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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 17 "Poincare Sections" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 111 "This wor
ksheet shows how to use Maple to obtain a Poincare section. Note that \+
these plots can take a long time." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 41 "Restart Maple and load the plots package.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 
"with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 79 "Define the system. Note that we do not need to make th
e system autonomous here." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "f:=y(t):g:=-k*y(t)-x(t)^3+B*cos(t):
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
28 "Choose the parameter values." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "k:=0.1:B:=11:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 34 "Define th
e differential equations." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de1:=diff(x(t),t)=f:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "de2:=diff(y(t),t)=g:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 202 "We need \+
to use the numeric option in dsolve not DEplot. It is also best to use
 the gear method rather than the default numerical method. We have inc
luded the intial conditions explicitly in the command." }}{PARA 0 "" 
0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "ueda \+
:= dsolve(\{de1, de2, y(0)=1, x(0)=1\}, \{x(t), y(t)\},type=numeric, m
ethod=gear): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 352 "We define the Poincare section to be a  function of t
he number of points being plotted. The t-range for the plot is over on
e complete period of the plot time the number of points to be plotted.
 This ensures that a point will only be plotted when t reaches a compl
ete period. Notice that in this case, since the frequency is one, one \+
complete period is " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }
{TEXT -1 107 ". We start at 2*Pi*points to exclude transients and plot
 up to 4*Pi*points. In general if the frequency is " }{XPPEDIT 18 0 "o
mega;" "6#%&omegaG" }{TEXT -1 39 " we would need to plot over a range \+
of " }{XPPEDIT 18 0 "2*Pi/omega;" "6#*&*&\"\"#\"\"\"%#PiGF&F&%&omegaG!
\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 131 "poincare:=(points)->odeplot(ueda, [x(t),
y(t)], (2*Pi*points)..(4*Pi*points),numpoints=points,style=point,symbo
l=point,colour=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "po
incare(2000);" }}}}{MARK "15" 0 }{VIEWOPTS 1 1 0 1 1 1803 }