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* Introduction to plotdf::
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* Definitions for plotdf::
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@node Introduction to plotdf, Definitions for plotdf, plotdf, plotdf
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@section Introduction to plotdf
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The function @code{plotdf} creates a plot of the direction field of a
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first-order Ordinary Differential Equation (ODE) or a system of two
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autonomous first-order ODE's.
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To plot the direction field of a single ODE, the ODE must be written in
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$${{dy}\over{dx}} = F(x,y)$$
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and the function @var{F} should be given as the argument for
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@code{plotdf}. The independent variable is always identified as @var{x},
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and the dependent variable as @var{y}. Those two variables should not
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have any values assigned to them.
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To plot the direction field of a set of two autonomous ODE's, they must
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be written in the form
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-- = G(x,y) -- = F(x,y)
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$${{dx}\over{dt}} = G(x,y) \qquad {{dy}\over{dt}} = F(x,y)$$
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and the argument for @code{plotdf} should be a list with the two
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functions @var{F} and @var{G}, in any order.
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If only one ODE is given, @code{plotdf} will implicitly admit
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@code{x=t}, and @code{G(x,y)=1}, transforming the non-autonomous
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equation into a system of two autonomous equations.
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@node Definitions for plotdf, , Introduction to plotdf, plotdf
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@section Definitions for plotdf
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@deffn {Function} plotdf (@var{dydx},...options...)
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@deffnx {Function} plotdf (@code{[}@var{dxdt},@var{dydt}@code{]},...options...)
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Displays a direction field in two dimensions @var{x} and @var{y}.
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@var{dydx}, @var{dxdt} and @var{dydt} are expressions that depend
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on @var{x} and @var{y}. In addition to those two variables, the
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expressions can also depend on a set of parameters, with numerical
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values given with the @code{parameters} option (the option syntax is
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given below), or with an range of allowed values specified by a
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Several other options can be given within the command, or entered into
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the menu that will appear when the upper-left corner of the plot window
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is clicked. Integral curves can be obtained by clicking on the plot, or
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with the option @code{trajectory_at}. The direction of the integration
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can be controlled with the @code{direction} option, which can have values
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of "forward", "backward" or "both". The number of integration steps is
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given by @code{nsteps} and the time interval between them is set up with
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the @code{tstep} option. The Adams Moulton method is used for the
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integration; it is also possible to switch to an adaptive Runge-Kutta
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The menu in the plot window has the following options: "Zoom", will
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change the behavior of the mouse so that it will allow you to zoom in
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on a region of the plot by clicking with the left button. Each click
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near a point magnifies the plot, keeping the center at the point where
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you clicked. Holding the SHIFT key while clicking, zooms out to the
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previous magnification. To resume computing trajectories when you
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click on a point, select "Integrate" from the menu.
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The option "Config" in the menu can be used to change the ODE(s) in use
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and various other settings. After configuration changes are made, the
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menu option "Replot" should be selected, to activate the new settings.
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If a pair of coordinates are entered in the field "Trajectory at" in the
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"Config" dialog menu, and the "enter" key is pressed, a new integral
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curve will be shown, in addition to the ones already shown. When
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"Replot" is selected, only the last integral curve entered will be
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Holding the right mouse button down while the cursor is moved, can be
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used to drag the plot sideways or up and down. Additional
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parameters such as the number of steps, the initial value of @var{t}
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and the x and y centers and radii, may be set in the Config menu.
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A copy of the plot can be printed to a Postscript printer, or saved as a
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postscript file, using the menu option "Save". To switch between
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printing and saving to a Postscript file, "Print Options" should be
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selected in the dialog window of "Config". After the settings in the
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"Save" dialog window are entered, "Save'' must be selected in the first
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menu, to create the file or print the plot.
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The @code{plotdf} command may include several commands, each command is
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a list of two or more items. The first item is the name of the option,
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and the remainder comprises the value or values assigned to the option.
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The options which are recognized by @code{plotdf} are the following:
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Option: @code{tstep} defines the length of the increments on the
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independent variable @var{t}, used to compute an integral curve. If only
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one expression @var{dydx} is given to @code{plotdf}, the @var{x}
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variable will be directly proportional to @var{t}: @code{x - xinitial =
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The default value is 0.1.
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Option: @code{nsteps} defines the number of steps of length @code{tstep}
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that will be used for the independent variable, to compute an integral
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The default value is 100.
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Option: @code{direction} defines the direction of the independent
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variable that will be followed to compute an integral curve. Possible
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values are @code{forward}, to make the independent variable increase
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@code{nsteps} times, with increments @code{tstep}, @code{backward}, to
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make the independent variable decrease, or @code{both} that will lead to
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an integral curve that extends @code{nsteps} forward, and @code{nsteps}
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backward. The keywords @code{right} and @code{left} can be used as
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synonyms for @code{forward} and @code{backward}.
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The defaul value is @code{both}.
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Option: @code{tinitial} defines the initial value of variable @var{t} used
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to compute integral curves. Since the differential equations are
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autonomous, that setting will only appear in the plot of the curves as
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functions of @var{t}.
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The default value is 0.
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Option: @code{versus_t} is used to create a second plot window, with a
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plot of an integral curve, as two functions @var{x}, @var{y}, of the
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independent variable @var{t}. If @code{versus_t} is given any value
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different from 0, the second plot window will be displayed. The second
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plot window includes another menu, similar to the menu of the main plot
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The default value is 0.
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Option: @code{trajectory_at} defines the coordinates @var{xinitial} and
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@var{yinitial} for the starting point of an integral curve.
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[trajectory_at,0.1,3.2]
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The option is empty by default.
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Option @code{parameters} defines a list of parameters, and their
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numerical values, used in the definition of the differential
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equations. The name and values of the parameters must be given in a
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string with a comma-separated sequence of pairs @code{name=value}.
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[parameters,"k=1.1,m=2.5"]
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Option: @code{sliders} defines a list o parameters that will be changed
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interactively using slider buttons, and the range of variation of those
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parameters. The names and ranges of the parameters must be given in a
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string with a comma-separated sequence of elements @code{name=min:max}
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[sliders,"k=0:4,m=1:3"]
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Option: @code{xfun} defines a string with semi-colon-separated sequence
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of functions of @var{x} to be displayed, on top of the direction field.
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Those functions will be parsed by Tcl and not by Maxima.
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[xfun,"x^2;sin(x);exp(x)"]
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Option: @code{xradius} is half of the length of the range of values that
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will be shown in the x direction.
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the default value is 10.
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Option: @code{yradius} is half of the length of the range of values that
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will be shown in the y direction.
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the default value is 10.
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Option: @code{xcenter} is the x coordinate of the point at the center of
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The default value is 0.
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Option: @code{ycenter} is the y coordinate of the point at the center of
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The default value is 0.
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Option: @code{width} defines the width of the plot window, in pixels.
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The default value is 500.
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Option: @code{height} defines the height of the plot window, in pixels.
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The default value is 500.
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NOTE: Due to a bug in @code{openmath}, all commands that use it, in particular
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@code{plotdf}, must end with a semicolon and not with a dollar sign. The
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dollar sign might work in some of the graphical interfaces to Maxima, but
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to avoid problems we will use a semicolon in all the examples below.
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To show the direction field of the differential equation @math{y' = exp(-x) + y} and the solution that goes through @math{(2, -0.1)}:
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(%i1) load("plotdf")$
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(%i2) plotdf(exp(-x)+y,[trajectory_at,2,-0.1]);
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@image{figures/plotdf1,8cm}
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To obtain the direction field for the equation @math{diff(y,x) = x - y^2} and the solution with initial condition @math{y(-1) = 3}, we can use the command:
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(%i3) plotdf(x-y^2,[xfun,"sqrt(x);-sqrt(x)"],
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[trajectory_at,-1,3], [direction,forward],
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[yradius,5],[xcenter,6]);
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The graph also shows the function @math{y = sqrt(x)}.
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@image{figures/plotdf2,8cm}
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The following example shows the direction field of a harmonic oscillator,
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defined by the two equations @math{dx/dt = y} and @math{dy/dt = -k*x/m},
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and the integral curve through @math{(x,y) = (6,0)}, with a slider that
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will allow you to change the value of @math{m} interactively (@math{k} is
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(%i4) plotdf([y,-k*x/m],[parameters,"m=2,k=2"],
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[sliders,"m=1:5"], [trajectory_at,6,0]);
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@image{figures/plotdf3,8cm}
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To plot the direction field of the Duffing equation, @math{m*x''+c*x'+k*x+b*x^3 = 0}, we introduce the variable @math{y=x'} and use:
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(%i5) plotdf([y,-(k*x + c*y + b*x^3)/m],
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[parameters,"k=-1,m=1.0,c=0,b=1"],
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[sliders,"k=-2:2,m=-1:1"],[tstep,0.1]);
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@image{figures/plotdf4,8cm}
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The direction field for a damped pendulum, including the
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solution for the given initial conditions, with a slider that
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can be used to change the value of the mass @math{m}, and with a plot of
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the two state variables as a function of time:
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(%i6) plotdf([y,-g*sin(x)/l - b*y/m/l],
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[parameters,"g=9.8,l=0.5,m=0.3,b=0.05"],
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[trajectory_at,1.05,-9],[tstep,0.01],
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[xradius,6],[yradius,14],
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[xcenter,-4],[direction,forward],[nsteps,300],
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[sliders,"m=0.1:1"], [versus_t,1]);
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@image{figures/plotdf5,8cm}@image{figures/plotdf6,8cm}
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To use this function write first @code{load("plotdf")}.
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doc/info/plotdf.texi
