@menu
* Introduction to dynamics::
* Functions and Variables for dynamics::
@end menu

@node Introduction to dynamics, Functions and Variables for dynamics, dynamics, dynamics
@section Introduction to dynamics

The additional package @code{dynamics} 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.

To use the functions in this package you must first load it with
@code{load("dynamics")}.

The commands that produce graphics accept the same options accepted by @code{plot2d} and some additional ones.

@opencatbox
@category{Dynamical systems} @category{Share packages} @category{Package dynamics}
@closecatbox

@node Functions and Variables for dynamics,  , Introduction to dynamics, dynamics
@section Functions and Variables for dynamics

@deffn {Function} chaosgame (@code{[[}@var{x1}, @var{y1}@code{]}...@code{[}@var{xm}, @var{ym}@code{]]}, @code{[}@var{x0}, @var{y0}@code{]}, @var{b}, @var{n}, ..., options, ...);

Implements the so-called chaos game: the initial point (@var{x0},
@var{y0}) is plotted and then one of the @var{m} points
@code{[}@var{x1}, @var{y1}@code{]}...@code{[}@var{xm}, @var{ym}@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} times that segment's
length. The procedure is repeated @var{n} times.

@opencatbox
@category{Package dynamics} @category{Plotting}
@closecatbox

@end deffn

@deffn {Function} evolution (@var{F}, @var{y0}, @var{n}, ..., options, ...);

Draws @var{n+1} points in a two-dimensional graph, where the horizontal
coordinates of the points are the integers 0, 1, 2, ..., @var{n}, and
the vertical coordinates are the corresponding values @var{y(n)} of the
sequence defined by the recurrence relation
@ifnottex
@example
        y(n+1) = F(y(n))
@end example
@end ifnottex
@tex
$$y_{n+1} = F(y_n)$$
@end tex

With initial value @var{y(0)} equal to @var{y0}. @var{F} must be an
expression that depends only on one variable (in the example, it
depend on @var{y}, but any other variable can be used),
@var{y0} must be a real number and @var{n} must be a positive integer.

@opencatbox
@category{Package dynamics} @category{Plotting}
@closecatbox

@end deffn

@deffn {Function} evolution2d (@code{[}@var{F}, @var{G}@code{]}, @code{[}@var{u}, @var{v}@code{]}, @code{[}@var{u0}, @var{y0}@code{]}, @var{n}, ..., options, ...);

Shows, in a two-dimensional plot, the first @var{n+1} points in the
sequence of points defined by the two-dimensional discrete dynamical
system with recurrence relations
@ifnottex
@example
        u(n+1) = F(u(n), v(n))    v(n+1) = G(u(n), v(n))
@end example
@end ifnottex
@tex
$$\cases{u_{n+1} = F(u_n, v_n) &\cr v_{n+1} = G(u_n, v_n)}$$
@end tex

With initial values @var{u0} and @var{v0}. @var{F} and @var{G} must be
two expressions that depend only on two variables, @var{u} and
@var{v}, which must be named explicitely in a list. 

@opencatbox
@category{Package dynamics} @category{Plotting}
@closecatbox

@end deffn

@deffn {Function} ifs (@code{[}@var{r1}, ..., @var{rm}@code{]}, @code{[}@var{A1}, ..., @var{Am}@code{]}, @code{[[}@var{x1}, @var{y1}@code{]}, ..., @code{[}@var{xm}, @var{ym}@code{]]}, @code{[}@var{x0}, @var{y0}@code{]}, @var{n}, ..., options, ...);

Implements the Iterated Function System method. This method is similar
to the method described in the function @code{chaosgame}, 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} that corresponds to the point chosen randomly.

The random choice of one of the @var{m} attractive points can be made with
a non-uniform probability distribution defined by the weights
@var{r1},...,@var{rm}. 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{r2} and @var{r3} could be 2, 7 and 10.

@opencatbox
@category{Package dynamics} @category{Plotting}
@closecatbox

@end deffn

@deffn {Function} orbits (@var{F}, @var{y0}, @var{n1}, @var{n2}, [@var{x}, @var{x0}, @var{xf}, @var{xstep}], ...options...);

Draws the orbits diagram for a family of one-dimensional
discrete dynamical systems, with one parameter @var{x}; that kind of
diagram is used to study the bifurcations of a one-dimensional discrete
system.

The function @var{F(y)} defines a sequence with a starting value of
@var{y0}, as in the case of the function @code{evolution}, but in this
case that function will also depend on a parameter @var{x} that will
take values in the interval from @var{x0} to @var{xf} with increments of
@var{xstep}. Each value used for the parameter @var{x} is shown on the
horizontal axis. The vertical axis will show the @var{n2} values
of the sequence @var{y(n1+1)},..., @var{y(n1+n2+1)} obtained after letting
the sequence evolve @var{n1} iterations.

@opencatbox
@category{Package dynamics} @category{Plotting}
@closecatbox

@end deffn


@deffn {Function} staircase (@var{F}, @var{y0}, @var{n}, ...options...);

Draws a staircase diagram for the sequence defined by the recurrence
relation
@ifnottex
@example
        y(n+1) = F(y(n))
@end example
@end ifnottex
@tex
$$y_{n+1} = F(y_n)$$
@end tex

The interpretation and allowed values of the input parameters is the
same as for the function @code{evolution}. A staircase diagram consists
of a plot of the function @var{F(y)}, together with the line
@var{G(y)} @code{=} @var{y}. A vertical segment is drawn from the
point (@var{y0}, @var{y0}) on that line until the point where it
intersects the function @var{F}. From that point a horizontal segment is
drawn until it reaches the point (@var{y1}, @var{y1}) on the line, and
the procedure is repeated @var{n} times until the point (@var{yn}, @var{yn})
is reached.

@opencatbox
@category{Package dynamics} @category{Plotting}
@closecatbox

@end deffn

@b{Options}

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.

The options accepted by the functions @code{evolution}, @code{evolution2d},
@code{staircase}, @code{orbits}, @code{ifs} and @code{chaosgame} are the same as the options for
@code{plot2d}. In addition to those options, @code{orbits} accepts and
extra option @var{pixels} that sets up the maximum number of different
points that will be represented in the vertical direction.

@b{Examples}

Graphical representation and staircase diagram for the sequence:
2, cos(2), cos(cos(2)),...

@example
(%i1) load("dynamics")$

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

(%i3) staircase(cos(y), 1, 11, [y, 0, 1.2]);
@end example

@ifnotinfo
@image{figures/dynamics1,8cm}
@image{figures/dynamics2,8cm}
@end ifnotinfo

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{[}@var{style},@code{[}@var{points},0.8@code{]]}.

Orbits diagram for the quadratic map, with a parameter @var{a}.
@ifnottex
@example
        x(n+1) = a + x(n)^2
@end example
@end ifnottex
@tex
$$x_{n+1} = a + x_n^2$$
@end tex

@example
(%i4) orbits(x^2+a, 0, 50, 200, [a, -2, 0.25], [style, dots]);
@end example

@ifnotinfo
@image{figures/dynamics3,8cm}
@end ifnotinfo

To enlarge the region around the lower bifurcation near x @code{=} -1.25 use:
@example
(%i5) orbits(x^2+a, 0, 100, 400, [a,-1,-1.53], [x,-1.6,-0.8],
             [nticks, 400], [style,dots]);
@end example

@ifnotinfo
@image{figures/dynamics4,8cm}
@end ifnotinfo

Evolution of a two-dimensional system that leads to a fractal:

@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]);
@end example

@ifnotinfo
@image{figures/dynamics5,8cm}
@end ifnotinfo

And an enlargement of a small region in that fractal:

@example
(%i9) evolution2d([f,g], [x,y], [-0.5,0], 300000, [x,-0.8,-0.6],
                  [y,-0.4,-0.2], [style, dots]);
@end example

@ifnotinfo
@image{figures/dynamics6,8cm}
@end ifnotinfo

A plot of Sierpinsky's triangle, obtained with the chaos game:

@example
(%i9) chaosgame([[0, 0], [1, 0], [0.5, sqrt(3)/2]], [0.1, 0.1], 1/2,
                 30000, [style, dots]);
@end example

@ifnotinfo
@image{figures/dynamics7,8cm}
@end ifnotinfo

Barnsley's fern, obtained with an Iterated Function System:

@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]);
@end example

@ifnotinfo
@image{figures/dynamics8,8cm}
@end ifnotinfo

See also the documentation for packages @mrefdot{mandelbrot} and @mrefdot{julia} in the Plotting section.
