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[Maxima-commits] CVS: maxima/doc/info dynamics.texi,1.2,1.2.2.1
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From: Jaime E. V. <vi...@us...> - 2006-08-01 11:34:25
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Update of /cvsroot/maxima/maxima/doc/info In directory sc8-pr-cvs7.sourceforge.net:/tmp/cvs-serv17450 Modified Files: Tag: RELEASE-5_10_0-BRANCH dynamics.texi Log Message: Fixes a few errors and typos in the text. Index: dynamics.texi =================================================================== RCS file: /cvsroot/maxima/maxima/doc/info/dynamics.texi,v retrieving revision 1.2 retrieving revision 1.2.2.1 diff -u -d -r1.2 -r1.2.2.1 --- dynamics.texi 31 Jul 2006 23:14:21 -0000 1.2 +++ dynamics.texi 1 Aug 2006 11:34:22 -0000 1.2.2.1 @@ -6,7 +6,7 @@ @node Introduction to dynamics, Definitions for dynamics, dynamics, dynamics @section Introduction to dynamics -The additional package @code{dynamicalsystems} includes several +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. @@ -17,7 +17,7 @@ @node Definitions for dynamics, , Introduction to dynamics, dynamics @section Definitions for dynamics -@deffn {Function} chaosgame (@code{[}@code{[}@var{x1}, @var{y1}@code{]}...@code{[}@var{xm}, @var{ym}@code{]}@code{]}, @code{[}@var{x0}, @var{y0}@code{]}, @var{b}, @var{n}...options...); +@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 @@ -29,7 +29,7 @@ @end deffn -@deffn {Function} evolution (@var{F}, @var{y0}, @var{n}...options...); +@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 @@ -45,15 +45,15 @@ @end tex With initial value @var{y(0)} equal to @var{y0}. @var{F} must be an -expression that depends only on the variable @var{y} (and on @var{n}), +expression that depends only on the variable @var{y} (and not on @var{n}), @var{y0} must be a real number and @var{n} must be a positive integer. @end deffn -@deffn {Function} evolution2d (@code{[}@var{F}, @var{G}@code{]}, @code{[}@var{x0}, @var{y0}@code{]}, @var{n}...options...); +@deffn {Function} evolution2d (@code{[}@var{F}, @var{G}@code{]}, @code{[}@var{x0}, @var{y0}@code{]}, @var{n}, ...options...); Shows, in a two-dimensional plot, the first @var{n+1} points in the -sequence o points defined by the two-dimensional discrete dynamical +sequence of points defined by the two-dimensional discrete dynamical system with recurrence relations @ifnottex @example @@ -69,22 +69,22 @@ @end deffn -@deffn {Function} ifs (@code{[}@var{r1},...,@var{rm}@code{]},@code{[}@var{A1},...,@var{Am}@code{]}, @var{y1}@code{]}...@code{[}@var{xm}, @var{ym}@code{]}@code{]}, @code{[}@var{x0},@var{y0}@code{]}, @var{n}...options...); +@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 Functions System method. This method is similar -to the method described in the function @code{chaosgame}. But instead of +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 @var{m} attractive points are not chosen randomly with a uniform -probability but with probabilities defined bay 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, you can @var{r1}, @var{r2} and @var{r3} could be 2, 7 and 10. +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. @end deffn -@deffn {Function} orbits (@var{F}, @var{y0}, @var{n1}, @var{n2}, [@var{x}, @var{x0}, @var{xf}, @var{xstep}]...options...); +@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 @@ -96,9 +96,9 @@ 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; on the vertical axis will be shown the @var{n2} values -of the sequence @var{y(n1+1)},..., @var{y(n1+n2+1)} obtained after the -sequence is left to evolve during @var{n1} iterations. +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. @end deffn @@ -112,8 +112,8 @@ and dependent variables and represents the derivative of the dependent variable with respect to the independent variable. -The independent variable is specified in domain, which must be a list -with four elements such as: +The independent variable is specified with @code{domain}, which must be a +list of four elements such as: @example [t, 0, 10, 0.1] @end example @@ -122,24 +122,26 @@ variable, and the last element sets the increments that should be used within that interval. -If m equations are going to be solved, there should be m dependent -variables v1, v2, ..., vm. The initial values for those variables will -be init1, init2, ..., initm. There will still be just one independent -variable defined by domain, as in the previous case. ODE1, ..., ODEm +If @var{m} equations are going to be solved, there should be @var{m} +dependent variables @var{v1}, @var{v2}, ..., @var{vm}. The initial values +for those variables will be @var{init1}, @var{init2}, ..., @var{initm}. +There will still be just one independent variable defined by @code{domain}, +as in the previous case. @var{ODE1}, ..., @var{ODEm} are the expressions for the derivatives of each dependent variable in -terms of the independent variable. Those expressions can depend on the -independent variable and all of the dependent variables. +terms of the independent variable. The only variables that may appear in +those expressions are the independent variable and any of the dependent +variables. -The result will be a list of lists with m+1 elements. Those m+1 elements -will be the values of the independent variable, followed by the -dependent variables, at each of the points in the interval of -integration. If at some point one of the variables becomes too large, +The result will be a list of lists with @var{m}+1 elements. Those @var{m}+1 +elements will be the value of the independent variable, followed by the +values of the dependent variables corresponding to that point in the interval +of integration. If at some point one of the variables becomes too large, the list will stop there. Otherwise, the list will extend until the last -value of the independent variable specified by domain. +value of the independent variable specified by @cod{domain}. @end deffn -@deffn {Function} staircase (@var{F}, @var{y0}, @var{n}...options...); +@deffn {Function} staircase (@var{F}, @var{y0}, @var{n}, ...options...); Draws a staircase diagram for the sequence defined by the recurrence relation @@ -155,12 +157,12 @@ 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 horizontal segment is drawn from the +@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 vertical segment is +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 until the point (@var{yn}, @var{yn}) is -reached. +the procedure is repeated @var{n} times until the point (@var{yn}, @var{yn}) +is reached. @end deffn @@ -170,14 +172,15 @@ @itemize @bullet @item -Option: @code{domain} the minimum and maximum values for the plot of the +Option: @code{domain} sets the minimum and maximum values for the plot of the function @var{F} shown by @code{staircase}. @example [domain, -2, 3.5] @end example @item -Option: @code{pointsize} radius of each point plotted, in units of points. +Option: @code{pointsize} defines the radius of each point plotted, in units of +points. @example [pointsize, 1.5] @end example @@ -264,7 +267,7 @@ @image{figures/dynamics3,8cm} @end ifnotinfo -To enlarge the region around the lower bifurcation near x @code{=} 1.25 use: +To enlarge the region around the lower bifurcation near x @code{=} -1.25 use: @example (%i5) orbits(x+y^2, 0, 100, 400, [x,-1,-1.53,-0.001], [pointsize,0.9], [ycenter,-1.2], [yradius,0.4]); @@ -308,7 +311,7 @@ @image{figures/dynamics7,8cm} @end ifnotinfo -Barnsley's fern, obtained with an Iterated Functions System: +Barnsley's fern, obtained with an Iterated Function System: @example (%i10) a1: matrix([0.85,0.04],[-0.04,0.85])$ @@ -319,8 +322,8 @@ (%i15) p2: [0,1.6]$ (%i16) p3: [0,0.44]$ (%i17) p4: [0,0]$ -(%i18) prob: [85,92,99,100]$ -(%i19) ifs(prob,[a1,a2,a3,a4],[p1,p1,p3,p4],[5,0],50000,[pointsize,0.9]); +(%i18) w: [85,92,99,100]$ +(%i19) ifs(w,[a1,a2,a3,a4],[p1,p2,p3,p4],[5,0],50000,[pointsize,0.9]); @end example @ifnotinfo @@ -355,7 +358,7 @@ @end example @end ifnottex @tex -$$\cases{{{dx}\over{dt}} = 4-x^2-4y^2 &\cr {{dy}\over{dt}} = y^2-x^2+1}$$ +$$\cases{{\displaystyle{dx}\over\displaystyle{dt}} = 4-x^2-4y^2 &\cr &\cr {\displaystyle{dy}\over\displaystyle{dt}} = y^2-x^2+1}$$ @end tex for t between 0 and 4, and with values of -1.25 and 0.75 for x and y at t=0 |