1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
|
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html401/loose.dtd">
<html>
<!-- Created on April, 24 2010 by texi2html 1.76 -->
<!--
Written by: Lionel Cons <Lionel.Cons@cern.ch> (original author)
Karl Berry <karl@freefriends.org>
Olaf Bachmann <obachman@mathematik.uni-kl.de>
and many others.
Maintained by: Many creative people <dev@texi2html.cvshome.org>
Send bugs and suggestions to <users@texi2html.cvshome.org>
-->
<head>
<title>Maxima 5.21.1 Manual: 68. plotdf</title>
<meta name="description" content="Maxima 5.21.1 Manual: 68. plotdf">
<meta name="keywords" content="Maxima 5.21.1 Manual: 68. plotdf">
<meta name="resource-type" content="document">
<meta name="distribution" content="global">
<meta name="Generator" content="texi2html 1.76">
<meta http-equiv="Content-Type" content="text/html; charset=us-ascii">
<style type="text/css">
<!--
a.summary-letter {text-decoration: none}
pre.display {font-family: serif}
pre.format {font-family: serif}
pre.menu-comment {font-family: serif}
pre.menu-preformatted {font-family: serif}
pre.smalldisplay {font-family: serif; font-size: smaller}
pre.smallexample {font-size: smaller}
pre.smallformat {font-family: serif; font-size: smaller}
pre.smalllisp {font-size: smaller}
span.sansserif {font-family:sans-serif; font-weight:normal;}
ul.toc {list-style: none}
body
{
color: black;
background: white;
margin-left: 8%;
margin-right: 13%;
}
h1
{
margin-left: +8%;
font-size: 150%;
font-family: sans-serif
}
h2
{
font-size: 125%;
font-family: sans-serif
}
h3
{
font-size: 100%;
font-family: sans-serif
}
h2,h3,h4,h5,h6 { margin-left: +4%; }
div.textbox
{
border: solid;
border-width: thin;
/* width: 100%; */
padding-top: 1em;
padding-bottom: 1em;
padding-left: 2em;
padding-right: 2em
}
div.titlebox
{
border: none;
padding-top: 1em;
padding-bottom: 1em;
padding-left: 2em;
padding-right: 2em;
background: rgb(200,255,255);
font-family: sans-serif
}
div.synopsisbox
{
border: none;
padding-top: 1em;
padding-bottom: 1em;
padding-left: 2em;
padding-right: 2em;
background: rgb(255,220,255);
/*background: rgb(200,255,255); */
/* font-family: fixed */
}
pre.example
{
border: 1px solid gray;
padding-top: 1em;
padding-bottom: 1em;
padding-left: 1em;
padding-right: 1em;
/* background: rgb(247,242,180); */ /* kind of sandy */
/* background: rgb(200,255,255); */ /* sky blue */
background-color: #F1F5F9; /* light blue-gray */
/* font-family: "Lucida Console", monospace */
}
div.spacerbox
{
border: none;
padding-top: 2em;
padding-bottom: 2em
}
div.image
{
margin: 0;
padding: 1em;
text-align: center;
}
div.categorybox
{
border: 1px solid gray;
padding-top: 0px;
padding-bottom: 0px;
padding-left: 1em;
padding-right: 1em;
background: rgb(247,242,220);
}
-->
</style>
<link rel="icon" href="http://maxima.sourceforge.net/favicon.ico"/>
</head>
<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
<a name="plotdf"></a>
<a name="SEC287"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="maxima_67.html#SEC286" title="Previous section in reading order"> < </a>]</td>
<td valign="middle" align="left">[<a href="#SEC288" title="Next section in reading order"> > </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="maxima_67.html#SEC278" title="Beginning of this chapter or previous chapter"> << </a>]</td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Up section"> Up </a>]</td>
<td valign="middle" align="left">[<a href="maxima_69.html#SEC290" title="Next chapter"> >> </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_79.html#SEC331" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<h1 class="chapter"> 68. plotdf </h1>
<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC288">68.1 Introduction to plotdf</a></td><td> </td><td align="left" valign="top">
</td></tr>
<tr><td align="left" valign="top"><a href="#SEC289">68.2 Functions and Variables for plotdf</a></td><td> </td><td align="left" valign="top">
</td></tr>
</table>
<p><a name="Item_003a-Introduction-to-plotdf"></a>
</p><hr size="6">
<a name="Introduction-to-plotdf"></a>
<a name="SEC288"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#SEC287" title="Previous section in reading order"> < </a>]</td>
<td valign="middle" align="left">[<a href="#SEC289" title="Next section in reading order"> > </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="#SEC287" title="Beginning of this chapter or previous chapter"> << </a>]</td>
<td valign="middle" align="left">[<a href="#SEC287" title="Up section"> Up </a>]</td>
<td valign="middle" align="left">[<a href="maxima_69.html#SEC290" title="Next chapter"> >> </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_79.html#SEC331" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<h2 class="section"> 68.1 Introduction to plotdf </h2>
<p>The function <code>plotdf</code> creates a plot of the direction field of a
first-order Ordinary Differential Equation (ODE) or a system of two
autonomous first-order ODE's.
</p>
<p>Since this is an additional package, in order to use it you must first
load it with <code>load("plotdf")</code>. Plotdf requires Openmath, which is
provided by the package Xmaxima (Xmaxima is not only used as a graphical
console for Maxima but also to plot graphs in the Openmath format).
</p>
<p>To plot the direction field of a single ODE, the ODE must be written in
the form:
</p><pre class="example"> dy
-- = F(x,y)
dx
</pre>
<p>and the function <var>F</var> should be given as the argument for
<code>plotdf</code>. If the independent and dependent variables are not <var>x</var>,
and <var>y</var>, as in the equation above, then those two variables should
be named explicitly in a list given as an argument to the plotdf command
(see the examples).
</p>
<p>To plot the direction field of a set of two autonomous ODE's, they must
be written in the form
</p><pre class="example"> dx dy
-- = G(x,y) -- = F(x,y)
dt dt
</pre>
<p>and the argument for <code>plotdf</code> should be a list with the two
functions <var>G</var> and <var>F</var>, 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 <var>x</var> and <var>y</var>, but if
they are not, then the second argument given to plotdf must be another
list naming the two variables, first the one on the horizontal axis and
then the one on the vertical axis.
</p>
<p>If only one ODE is given, <code>plotdf</code> will implicitly admit
<code>x=t</code>, and <code>G(x,y)=1</code>, transforming the non-autonomous
equation into a system of two autonomous equations.
</p>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Differential-equations">Differential equations</a>
·
<a href="maxima_95.html#Category_003a-Share-packages">Share packages</a>
·
<a href="maxima_95.html#Category_003a-Package-plotdf">Package plotdf</a>
</p>
</div>
<p><a name="Item_003a-Functions-and-Variables-for-plotdf"></a>
</p><hr size="6">
<a name="Functions-and-Variables-for-plotdf"></a>
<a name="SEC289"></a>
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#SEC288" title="Previous section in reading order"> < </a>]</td>
<td valign="middle" align="left">[<a href="maxima_69.html#SEC290" title="Next section in reading order"> > </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="#SEC287" title="Beginning of this chapter or previous chapter"> << </a>]</td>
<td valign="middle" align="left">[<a href="#SEC287" title="Up section"> Up </a>]</td>
<td valign="middle" align="left">[<a href="maxima_69.html#SEC290" title="Next chapter"> >> </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_79.html#SEC331" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<h2 class="section"> 68.2 Functions and Variables for plotdf </h2>
<dl>
<dt><u>Function:</u> <b>plotdf</b><i> (<var>dydx</var>, ...options...)</i>
<a name="IDX2425"></a>
</dt>
<dt><u>Function:</u> <b>plotdf</b><i> (<var>dvdu</var>, <code>[</code><var>u</var>,<var>v</var><code>]</code>, ...options...)</i>
<a name="IDX2426"></a>
</dt>
<dt><u>Function:</u> <b>plotdf</b><i> (<code>[</code><var>dxdt</var>,<var>dydt</var><code>]</code>, ...options...)</i>
<a name="IDX2427"></a>
</dt>
<dt><u>Function:</u> <b>plotdf</b><i> (<code>[</code><var>dudt</var>,<var>dvdt</var><code>]</code>, <code>[</code><var>u</var>,<var>v</var><code>]</code>, ...options...)</i>
<a name="IDX2428"></a>
</dt>
<dd><p>Displays a direction field in two dimensions <var>x</var> and <var>y</var>.
</p>
<p><var>dydx</var>, <var>dxdt</var> and <var>dydt</var> are expressions that depend on
<var>x</var> and <var>y</var>. <var>dvdu</var>, <var>dudt</var> and <var>dvdt</var> are
expressions that depend on <var>u</var> and <var>v</var>. In addition to those two
variables, the expressions can also depend on a set of parameters, with
numerical values given with the <code>parameters</code> option (the option
syntax is given below), or with a range of allowed values specified by a
<var>sliders</var> option.
</p>
<p>Several other options can be given within the command, or selected in
the menu. Integral curves can be obtained by clicking on the plot, or
with the option <code>trajectory_at</code>. The direction of the integration
can be controlled with the <code>direction</code> option, which can have
values of <em>forward</em>, <em>backward</em> or <em>both</em>. The number of
integration steps is given by <code>nsteps</code> and the time interval
between them is set up with the <code>tstep</code> option. The Adams Moulton
method is used for the integration; it is also possible to switch to an
adaptive Runge-Kutta 4th order method.
</p>
<p><b>Plot window menu:</b>
</p>
<p>The menu in the plot window has the following options: <em>Zoom</em>, will
change the behavior of the mouse so that it will allow you to zoom in on
a region of the plot by clicking with the left button. Each click near a
point magnifies the plot, keeping the center at the point where you
clicked. Holding the <kbd>Shift</kbd> key while clicking, zooms out to the
previous magnification. To resume computing trajectories when you click
on a point, select <em>Integrate</em> from the menu.
</p>
<p>The option <em>Config</em> in the menu can be used to change the ODE(s) in
use and various other settings. After configuration changes are made,
the menu option <em>Replot</em> should be selected, to activate the new
settings. If a pair of coordinates are entered in the field
<em>Trajectory at</em> in the <em>Config</em> dialog menu, and the
<kbd>enter</kbd> key is pressed, a new integral curve will be shown, in
addition to the ones already shown. When <em>Replot</em> is selected, only
the last integral curve entered will be shown.
</p>
<p>Holding the right mouse button down while the cursor is moved, can be
used to drag the plot sideways or up and down. Additional
parameters such as the number of steps, the initial value of <var>t</var>
and the x and y centers and radii, may be set in the Config menu.
</p>
<p>A copy of the plot can be saved as a postscript file, using the menu
option <em>Save</em>.
</p>
<p><b>Plot options:</b>
</p>
<p>The <code>plotdf</code> command may include several commands, each command is
a list of two or more items. The first item is the name of the option,
and the remainder comprises the value or values assigned to the option.
</p>
<p>The options which are recognized by <code>plotdf</code> are the following:
</p>
<ul>
<li>
<em>tstep</em> defines the length of the increments on the
independent variable <var>t</var>, used to compute an integral curve. If only
one expression <var>dydx</var> is given to <code>plotdf</code>, the <var>x</var>
variable will be directly proportional to <var>t</var>.
The default value is 0.1.
</li><li>
<em>nsteps</em> defines the number of steps of length <code>tstep</code>
that will be used for the independent variable, to compute an integral
curve.
The default value is 100.
</li><li>
<em>direction</em> defines the direction of the independent
variable that will be followed to compute an integral curve. Possible
values are <code>forward</code>, to make the independent variable increase
<code>nsteps</code> times, with increments <code>tstep</code>, <code>backward</code>, to
make the independent variable decrease, or <code>both</code> that will lead to
an integral curve that extends <code>nsteps</code> forward, and <code>nsteps</code>
backward. The keywords <code>right</code> and <code>left</code> can be used as
synonyms for <code>forward</code> and <code>backward</code>.
The default value is <code>both</code>.
</li><li>
<em>tinitial</em> defines the initial value of variable <var>t</var> used
to compute integral curves. Since the differential equations are
autonomous, that setting will only appear in the plot of the curves as
functions of <var>t</var>.
The default value is 0.
</li><li>
<em>versus_t</em> is used to create a second plot window, with a
plot of an integral curve, as two functions <var>x</var>, <var>y</var>, of the
independent variable <var>t</var>. If <code>versus_t</code> is given any value
different from 0, the second plot window will be displayed. The second
plot window includes another menu, similar to the menu of the main plot
window.
The default value is 0.
</li><li>
<em>trajectory_at</em> defines the coordinates <var>xinitial</var> and
<var>yinitial</var> for the starting point of an integral curve.
The option is empty by default.
</li><li>
<em>parameters</em> defines a list of parameters, and their
numerical values, used in the definition of the differential
equations. The name and values of the parameters must be given in a
string with a comma-separated sequence of pairs <code>name=value</code>.
</li><li>
<em>sliders</em> defines a list of parameters that will be changed
interactively using slider buttons, and the range of variation of those
parameters. The names and ranges of the parameters must be given in a
string with a comma-separated sequence of elements <code>name=min:max</code>
</li><li>
<em>xfun</em> defines a string with semi-colon-separated sequence
of functions of <var>x</var> to be displayed, on top of the direction field.
Those functions will be parsed by Tcl and not by Maxima.
</li><li>
<em>x</em> should be followed by two numbers, which will set up the minimum
and maximum values shown on the horizontal axis. If the variable on the
horizontal axis is not <var>x</var>, then this option should have the name of
the variable on the horizontal axis.
The default horizontal range is from -10 to 10.
</li><li>
<em>y</em> should be followed by two numbers, which will set up the minimum
and maximum values shown on the vertical axis. If the variable on the
vertical axis is not <var>y</var>, then this option should have the name of
the variable on the vertical axis.
The default vertical range is from -10 to 10.
</li></ul>
<p><b>Examples:</b>
</p>
<ul>
<li>
To show the direction field of the differential equation <em>y' = exp(-x) + y</em> and the solution that goes through <em>(2, -0.1)</em>:
<pre class="example">(%i1) load("plotdf")$
(%i2) plotdf(exp(-x)+y,[trajectory_at,2,-0.1])$
</pre>
<p><div class="image"><img src="./figures/plotdf1.gif" alt="figures/plotdf1"></div>
</p>
</li><li>
To obtain the direction field for the equation <em>diff(y,x) = x - y^2</em> and the solution with initial condition <em>y(-1) = 3</em>, we can use the command:
<pre class="example">(%i3) plotdf(x-y^2,[xfun,"sqrt(x);-sqrt(x)"],
[trajectory_at,-1,3], [direction,forward],
[y,-5,5], [x,-4,16])$
</pre><p>The graph also shows the function <em>y = sqrt(x)</em>.
</p>
<p><div class="image"><img src="./figures/plotdf2.gif" alt="figures/plotdf2"></div>
</p>
</li><li>
The following example shows the direction field of a harmonic oscillator,
defined by the two equations <em>dz/dt = v</em> and <em>dv/dt = -k*z/m</em>,
and the integral curve through <em>(z,v) = (6,0)</em>, with a slider that
will allow you to change the value of <em>m</em> interactively (<em>k</em> is
fixed at 2):
<pre class="example">(%i4) plotdf([v,-k*z/m], [z,v], [parameters,"m=2,k=2"],
[sliders,"m=1:5"], [trajectory_at,6,0])$
</pre>
<p><div class="image"><img src="./figures/plotdf3.gif" alt="figures/plotdf3"></div>
</p>
</li><li>
To plot the direction field of the Duffing equation, <em>m*x"+c*x'+k*x+b*x^3 = 0</em>, we introduce the variable <em>y=x'</em> and use:
<pre class="example">(%i5) plotdf([y,-(k*x + c*y + b*x^3)/m],
[parameters,"k=-1,m=1.0,c=0,b=1"],
[sliders,"k=-2:2,m=-1:1"],[tstep,0.1])$
</pre>
<p><div class="image"><img src="./figures/plotdf4.gif" alt="figures/plotdf4"></div>
</p>
</li><li>
The direction field for a damped pendulum, including the
solution for the given initial conditions, with a slider that
can be used to change the value of the mass <em>m</em>, and with a plot of
the two state variables as a function of time:
<pre class="example">(%i6) plotdf([w,-g*sin(a)/l - b*w/m/l], [a,w],
[parameters,"g=9.8,l=0.5,m=0.3,b=0.05"],
[trajectory_at,1.05,-9],[tstep,0.01],
[a,-10,2], [w,-14,14], [direction,forward],
[nsteps,300], [sliders,"m=0.1:1"], [versus_t,1])$
</pre>
<p><div class="image"><img src="./figures/plotdf5.gif" alt="figures/plotdf5"></div><div class="image"><img src="./figures/plotdf6.gif" alt="figures/plotdf6"></div>
</p>
</li></ul>
<div class=categorybox>
<p>Categories: <a href="maxima_95.html#Category_003a-Package-plotdf">Package plotdf</a>
·
<a href="maxima_95.html#Category_003a-Plotting">Plotting</a>
·
<a href="maxima_95.html#Category_003a-Numerical-methods">Numerical methods</a>
</p>
</div>
</dd></dl>
<p><a name="Item_003a-romberg"></a>
</p><hr size="6">
<table cellpadding="1" cellspacing="1" border="0">
<tr><td valign="middle" align="left">[<a href="#SEC287" title="Beginning of this chapter or previous chapter"> << </a>]</td>
<td valign="middle" align="left">[<a href="maxima_69.html#SEC290" title="Next chapter"> >> </a>]</td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left"> </td>
<td valign="middle" align="left">[<a href="maxima.html#SEC_Top" title="Cover (top) of document">Top</a>]</td>
<td valign="middle" align="left">[<a href="maxima_toc.html#SEC_Contents" title="Table of contents">Contents</a>]</td>
<td valign="middle" align="left">[<a href="maxima_79.html#SEC331" title="Index">Index</a>]</td>
<td valign="middle" align="left">[<a href="maxima_abt.html#SEC_About" title="About (help)"> ? </a>]</td>
</tr></table>
<p>
<font size="-1">
This document was generated by <em>Robert Dodier</em> on <em>April, 24 2010</em> using <a href="http://texi2html.cvshome.org/"><em>texi2html 1.76</em></a>.
</font>
<br>
</p>
</body>
</html>
|
