From: Jaime E. Villate <villate@us...>  20060928 10:02:54

Update of /cvsroot/maxima/maxima/doc/info In directory sc8prcvs7.sourceforge.net:/tmp/cvsserv2428 Modified Files: dynamics.texi Log Message: A few improvements to the text. Index: dynamics.texi =================================================================== RCS file: /cvsroot/maxima/maxima/doc/info/dynamics.texi,v retrieving revision 1.5 retrieving revision 1.6 diff u d r1.5 r1.6  dynamics.texi 26 Sep 2006 04:59:41 0000 1.5 +++ dynamics.texi 28 Sep 2006 10:02:48 0000 1.6 @@ 22,10 +22,10 @@ Implements the socalled 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{]}...@...{[}@var{xm}, @var{ym}@code{]} will be selected at random. The next point plotted will be in the segment from the previous point to the point chosen randomly, at a fraction @var{b} of the distance from the random point. The procedure is repeated @var{n} times. +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. @end deffn @@ 109,11 +109,11 @@ equation, and the second form solves a system of m of those equations, using the 4th order RungeKutta method. var represents the dependent variable. ODE must be an expression that depends only on the independent and dependent variables and represents the derivative of the dependent +and dependent variables and defines the derivative of the dependent variable with respect to the independent variable. The independent variable is specified with @code{domain}, which must be a list of four elements such as: +list of four elements as, for instance: @example [t, 0, 10, 0.1] @end example @@ 126,18 +126,23 @@ 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 +as in the previous case. @var{ODE1}, ..., @var{ODEm} 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. +variables. It is important to give the derivatives @var{ODE1}, ..., +@var{ODEm} 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. 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 @code{domain}. +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}+1 elements: the value of the independent variable, followed +by the values of the dependent variables corresponding to that point. @end deffn @@ 168,12 +173,13 @@ @b{Options} The options accepted by the functions that plot graphs are: +The options accepted by the functions in this package that plot graphs are: @itemize @bullet @item Option: @code{domain} sets the minimum and maximum values for the plot of the function @var{F} shown by @code{staircase}. +Option: @code{domain} sets the minimum and maximum values for the +independent variable in the plot of the function @var{F} shown by +@code{staircase}. @example [domain, 2, 3.5] @end example @@ 193,14 +199,15 @@ @end example @item Option: @code{xcenter} is the x coordinate of the point at the center of the plot. This option is not used by the function @code{orbits}. +Option: @code{xcenter} is the x coordinate of the point that should +appear at the center of the plot. This option is not used by the +function @code{orbits}. @example [xcenter,3.45] @end example @item Option: @code{xradius} is half of the length of the range of values that +Option: @code{xradius} is half of the length of the interval of values that will be shown in the x direction. This option is not used by the function @code{orbits}. @example @@ 214,8 +221,8 @@ @end example @item Option: @code{ycenter} is the y coordinate of the point at the center of the plot. +Option: @code{ycenter} is the y coordinate of the point that should +appear at the center of the plot. @example [ycenter,4.5] @end example @@ 247,7 +254,8 @@ If your system is slow, you'll have to reduce the number of iterations in the following examples. And the pointsize that gives the best results depends on the monitor and the resolution being used. +depends on the monitor and the resolution being used. You should +experiment using different values. Orbits diagram for the quadratic map @ifnottex @@ 361,7 +369,7 @@ $$\cases{{\displaystyle{dx}\over\displaystyle{dt}} = 4x^24y^2 &\cr &\cr {\displaystyle{dy}\over\displaystyle{dt}} = y^2x^2+1}$$ @end tex for t between 0 and 4, and with values of 1.25 and 0.75 for x and y at t=0 +for t between 0 and 4, and with values of 1.25 and 0.75 for x and y at t=0: @example (%i21) sol: rk([4x^24*y^2,y^2x^2+1],[x,y],[1.25,0.75],[t,0,4,0.02])$ 