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<a name="Graphical-analysis-of-discrete-dynamical-systems"></a>
<div class="header">
<p>
Next: <a href="maxima_259.html#Visualization-with-VTK" accesskey="n" rel="next">Visualization with VTK</a>, Previous: <a href="maxima_257.html#The-dynamics-package" accesskey="p" rel="previous">The dynamics package</a>, Up: <a href="maxima_256.html#dynamics_002dpkg" accesskey="u" rel="up">dynamics-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
</div>
<a name="Graphical-analysis-of-discrete-dynamical-systems-1"></a>
<h3 class="section">55.2 Graphical analysis of discrete dynamical systems</h3>
<a name="chaosgame"></a><a name="Item_003a-dynamics_002fdeffn_002fchaosgame"></a><dl>
<dt><a name="index-chaosgame"></a>Function: <strong>chaosgame</strong> <em>([[<var>x1</var>, <var>y1</var>]…[<var>xm</var>, <var>ym</var>]], [<var>x0</var>, <var>y0</var>], <var>b</var>, <var>n</var>, <var>options</var>, …);</em></dt>
<dd>
<p>Implements the so-called chaos game: the initial point (<var>x0</var>,
<var>y0</var>) is plotted and then one of the <var>m</var> points
[<var>x1</var>, <var>y1</var>]…<var>xm</var>, <var>ym</var>]
will be selected at random. The next point plotted will be on the
segment from the previous point plotted to the point chosen randomly, at a
distance from the random point which will be <var>b</var> times that segment’s
length. The procedure is repeated <var>n</var> times. The options are the
same as for <code><a href="maxima_68.html#plot2d">plot2d</a></code>.
</p>
<p><strong>Example</strong>. A plot of Sierpinsky’s triangle:
</p>
<div class="example">
<pre class="example">(%i1) chaosgame([[0, 0], [1, 0], [0.5, sqrt(3)/2]], [0.1, 0.1], 1/2,
30000, [style, dots]);
</pre></div>
<img src="./figures/dynamics7.png" alt="./figures/dynamics7">
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-dynamics">Package dynamics</a>
·<a href="maxima_424.html#Category_003a-Plotting">Plotting</a>
·</div>
</dd></dl>
<a name="evolution"></a><a name="Item_003a-dynamics_002fdeffn_002fevolution"></a><dl>
<dt><a name="index-evolution"></a>Function: <strong>evolution</strong> <em>(<var>F</var>, <var>y0</var>, <var>n</var>, …, <var>options</var>, …);</em></dt>
<dd>
<p>Draws <var>n+1</var> points in a two-dimensional graph, where the horizontal
coordinates of the points are the integers 0, 1, 2, ..., <var>n</var>, and
the vertical coordinates are the corresponding values <var>y(n)</var> of the
sequence defined by the recurrence relation
</p><div class="example">
<pre class="example"> y(n+1) = F(y(n))
</pre></div>
<p>With initial value <var>y(0)</var> equal to <var>y0</var>. <var>F</var> must be an
expression that depends only on one variable (in the example, it
depend on <var>y</var>, but any other variable can be used),
<var>y0</var> must be a real number and <var>n</var> must be a positive integer.
This function accepts the same options as <code><a href="maxima_68.html#plot2d">plot2d</a></code>.
</p>
<p><strong>Example</strong>.
</p>
<div class="example">
<pre class="example">(%i1) evolution(cos(y), 2, 11);
</pre></div>
<img src="./figures/dynamics1.png" alt="./figures/dynamics1">
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-dynamics">Package dynamics</a>
·<a href="maxima_424.html#Category_003a-Plotting">Plotting</a>
·</div>
</dd></dl>
<a name="evolution2d"></a><a name="Item_003a-dynamics_002fdeffn_002fevolution2d"></a><dl>
<dt><a name="index-evolution2d"></a>Function: <strong>evolution2d</strong> <em>([<var>F</var>, <var>G</var>], [<var>u</var>, <var>v</var>], [<var>u0</var>, <var>y0</var>], <var>n</var>, <var>options</var>, …);</em></dt>
<dd>
<p>Shows, in a two-dimensional plot, the first <var>n+1</var> points in the
sequence of points defined by the two-dimensional discrete dynamical
system with recurrence relations
</p><div class="example">
<pre class="example"> u(n+1) = F(u(n), v(n)) v(n+1) = G(u(n), v(n))
</pre></div>
<p>With initial values <var>u0</var> and <var>v0</var>. <var>F</var> and <var>G</var> must be
two expressions that depend only on two variables, <var>u</var> and
<var>v</var>, which must be named explicitly in a list. The options are the
same as for <code><a href="maxima_68.html#plot2d">plot2d</a></code>.
</p>
<p><strong>Example</strong>. Evolution of a two-dimensional discrete dynamical system:
</p>
<div class="example">
<pre class="example">(%i1) f: 0.6*x*(1+2*x)+0.8*y*(x-1)-y^2-0.9$
(%i2) g: 0.1*x*(1-6*x+4*y)+0.1*y*(1+9*y)-0.4$
(%i3) evolution2d([f,g], [x,y], [-0.5,0], 50000, [style,dots]);
</pre></div>
<img src="./figures/dynamics5.png" alt="./figures/dynamics5">
<p>And an enlargement of a small region in that fractal:
</p>
<div class="example">
<pre class="example">(%i9) evolution2d([f,g], [x,y], [-0.5,0], 300000, [x,-0.8,-0.6],
[y,-0.4,-0.2], [style, dots]);
</pre></div>
<img src="./figures/dynamics6.png" alt="./figures/dynamics6">
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-dynamics">Package dynamics</a>
·<a href="maxima_424.html#Category_003a-Plotting">Plotting</a>
·</div>
</dd></dl>
<a name="ifs"></a><a name="Item_003a-dynamics_002fdeffn_002fifs"></a><dl>
<dt><a name="index-ifs"></a>Function: <strong>ifs</strong> <em>([<var>r1</var>, …, <var>rm</var>], [<var>A1</var>,…, <var>Am</var>], [[<var>x1</var>, <var>y1</var>], …, [<var>xm</var>, <var>ym</var>]], [<var>x0</var>, <var>y0</var>], <var>n</var>, <var>options</var>, …);</em></dt>
<dd>
<p>Implements the Iterated Function System method. This method is similar
to the method described in the function <code><a href="#chaosgame">chaosgame</a></code>. but instead of
shrinking the segment from the current point to the randomly chosen
point, the 2 components of that segment will be multiplied by the 2 by 2
matrix <var>Ai</var> that corresponds to the point chosen randomly.
</p>
<p>The random choice of one of the <var>m</var> attractive points can be made
with a non-uniform probability distribution defined by the weights
<var>r1</var>,...,<var>rm</var>. Those weights are given in cumulative form; for
instance if there are 3 points with probabilities 0.2, 0.5 and 0.3, the
weights <var>r1</var>, <var>r2</var> and <var>r3</var> could be 2, 7 and 10. The
options are the same as for <code><a href="maxima_68.html#plot2d">plot2d</a></code>.
</p>
<p><strong>Example</strong>. Barnsley’s fern, obtained with 4 matrices and 4 points:
</p>
<div class="example">
<pre class="example">(%i1) a1: matrix([0.85,0.04],[-0.04,0.85])$
(%i2) a2: matrix([0.2,-0.26],[0.23,0.22])$
(%i3) a3: matrix([-0.15,0.28],[0.26,0.24])$
(%i4) a4: matrix([0,0],[0,0.16])$
(%i5) p1: [0,1.6]$
(%i6) p2: [0,1.6]$
(%i7) p3: [0,0.44]$
(%i8) p4: [0,0]$
(%i9) w: [85,92,99,100]$
(%i10) ifs(w, [a1,a2,a3,a4], [p1,p2,p3,p4], [5,0], 50000, [style,dots]);
</pre></div>
<img src="./figures/dynamics8.png" alt="./figures/dynamics8">
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-dynamics">Package dynamics</a>
·<a href="maxima_424.html#Category_003a-Plotting">Plotting</a>
·</div>
</dd></dl>
<a name="orbits"></a><a name="Item_003a-dynamics_002fdeffn_002forbits"></a><dl>
<dt><a name="index-orbits"></a>Function: <strong>orbits</strong> <em>(<var>F</var>, <var>y0</var>, <var>n1</var>, <var>n2</var>, [<var>x</var>, <var>x0</var>, <var>xf</var>, <var>xstep</var>], <var>options</var>, …);</em></dt>
<dd>
<p>Draws the orbits diagram for a family of one-dimensional
discrete dynamical systems, with one parameter <var>x</var>; that kind of
diagram is used to study the bifurcations of an one-dimensional discrete
system.
</p>
<p>The function <var>F(y)</var> defines a sequence with a starting value of
<var>y0</var>, as in the case of the function <code>evolution</code>, but in this
case that function will also depend on a parameter <var>x</var> that will
take values in the interval from <var>x0</var> to <var>xf</var> with increments of
<var>xstep</var>. Each value used for the parameter <var>x</var> is shown on the
horizontal axis. The vertical axis will show the <var>n2</var> values
of the sequence <var>y(n1+1)</var>,..., <var>y(n1+n2+1)</var> obtained after letting
the sequence evolve <var>n1</var> iterations. In addition to the options
accepted by <code><a href="maxima_68.html#plot2d">plot2d</a></code>, it accepts an option <var>pixels</var> that
sets up the maximum number of different points that will be represented
in the vertical direction.
</p>
<p><strong>Example</strong>. Orbits diagram of the quadratic map, with a parameter
<var>a</var>:
</p>
<div class="example">
<pre class="example">(%i1) orbits(x^2+a, 0, 50, 200, [a, -2, 0.25], [style, dots]);
</pre></div>
<img src="./figures/dynamics3.png" alt="./figures/dynamics3">
<p>To enlarge the region around the lower bifurcation near x <code>=</code> -1.25 use:
</p><div class="example">
<pre class="example">(%i2) orbits(x^2+a, 0, 100, 400, [a,-1,-1.53], [x,-1.6,-0.8],
[nticks, 400], [style,dots]);
</pre></div>
<img src="./figures/dynamics4.png" alt="./figures/dynamics4">
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-dynamics">Package dynamics</a>
·<a href="maxima_424.html#Category_003a-Plotting">Plotting</a>
·</div>
</dd></dl>
<a name="staircase"></a><a name="Item_003a-dynamics_002fdeffn_002fstaircase"></a><dl>
<dt><a name="index-staircase"></a>Function: <strong>staircase</strong> <em>(<var>F</var>, <var>y0</var>, <var>n</var>,<var>options</var>,…);</em></dt>
<dd>
<p>Draws a staircase diagram for the sequence defined by the recurrence
relation
</p><div class="example">
<pre class="example"> y(n+1) = F(y(n))
</pre></div>
<p>The interpretation and allowed values of the input parameters is the
same as for the function <code><a href="#evolution">evolution</a></code>. A staircase diagram consists
of a plot of the function <var>F(y)</var>, together with the line <var>G(y)</var>
<code>=</code> <var>y</var>. A vertical segment is drawn from the point (<var>y0</var>,
<var>y0</var>) on that line until the point where it intersects the function
<var>F</var>. From that point a horizontal segment is drawn until it reaches
the point (<var>y1</var>, <var>y1</var>) on the line, and the procedure is
repeated <var>n</var> times until the point (<var>yn</var>, <var>yn</var>) is
reached. The options are the same as for <code><a href="maxima_68.html#plot2d">plot2d</a></code>.
</p>
<p><strong>Example</strong>.
</p>
<div class="example">
<pre class="example">(%i1) staircase(cos(y), 1, 11, [y, 0, 1.2]);
</pre></div>
<img src="./figures/dynamics2.png" alt="./figures/dynamics2">
<div class=categorybox>
Categories:<a href="maxima_424.html#Category_003a-Package-dynamics">Package dynamics</a>
·<a href="maxima_424.html#Category_003a-Plotting">Plotting</a>
·</div>
</dd></dl>
<a name="Item_003a-dynamics_002fnode_002fVisualization-with-VTK"></a><hr>
<div class="header">
<p>
Next: <a href="maxima_259.html#Visualization-with-VTK" accesskey="n" rel="next">Visualization with VTK</a>, Previous: <a href="maxima_257.html#The-dynamics-package" accesskey="p" rel="previous">The dynamics package</a>, Up: <a href="maxima_256.html#dynamics_002dpkg" accesskey="u" rel="up">dynamics-pkg</a> [<a href="maxima_toc.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="maxima_423.html#Function-and-Variable-Index" title="Index" rel="index">Index</a>]</p>
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