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Next: Functions and Variables for drawdf, Previous: Package drawdf, Up: Package drawdf [Contents][Index]
The function drawdf draws the direction field of a first-order
Ordinary Differential Equation (ODE) or a system of two autonomous
first-order ODE’s.
Since this is an additional package, in order to use it you must first
load it with load("drawdf"). Drawdf is built upon the draw
package, which requires Gnuplot 4.2.
To plot the direction field of a single ODE, the ODE must be written in the form:
dy
-- = F(x,y)
dx
and the function F should be given as the argument for
drawdf. If the independent and dependent variables are not x,
and y, as in the equation above, then those two variables should
be named explicitly in a list given as an argument to the drawdf command
(see the examples).
To plot the direction field of a set of two autonomous ODE’s, they must be written in the form
dx dy
-- = G(x,y) -- = F(x,y)
dt dt
and the argument for drawdf should be a list with the two
functions G and F, in that order; namely, the first
expression in the list will be taken to be the time derivative of the
variable represented on the horizontal axis, and the second expression
will be the time derivative of the variable represented on the vertical
axis. Those two variables do not have to be x and y, but if
they are not, then the second argument given to drawdf must be another
list naming the two variables, first the one on the horizontal axis and
then the one on the vertical axis.
If only one ODE is given, drawdf will implicitly admit
x=t, and G(x,y)=1, transforming the non-autonomous
equation into a system of two autonomous equations.
Previous: Introduction to drawdf, Up: Package drawdf [Contents][Index]
Function drawdf draws a 2D direction field with optional
solution curves and other graphics using the draw package.
The first argument specifies the derivative(s), and must be either an expression or a list of two expressions. dydx, dxdt and dydt are expressions that depend on x and y. dvdu, dudt and dvdt are expressions that depend on u and v.
If the independent and dependent variables are not x and
y, then their names must be specified immediately following the
derivative(s), either as a list of two names
[u,v], or as two lists of the form
[u,umin,umax] and
[v,vmin,vmax].
The remaining arguments are graphic options, graphic objects,
or lists containing graphic options and objects, nested to arbitrary
depth. The set of graphic options and objects supported by
drawdf is a superset of those supported by draw2d and
gr2d from the draw package.
The arguments are interpreted sequentially: graphic options affect all following graphic objects. Furthermore, graphic objects are drawn on the canvas in order specified, and may obscure graphics drawn earlier. Some graphic options affect the global appearance of the scene.
The additional graphic objects supported by drawdf include:
solns_at, points_at, saddles_at, soln_at,
point_at, and saddle_at.
The additional graphic options supported by drawdf include:
field_degree, soln_arrows, field_arrows,
field_grid, field_color, show_field,
tstep, nsteps, duration, direction,
field_tstep, field_nsteps, and field_duration.
Commonly used graphic objects inherited from the draw
package include: explicit, implicit, parametric,
polygon, points, vector, label, and all
others supported by draw2d and gr2d.
Commonly used graphic options inherited from the draw
package include:
points_joined, color,
point_type, point_size, line_width,
line_type, key, title, xlabel,
ylabel, user_preamble, terminal,
dimensions, file_name, and all
others supported by draw2d and gr2d.
See also draw2d, rk, desolve and
ode2.
Users of wxMaxima or Imaxima may optionally use wxdrawdf, which
is identical to drawdf except that the graphics are drawn
within the notebook using wxdraw.
To make use of this function, write first load("drawdf").
Examples:
(%i1) load("drawdf")$
(%i2) drawdf(exp(-x)+y)$ /* default vars: x,y */
(%i3) drawdf(exp(-t)+y, [t,y])$ /* default range: [-10,10] */
(%i4) drawdf([y,-9*sin(x)-y/5], [x,1,5], [y,-2,2])$
For backward compatibility, drawdf accepts
most of the parameters supported by plotdf.
(%i5) drawdf(2*cos(t)-1+y, [t,y], [t,-5,10], [y,-4,9],
[trajectory_at,0,0])$
soln_at and solns_at draw solution curves
passing through the specified points, using a slightly
enhanced 4th-order Runge Kutta numerical integrator.
(%i6) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
solns_at([0,0.1],[0,-0.1]),
color=blue, soln_at(0,0))$
field_degree=2 causes the field to be composed of quadratic
splines, based on the first and second derivatives at each grid point.
field_grid=[COLS,ROWS] specifies the number
of columns and rows in the grid.
(%i7) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
field_degree=2, field_grid=[20,15],
solns_at([0,0.1],[0,-0.1]),
color=blue, soln_at(0,0))$
soln_arrows=true adds arrows to the solution curves, and (by
default) removes them from the direction field. It also changes the
default colors to emphasize the solution curves.
(%i8) drawdf(2*cos(t)-1+y, [t,-5,10], [y,-4,9],
soln_arrows=true,
solns_at([0,0.1],[0,-0.1],[0,0]))$
duration=40 specifies the time duration of numerical
integration (default 10). Integration will also stop automatically if
the solution moves too far away from the plotted region, or if the
derivative becomes complex or infinite. Here we also specify
field_degree=2 to plot quadratic splines. The equations below
model a predator-prey system.
(%i9) drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1],
field_degree=2, duration=40,
soln_arrows=true, point_at(1/2,1/2),
solns_at([0.1,0.2], [0.2,0.1], [1,0.8], [0.8,1],
[0.1,0.1], [0.6,0.05], [0.05,0.4],
[1,0.01], [0.01,0.75]))$
field_degree='solns causes the field to be composed
of many small solution curves computed by 4th-order
Runge Kutta, with better results in this case.
(%i10) drawdf([x*(1-x-y), y*(3/4-y-x/2)], [x,0,1.1], [y,0,1],
field_degree='solns, duration=40,
soln_arrows=true, point_at(1/2,1/2),
solns_at([0.1,0.2], [0.2,0.1], [1,0.8],
[0.8,1], [0.1,0.1], [0.6,0.05],
[0.05,0.4], [1,0.01], [0.01,0.75]))$
saddles_at attempts to automatically linearize the equation at
each saddle, and to plot a numerical solution corresponding to each
eigenvector, including the separatrices. tstep=0.05 specifies
the maximum time step for the numerical integrator (the default is
0.1). Note that smaller time steps will sometimes be used in order to
keep the x and y steps small. The equations below model a damped
pendulum.
(%i11) drawdf([y,-9*sin(x)-y/5], tstep=0.05,
soln_arrows=true, point_size=0.5,
points_at([0,0], [2*%pi,0], [-2*%pi,0]),
field_degree='solns,
saddles_at([%pi,0], [-%pi,0]))$
show_field=false suppresses the field entirely.
(%i12) drawdf([y,-9*sin(x)-y/5], tstep=0.05,
show_field=false, soln_arrows=true,
point_size=0.5,
points_at([0,0], [2*%pi,0], [-2*%pi,0]),
saddles_at([3*%pi,0], [-3*%pi,0],
[%pi,0], [-%pi,0]))$
drawdf passes all unrecognized parameters to draw2d or
gr2d, allowing you to combine the full power of the draw
package with drawdf.
(%i13) drawdf(x^2+y^2, [x,-2,2], [y,-2,2], field_color=gray,
key="soln 1", color=black, soln_at(0,0),
key="soln 2", color=red, soln_at(0,1),
key="isocline", color=green, line_width=2,
nticks=100, parametric(cos(t),sin(t),t,0,2*%pi))$
drawdf accepts nested lists of graphic options and objects,
allowing convenient use of makelist and other function calls to
generate graphics.
(%i14) colors : ['red,'blue,'purple,'orange,'green]$
(%i15) drawdf([x-x*y/2, (x*y - 3*y)/4],
[x,2.5,3.5], [y,1.5,2.5],
field_color = gray,
makelist([ key = concat("soln",k),
color = colors[k],
soln_at(3, 2 + k/20) ],
k,1,5))$
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