">DiGraph</span><span class="p">()</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<span class="n">k1</span> <span class="o">=</span> <span class="n">k</span>
<span class="n">G</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="n">k1</span><span class="p">)</span> <span class="c1"># case k1==knext, at least add node</span>
<span class="n">knext</span> <span class="o">=</span> <span class="n">powersum</span><span class="p">(</span><span class="n">k1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">k1</span><span class="p">,</span> <span class="n">knext</span><span class="p">)</span>
<span class="k">while</span> <span class="n">k1</span> <span class="o">!=</span> <span class="n">knext</span><span class="p">:</span> <span class="c1"># stop if fixed point</span>
<span class="n">k1</span> <span class="o">=</span> <span class="n">knext</span>
<span class="n">knext</span> <span class="o">=</span> <span class="n">powersum</span><span class="p">(</span><span class="n">k1</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">k1</span><span class="p">,</span> <span class="n">knext</span><span class="p">)</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">out_degree</span><span class="p">(</span><span class="n">knext</span><span class="p">)</span> <span class="o">>=</span> <span class="mi">1</span><span class="p">:</span>
<span class="c1"># knext has already been iterated in and out</span>
<span class="k">break</span>
<span class="k">return</span> <span class="n">G</span>
<span class="k">def</span> <span class="nf">sum_of_digits_graph</span><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">nmax</span></a><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">10</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="k">return</span> <span class="n">powersum</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="k">return</span> <span class="n">discrete_dynamics_digraph</span><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">nmax</span></a><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">squaring_cycle_digraph</span><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">nmax</span></a><span class="p">,</span> <span class="n">b</span><span class="o">=</span><span class="mi">10</span><span class="p">):</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="k">return</span> <span class="n">powersum</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="n">b</span><span class="p">)</span>
<span class="k">return</span> <span class="n">discrete_dynamics_digraph</span><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">nmax</span></a><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">cubing_153_digraph</span><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">nmax</span></a><span class="p">):</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="k">return</span> <span class="n">powersum</span><span class="p">(</span><span class="n">n</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="k">return</span> <span class="n">discrete_dynamics_digraph</span><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">nmax</span></a><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">discrete_dynamics_digraph</span><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">nmax</span></a><span class="p">,</span> <span class="n">f</span><span class="p">,</span> <span class="n">itermax</span><span class="o">=</span><span class="mi">50000</span><span class="p">):</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">nx</span><span class="o">.</span><span class="n">DiGraph</span><span class="p">()</span>
<span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">nmax</span></a> <span class="o">+</span> <span class="mi">1</span><span class="p">):</span>
<span class="n">kold</span> <span class="o">=</span> <span class="n">k</span>
<span class="n">G</span><span class="o">.</span><span class="n">add_node</span><span class="p">(</span><span class="n">kold</span><span class="p">)</span>
<span class="n">knew</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">kold</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">kold</span><span class="p">,</span> <span class="n">knew</span><span class="p">)</span>
<span class="k">while</span> <span class="n">kold</span> <span class="o">!=</span> <span class="n">knew</span> <span class="ow">and</span> <span class="n">kold</span> <span class="o"><<</span> <span class="n">itermax</span><span class="p">:</span>
<span class="c1"># iterate until fixed point reached or itermax is exceeded</span>
<span class="n">kold</span> <span class="o">=</span> <span class="n">knew</span>
<span class="n">knew</span> <span class="o">=</span> <span class="n">f</span><span class="p">(</span><span class="n">kold</span><span class="p">)</span>
<span class="n">G</span><span class="o">.</span><span class="n">add_edge</span><span class="p">(</span><span class="n">kold</span><span class="p">,</span> <span class="n">knew</span><span class="p">)</span>
<span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">out_degree</span><span class="p">(</span><span class="n">knew</span><span class="p">)</span> <span class="o">>=</span> <span class="mi">1</span><span class="p">:</span>
<span class="c1"># knew has already been iterated in and out</span>
<span class="k">break</span>
<span class="k">return</span> <span class="n">G</span>
<span class="k">def</span> <span class="nf">collatz_problem_digraph</span><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">nmax</span></a><span class="p">):</span>
<span class="k">def</span> <span class="nf">f</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
<span class="k">if</span> <span class="n">n</span> <span class="o">%</span> <span class="mi">2</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="k">return</span> <span class="n">n</span> <span class="o">//</span> <span class="mi">2</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="mi">3</span> <span class="o">*</span> <span class="n">n</span> <span class="o">+</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">discrete_dynamics_digraph</span><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">nmax</span></a><span class="p">,</span> <span class="n">f</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">fixed_points</span><span class="p">(</span><span class="n">G</span><span class="p">):</span>
<span class="w"> </span><span class="sd">"""Return a list of fixed points for the discrete dynamical</span>
<span class="sd"> system represented by the digraph G.</span>
<span class="sd"> """</span>
<span class="k">return</span> <span class="p">[</span><span class="n">n</span> <span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="n">G</span> <span class="k">if</span> <span class="n">G</span><span class="o">.</span><span class="n">out_degree</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="mi">0</span><span class="p">]</span>
<a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">nmax</span></a> <span class="o">=</span> <span class="mi">10000</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"Building cubing_153_digraph(</span><span class="si">{</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">nmax</span></a><span class="si">}</span><span class="s2">)"</span><span class="p">)</span>
<span class="n">G</span> <span class="o">=</span> <span class="n">cubing_153_digraph</span><span class="p">(</span><a href="https://docs.python.org/3/library/functions.html#int" title="builtins.int" class="sphx-glr-backref-module-builtins sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">nmax</span></a><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Resulting digraph has"</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">G</span><span class="p">),</span> <span class="s2">"nodes and"</span><span class="p">,</span> <span class="n">G</span><span class="o">.</span><span class="n">size</span><span class="p">(),</span> <span class="s2">" edges"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"Shortest path from 177 to 153 is:"</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="n">nx</span><span class="o">.</span><span class="n">shortest_path</span><span class="p">(</span><span class="n">G</span><span class="p">,</span> <span class="mi">177</span><span class="p">,</span> <span class="mi">153</span><span class="p">))</span>
<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"fixed points are </span><span class="si">{</span><span class="n">fixed_points</span><span class="p">(</span><span class="n">G</span><span class="p">)</span><span class="si">}</span><span class="s2">"</span><span class="p">)</span>
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