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
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--Converted with LaTeX2HTML 98.1p1 release (March 2nd, 1998)
originally by Nikos Drakos (nikos@cbl.leeds.ac.uk), CBLU, University of Leeds
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
* with significant contributions from:
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
<HTML>
<HEAD>
<TITLE>A Few Remarks on the Definition of Duffing's Equations</TITLE>
<META NAME="description" CONTENT="A Few Remarks on the Definition of Duffing's Equations">
<META NAME="keywords" CONTENT="userman">
<META NAME="resource-type" CONTENT="document">
<META NAME="distribution" CONTENT="global">
<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=iso-8859-1">
<LINK REL="STYLESHEET" HREF="userman.css">
<LINK REL="previous" HREF="node64.html">
<LINK REL="up" HREF="node64.html">
<LINK REL="next" HREF="node66.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html1335"
HREF="node66.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="/usr/lib/latex2html/icons.gif/next_motif.gif"></A>
<A NAME="tex2html1332"
HREF="node64.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="/usr/lib/latex2html/icons.gif/up_motif.gif"></A>
<A NAME="tex2html1328"
HREF="node64.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="/usr/lib/latex2html/icons.gif/previous_motif.gif"></A>
<A NAME="tex2html1334"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="/usr/lib/latex2html/icons.gif/contents_motif.gif"></A>
<BR>
<B> Next:</B> <A NAME="tex2html1336"
HREF="node66.html">Deleting Dynamical Systems</A>
<B> Up:</B> <A NAME="tex2html1333"
HREF="node64.html">A Vector Field Example:</A>
<B> Previous:</B> <A NAME="tex2html1329"
HREF="node64.html">A Vector Field Example:</A>
<BR>
<BR>
<!--End of Navigation Panel-->
<H2><A NAME="SECTION00631000000000000000">
A Few Remarks on the Definition of Duffing's Equations</A>
</H2>
Recall that when we installed Duffing's equations as a time-dependent vector field,
we defined
<!-- MATH: $\sin \omega t$ -->
92#92
as an auxiliary function and claimed that we could
use it to study the time-
<!-- MATH: $2 \pi/\omega$ -->
89#89
stroboscopic map.
In theory, there is nothing wrong with
this, however in practice we will encounter numerical errors in the evaluation of
transcendental functions such as
<!-- MATH: $\sin \omega t$ -->
92#92
for large values of <I>t</I>.
Since we are often interested in generating
Poincaré maps for extremely long times, and since the function
<!-- MATH: $\cos \omega t$ -->
93#93also appears in the definition of our vector field, the user may want to extend phase space
by introducing the variable 94#94.
Then we can rewrite Duffing's
equations in the form
<BR>
95#95
<BR>
where
<!-- MATH: $(u,v,\theta) \in \real^2 \times S^1$ -->
96#96,
and <I>S</I><SUP>1</SUP> is the circle of length
<!-- MATH: $2 \pi/ \omega$ -->
89#89.
That is, 94#94
takes values in
<!-- MATH: $[0, 2 \pi/ \omega)$ -->
97#97.
The problem with this formulation is that DsTool <EM>cannot handle periodic variables
whose length depends on a parameter!</EM> To overcome this difficulty, we change coordinates
via the transformation
<!-- MATH: $\psi = \omega \theta$ -->
98#98.
Thus we could study Duffing's equations
on extended phase space in the form
<A NAME="duff2"> </A>
<BR>
99#99
<BR>
where
<!-- MATH: $(u,v,\psi) \in \real^2 \times S^1$ -->
100#100,
and <I>S</I><SUP>1</SUP> is now the circle of length 82#82.
<P>
The advantage to an extended phase space such as we have for Equation <A HREF="node65.html#duff2"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="/usr/lib/latex2html/icons.gif/cross_ref_motif.gif"></A>
is that it is trivial to plot Poincaré sections for this set of equations because we
can make 101#101
a periodic variable. This allows us to request that DsTool only plot
points when
<!-- MATH: $\theta=\theta_0$ -->
102#102
for some 103#103.
In contrast, DsTool <EM>never treats time as a periodic variable</EM>,
so we needed to define the auxiliary function
<!-- MATH: $\sin \omega t$ -->
92#92
in order to be able to generate a
Poincaré map for Duffing's equations.
<P>
<HR>
<!--Navigation Panel-->
<A NAME="tex2html1335"
HREF="node66.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="/usr/lib/latex2html/icons.gif/next_motif.gif"></A>
<A NAME="tex2html1332"
HREF="node64.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="/usr/lib/latex2html/icons.gif/up_motif.gif"></A>
<A NAME="tex2html1328"
HREF="node64.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="/usr/lib/latex2html/icons.gif/previous_motif.gif"></A>
<A NAME="tex2html1334"
HREF="node1.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="/usr/lib/latex2html/icons.gif/contents_motif.gif"></A>
<BR>
<B> Next:</B> <A NAME="tex2html1336"
HREF="node66.html">Deleting Dynamical Systems</A>
<B> Up:</B> <A NAME="tex2html1333"
HREF="node64.html">A Vector Field Example:</A>
<B> Previous:</B> <A NAME="tex2html1329"
HREF="node64.html">A Vector Field Example:</A>
<!--End of Navigation Panel-->
<ADDRESS>
<I>John Lapeyre</I>
<BR><I>1998-09-04</I>
</ADDRESS>
</BODY>
</HTML>
|
