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<a name="dynamics"></a>
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<h1 class="chapter"> 49. dynamics </h1>

<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC215">49.1 Introduction to dynamics</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC216">49.2 Functions and Variables for dynamics</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">
</td></tr>
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<p><a name="Item_003a-Introduction-to-dynamics"></a>
</p><hr size="6">
<a name="Introduction-to-dynamics"></a>
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<h2 class="section"> 49.1 Introduction to dynamics </h2>

<p>The additional package <code>dynamics</code> includes several
functions to create various graphical representations of discrete
dynamical systems and fractals, and an implementation of the Runge-Kutta
4th-order numerical method for solving systems of differential equations.
</p>
<p>To use the functions in this package you must first load it with
<code>load(&quot;dynamics&quot;)</code>.
</p>
<p><b>Changes introduced in Maxima 5.12</b>
</p>
<p>Starting with Maxima 5.12, the dynamics package now uses the function
<code>plot2d</code> to do the graphs. The commands that produce graphics
(with the exception of <code>julia</code> and <code>mandelbrot</code>) now accept
any options of <code>plot2d</code>, including the option to change among the
various graphical interfaces, using different plot styles and colors,
and representing one or both axes in a logarithmic scale. The old
options <var>domain</var>, <var>pointsize</var>, <var>xcenter</var>, <var>xradius</var>,
<var>ycenter</var>, <var>yradius</var>, <var>xaxislabel</var> and <var>yaxislabel</var>
are not accepted in this new version.
</p>
<p>All programs will now accept any variables names, and not just <var>x</var>
and <var>y</var> as in the older versions. Two required parameters have
changes in two of the programs: <code>evolution2d</code> now requires a list
naming explicitely the two independent variables, and the horizontal
range for <code>orbits</code> no longer requires a step size; the range
should only specify the variable name, and the minimum and maximum
values; the number of steps can now be changed with the option
<var>nticks</var>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Dynamical-systems">Dynamical systems</a>
 &middot;
<a href="maxima_95.html#Category_003a-Share-packages">Share packages</a>
 &middot;
<a href="maxima_95.html#Category_003a-Package-dynamics">Package dynamics</a>
</p>
</div>


<p><a name="Item_003a-Functions-and-Variables-for-dynamics"></a>
</p><hr size="6">
<a name="Functions-and-Variables-for-dynamics"></a>
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</tr></table>
<h2 class="section"> 49.2 Functions and Variables for dynamics </h2>

<p><a name="Item_003a-chaosgame"></a>
</p><dl>
<dt><u>Function:</u> <b>chaosgame</b><i> (<code>[[</code><var>x1</var>, <var>y1</var><code>]</code>...<code>[</code><var>xm</var>, <var>ym</var><code>]]</code>, <code>[</code><var>x0</var>, <var>y0</var><code>]</code>, <var>b</var>, <var>n</var>, ..., options, ...);</i>
<a name="IDX2017"></a>
</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
<code>[</code><var>x1</var>, <var>y1</var><code>]</code>...<code>[</code><var>xm</var>, <var>ym</var><code>]</code>
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.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-dynamics">Package dynamics</a>
 &middot;
<a href="maxima_95.html#Category_003a-Plotting">Plotting</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-evolution"></a>
</p><dl>
<dt><u>Function:</u> <b>evolution</b><i> (<var>F</var>, <var>y0</var>, <var>n</var>, ..., options, ...);</i>
<a name="IDX2018"></a>
</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><pre class="example">        y(n+1) = F(y(n))
</pre>
<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.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-dynamics">Package dynamics</a>
 &middot;
<a href="maxima_95.html#Category_003a-Plotting">Plotting</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-evolution2d"></a>
</p><dl>
<dt><u>Function:</u> <b>evolution2d</b><i> (<code>[</code><var>F</var>, <var>G</var><code>]</code>, <code>[</code><var>u</var>, <var>v</var><code>]</code>, <code>[</code><var>u0</var>, <var>y0</var><code>]</code>, <var>n</var>, ..., options, ...);</i>
<a name="IDX2019"></a>
</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><pre class="example">        u(n+1) = F(u(n), v(n))    v(n+1) = G(u(n), v(n))
</pre>
<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 explicitely in a list. 
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-dynamics">Package dynamics</a>
 &middot;
<a href="maxima_95.html#Category_003a-Plotting">Plotting</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-ifs"></a>
</p><dl>
<dt><u>Function:</u> <b>ifs</b><i> (<code>[</code><var>r1</var>, ..., <var>rm</var><code>]</code>, <code>[</code><var>A1</var>, ..., <var>Am</var><code>]</code>, <code>[[</code><var>x1</var>, <var>y1</var><code>]</code>, ..., <code>[</code><var>xm</var>, <var>ym</var><code>]]</code>, <code>[</code><var>x0</var>, <var>y0</var><code>]</code>, <var>n</var>, ..., options, ...);</i>
<a name="IDX2020"></a>
</dt>
<dd><p>Implements the Iterated Function System method. This method is similar
to the method described in the function <code>chaosgame</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.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-dynamics">Package dynamics</a>
 &middot;
<a href="maxima_95.html#Category_003a-Plotting">Plotting</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-julia"></a>
</p><dl>
<dt><u>Function:</u> <b>julia</b><i> (<var>x</var>, <var>y</var>, ...<var>options</var>...)</i>
<a name="IDX2021"></a>
</dt>
<dd><p>Creates a graphics file with the representation of the Julia set for the
complex number (<var>x</var> + i <var>y</var>). The parameters <var>x</var> and <var>y</var>
must be real. The file is created in the current directory or in the user's
directory, using the XPM graphics format. The program may take several
seconds to run and after it is finished, a message will be printed with
the name of the file created.
</p>
<p>The points which do not belong to the Julia set are assigned different
colors, according to the number of iterations it takes the sequence
starting at that point to move out of the convergence circle of radius
2. The maximum number of iterations is set with the option <var>levels</var>;
after that number of iterations, if the sequence is still inside the
convergence circle, the point will be painted with the color defined by
the option <var>color</var>.
</p>
<p>All the colors used for the points that do not belong to the Julia set
will have the same <var>saturation</var> and <var>value</var>, but with different
hue angles distributed uniformly between <var>hue</var> and (<var>hue</var> +
<var>huerange</var>).
</p>
<p><var>options</var> is an optional sequence of options. The list of accepted
options is given in a section below.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-dynamics">Package dynamics</a>
 &middot;
<a href="maxima_95.html#Category_003a-Plotting">Plotting</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-mandelbrot"></a>
</p><dl>
<dt><u>Function:</u> <b>mandelbrot</b><i> (<var>options</var>)</i>
<a name="IDX2022"></a>
</dt>
<dd><p>Creates a graphics file with the representation of the Mandelbrot
set. The file is created in the current directory or in the user's
directory, using the XPM graphics format. The program may take several
seconds to run and after it is finished, a message will be printed with
the name of the file created.
</p>
<p>The points which do not belong to the Mandelbrot set are
assigned different colors, according to the number of iterations it
takes the sequence generated with that point to move out of the
convergence circle o radius 2. The maximum number of iterations is set with
the option <var>levels</var>; after that number of iterations, if the
sequence is still inside the convergence circle, the point will be
painted with the color defined by the option <var>color</var>.
</p>
<p>All the colors used for the points that do not belong to the Mandelbrot
set will have the same <var>saturation</var> and <var>value</var>, but with
different hue angles distributed uniformly between <var>hue</var> and
(<var>hue</var> + <var>huerange</var>).
</p>
<p><var>options</var> is an optional sequence of options. The list of accepted
options is given in a section below.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-dynamics">Package dynamics</a>
 &middot;
<a href="maxima_95.html#Category_003a-Plotting">Plotting</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-orbits"></a>
</p><dl>
<dt><u>Function:</u> <b>orbits</b><i> (<var>F</var>, <var>y0</var>, <var>n1</var>, <var>n2</var>, [<var>x</var>, <var>x0</var>, <var>xf</var>, <var>xstep</var>], ...options...);</i>
<a name="IDX2023"></a>
</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 a 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.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-dynamics">Package dynamics</a>
 &middot;
<a href="maxima_95.html#Category_003a-Plotting">Plotting</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-rk"></a>
</p><dl>
<dt><u>Function:</u> <b>rk</b><i> (<var>ODE</var>, <var>var</var>, <var>initial</var>, <var>domain</var>)</i>
<a name="IDX2024"></a>
</dt>
<dt><u>Function:</u> <b>rk</b><i> ([<var>ODE1</var>,...,<var>ODEm</var>], [<var>v1</var>,...,<var>vm</var>], [<var>init1</var>,...,<var>initm</var>], <var>domain</var>)</i>
<a name="IDX2025"></a>
</dt>
<dd><p>The first form solves numerically one first-order ordinary differential
equation, and the second form solves a system of m of those equations,
using the 4th order Runge-Kutta method. <var>var</var> represents the dependent
variable. <var>ODE</var> must be an expression that depends only on the independent
and dependent variables and defines the derivative of the dependent
variable with respect to the independent variable.
</p>
<p>The independent variable is specified with <code>domain</code>, which must be a
list of four elements as, for instance:
</p><pre class="example">[t, 0, 10, 0.1]
</pre><p>the first element of the list identifies the independent variable, the
second and third elements are the initial and final values for that
variable, and the last element sets the increments that should be used
within that interval.
</p>
<p>If <var>m</var> equations are going to be solved, there should be <var>m</var>
dependent variables <var>v1</var>, <var>v2</var>, ..., <var>vm</var>. The initial values
for those variables will be <var>init1</var>, <var>init2</var>, ..., <var>initm</var>.
There will still be just one independent variable defined by <code>domain</code>,
as in the previous case. <var>ODE1</var>, ..., <var>ODEm</var> are the expressions
that define the derivatives of each dependent variable in
terms of the independent variable. The only variables that may appear in
those expressions are the independent variable and any of the dependent
variables. It is important to give the derivatives <var>ODE1</var>, ...,
<var>ODEm</var> in the list in exactly the same order used for the dependent
variables; for instance, the third element in the list will be interpreted
as the derivative of the third dependent variable.
</p>
<p>The program will try to integrate the equations from the initial value
of the independent variable until its last value, using constant
increments. If at some step one of the dependent variables takes an
absolute value too large, the integration will be interrupted at that
point. The result will be a list with as many elements as the number of
iterations made. Each element in the results list is itself another list
with <var>m</var>+1 elements: the value of the independent variable, followed
by the values of the dependent variables corresponding to that point.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-dynamics">Package dynamics</a>
 &middot;
<a href="maxima_95.html#Category_003a-Differential-equations">Differential equations</a>
 &middot;
<a href="maxima_95.html#Category_003a-Numerical-methods">Numerical methods</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-staircase"></a>
</p><dl>
<dt><u>Function:</u> <b>staircase</b><i> (<var>F</var>, <var>y0</var>, <var>n</var>, ...options...);</i>
<a name="IDX2026"></a>
</dt>
<dd><p>Draws a staircase diagram for the sequence defined by the recurrence
relation
</p><pre class="example">        y(n+1) = F(y(n))
</pre>
<p>The interpretation and allowed values of the input parameters is the
same as for the function <code>evolution</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.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Package-dynamics">Package dynamics</a>
 &middot;
<a href="maxima_95.html#Category_003a-Plotting">Plotting</a>
</p>
</div>


</dd></dl>

<p><b>Options</b>
</p>
<p>Each option is a list of two or more items. The first item is the name
of the option, and the remainder comprises the arguments for the option.
</p>
<p>The options accepted by the functions <code>evolution</code>, <code>evolution2d</code>,
<code>staircase</code>, <code>orbits</code>, <code>ifs</code> and <code>chaosgame</code> are the same as the options for
<code>plot2d</code>. In addition to those options, <code>orbits</code> accepts and
extra option <var>pixels</var> that sets up the maximum number of different
points that will be represented in the vertical direction.
</p>
<p>The following options are accepted by the functions <code>julia</code> and <code>mandelbrot</code>:
</p>
<ul>
<li>
<em>size</em> takes either one or two arguments. If only one argument is
given, the width and height of the graphic file created will be equal to
that value, in pixels. If two arguments are given, they will define the
width and height. The default value is 400 pixels for both the width and
height. If the two values are not equal, the set will appear distorted.

</li><li>
<em>levels</em> defines the maximum number of iterations, which is also
equal to the number of colors used for points not belonging to the
set. The default value is 12; larger values mean much longer processing
times.

</li><li>
<em>huerange</em> defines the range of hue angles used for the hue of
points not belonging to the set. The default value is 360,
which means that the colors will expand all the range of hues. Values
bigger than 360, will mean repeated ranges of the hue, and negative
values can be used to make the hue angle decrease as the number of
iterations increases.

</li><li>
<em>hue</em> sets the hue, in degrees, of the first color used for the
points which do not belong to the set. Its default value is 300 degrees,
which corresponds to magenta; the values for other standard colors are 0
for red, 45 for orange, 60 for yellow, 120 for green, 180 for cyan and
240 for blue. See also option <var>huerange</var>.

</li><li>
<em>saturation</em> sets the value of the saturation used for points not
belonging to the set. It must be between 0 and 1. The default is 0.46.

</li><li>
<em>value</em> sets the value of the colors used for points not belonging
to the set. It must be between 0 and 1; the higher the value, the
brighter the colors. The default is 0.96

</li><li>
<em>color</em> must be followed by three parameters that define the hue,
saturation and value, for the color used to represent the points of the
set. The default value is 0 for the three parameters, which corresponds
to black. For an explanation of the range of allowed values, see options
<var>hue</var>, <var>saturation</var> and <var>value</var>.

</li><li>
<em>center</em> must be followed by two real parameters, which give the
coordinates, on the complex plane, of the point in the center of the
region shown. The default value is 0 for both coordinates (the origin).

</li><li>
<em>radius</em> sets the radius of the biggest circle inside the square
region that will be displayed. The default value is 2.

</li><li>
<em>filename</em> gives the name of the file where the resulting graph will
be saved. The extension .xpm will be added to that name. If the file
already exists, it will be replaced by the file generated by the
function. The default values are julia for the Julia set, and mandelbrot
for the Mandelbrot set.

</li></ul>

<p><b>Examples</b>
</p>
<p>Graphical representation and staircase diagram for the sequence:
2, cos(2), cos(cos(2)),...
</p>
<pre class="example">(%i1) load(&quot;dynamics&quot;)$

(%i2) evolution(cos(y), 2, 11);

(%i3) staircase(cos(y), 1, 11, [y, 0, 1.2]);
</pre>
<p><div class="image"><img src="./figures/dynamics1.gif" alt="figures/dynamics1"></div>
<div class="image"><img src="./figures/dynamics2.gif" alt="figures/dynamics2"></div>
</p>
<p>If your system is slow, you'll have to reduce the number of iterations in
the following examples. And if the dots appear too small in your
monitor, you might want to try a different style, such as
<code>[</code><var>style</var>,<code>[</code><var>points</var>,0.8<code>]]</code>.
</p>
<p>Orbits diagram for the quadratic map, with a parameter <var>a</var>.
</p><pre class="example">        x(n+1) = a + x(n)^2
</pre>
<pre class="example">(%i4) orbits(x^2+a, 0, 50, 200, [a, -2, 0.25], [style, dots]);
</pre>
<p><div class="image"><img src="./figures/dynamics3.gif" alt="figures/dynamics3"></div>
</p>
<p>To enlarge the region around the lower bifurcation near x <code>=</code> -1.25 use:
</p><pre class="example">(%i5) orbits(x^2+a, 0, 100, 400, [a,-1,-1.53], [x,-1.6,-0.8],
             [nticks, 400], [style,dots]);
</pre>
<p><div class="image"><img src="./figures/dynamics4.gif" alt="figures/dynamics4"></div>
</p>
<p>Evolution of a two-dimensional system that leads to a fractal:
</p>
<pre class="example">(%i6) f: 0.6*x*(1+2*x)+0.8*y*(x-1)-y^2-0.9$

(%i7) g: 0.1*x*(1-6*x+4*y)+0.1*y*(1+9*y)-0.4$

(%i8) evolution2d([f,g], [x,y], [-0.5,0], 50000, [style,dots]);
</pre>
<p><div class="image"><img src="./figures/dynamics5.gif" alt="figures/dynamics5"></div>
</p>
<p>And an enlargement of a small region in that fractal:
</p>
<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>
<p><div class="image"><img src="./figures/dynamics6.gif" alt="figures/dynamics6"></div>
</p>
<p>A plot of Sierpinsky's triangle, obtained with the chaos game:
</p>
<pre class="example">(%i9) chaosgame([[0, 0], [1, 0], [0.5, sqrt(3)/2]], [0.1, 0.1], 1/2,
                 30000, [style, dots]);
</pre>
<p><div class="image"><img src="./figures/dynamics7.gif" alt="figures/dynamics7"></div>
</p>
<p>Barnsley's fern, obtained with an Iterated Function System:
</p>
<pre class="example">(%i10) a1: matrix([0.85,0.04],[-0.04,0.85])$

(%i11) a2: matrix([0.2,-0.26],[0.23,0.22])$

(%i12) a3: matrix([-0.15,0.28],[0.26,0.24])$

(%i13) a4: matrix([0,0],[0,0.16])$

(%i14) p1: [0,1.6]$

(%i15) p2: [0,1.6]$

(%i16) p3: [0,0.44]$

(%i17) p4: [0,0]$

(%i18) w: [85,92,99,100]$

(%i19) ifs(w, [a1,a2,a3,a4], [p1,p2,p3,p4], [5,0], 50000, [style,dots]);
</pre>
<p><div class="image"><img src="./figures/dynamics8.gif" alt="figures/dynamics8"></div>
</p>
<p>To create a file named <em>dynamics9.xpm</em> with a graphical
representation of the Mandelbrot set, with 12 colors, use:
</p>
<pre class="example">mandelbrot([filename,&quot;dynamics9&quot;])$
</pre>
<p><div class="image"><img src="./figures/dynamics9.gif" alt="figures/dynamics9"></div>
</p>
<p>and the Julia set for the number (-0.55 + i 0.6) can be obtained with:
</p><pre class="example">julia(-0.55, 0.6, [levels, 36], [center, 0, 0.6], [radius, 0.3],
      [hue, 240], [huerange, -180], [filename, &quot;dynamics10&quot;])$
</pre>
<p>the graph will be saved in the file <em>dynamics10.xpm</em> and will show
the region from -0.3 to 0.3 in the x direction, and from 0.3 to 0.9 in
the y direction. 36 colors will be used, starting with blue and ending
with yellow.
</p>
<p><div class="image"><img src="./figures/dynamics10.gif" alt="figures/dynamics10"></div>
</p>
<p>To solve numerically the differential equation
</p>
<pre class="example">          dx/dt = t - x^2
</pre>
<p>With initial value x(t=0) = 1, in the interval of t from 0 to 8 and with
increments of 0.1 for t, use:
</p>
<pre class="example">(%i20) results: rk(t-x^2,x,1,[t,0,8,0.1])$
</pre>
<p>the results will be saved in the list <code>results</code>.
</p>
<p>To solve numerically the system:
</p>
<pre class="example">        dx/dt = 4-x^2-4*y^2     dy/dt = y^2-x^2+1
</pre>
<p>for t between 0 and 4, and with values of -1.25 and 0.75 for x and y at t=0:
</p>
<pre class="example">(%i21) sol: rk([4-x^2-4*y^2,y^2-x^2+1],[x,y],[-1.25,0.75],[t,0,4,0.02])$
</pre>
<p><a name="Item_003a-ezunits"></a>
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