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{SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 14 "k:='k':c:='c':" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "#defini
ng these saves typing later" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "fxns:=\{y1(t),y2(t)\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%fxnsG
<$-%#y1G6#%\"tG-%#y2GF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "
#these are the equations " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
52 "sys:=D(y1)(t)=y2(t), D(y2)(t)=-c*y2(t)-k*sin(y1(t));" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%$sysG6$/--%\"DG6#%#y1G6#%\"tG-%#y2GF,/--F)6#F/
F,,&*&%\"cG\"\"\"F.F7!\"\"*&%\"kGF7-%$sinG6#-F+F,F7F8" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "c:=3;k:=1;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"cG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG\"
\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "#trajectories start
ing on the positive y2 axis" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
164 "for i from 1 to 14 do p[i]:=dsolve(\{sys,y1(0)=0,y2(0)=i*Pi/7\},f
xns,type=numeric): pl[i]:=odeplot(p[i], [y1(t),y2(t)],0...15, numpoint
s=250,view=[0...6,-3...3]): od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 313 "#so the way it \+
works is, the p[i] is defined to be the numerical solution of the diff
erential equations with the initial data given in the curly brackets, \+
the pl[i] is then defined as the odeplot of this. the do loop runs thr
ough various initial data, the numpoints gives the number of points us
ed by the numerics" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "#trajectories starting on th
e negative y2 axis" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 166 "for \+
i from 1 to 14 do q[i]:=dsolve(\{sys,y1(0)=i*Pi/8,y2(0)=0\},fxns,type=
numeric): ql[i]:=odeplot(q[i], [y1(t),y2(t)],-15...15, numpoints=150,v
iew=[0...6,-3...3]): od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "#trajectories starting \+
at the first saddle point" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
56 "s:=dsolve(\{sys,y1(0)=Pi-.01,y2(0)=0\},fxns,type=numeric):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "sl1:=odeplot(s, [y1(t),y2(t)
],0...40, numpoints=100,view=[0...6,-3...3]):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 74 "sl2:=odeplot(s, [y1(t),y2(t)],-12...0, numpoints
=100,view=[0...6,-3...3]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
56 "z:=dsolve(\{sys,y1(0)=Pi+.01,y2(0)=0\},fxns,type=numeric):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "sl3:=odeplot(z, [y1(t),y2(t)
],0...40, numpoints=100,view=[0...6,-3...3]):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 74 "sl4:=odeplot(z, [y1(t),y2(t)],-12...0, numpoints
=100,view=[0...6,-3...3]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
27 "display(\{sl1,sl2,sl3,sl4\});" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "#trajectories starting at the 0 point" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 408 "t:=dsolve(\{sys,y1(0)=-.01,y2(0)=0
\},fxns,type=numeric):tl1:=odeplot(t, [y1(t),y2(t)],0...12, numpoints=
100,view=[0...6,-3...3]):\ntl2:=odeplot(t, [y1(t),y2(t)],-12...0, nump
oints=100,view=[0...6,-3...3]):\nzt:=dsolve(\{sys,y1(0)=+.01,y2(0)=0\}
,fxns,type=numeric):\ntl3:=odeplot(zt, [y1(t),y2(t)],0...12, numpoints
=75,view=[0...6,-3...3]):\ntl4:=odeplot(zt, [y1(t),y2(t)],-12...0, num
points=75,view=[0...6,-3...3]):\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "display(\{tl1,tl2,tl3,tl4\});" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 118 "#display the plots, after trying various comb
inations, it seems using only the odd trajectories gave the nicest out
put" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 151 "display(\{ql[1],ql[3],ql[5],ql[7],ql[9],ql[
11],ql[13],pl[1],pl[3],pl[5],pl[7],pl[9],pl[11],pl[13],sl1,sl2,sl3,sl4
,tl1,tl2,tl3,tl4\},scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 237 "#traje
ctories starting on the negative y2 axis\nfor i from 1 to 14 do r[i]:=
dsolve(\{sys,y1(0)=1.9*sin(Pi*i/7),y2(0)=1.9*cos(Pi*i/7)\},fxns,type=n
umeric): rl[i]:=odeplot(r[i], [y1(t),y2(t)],0...15, numpoints=350,view
=[-6...6,-3...3]): od:\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
55 "display(\{rl[1],rl[3],rl[5],rl[7],rl[9],rl[11],rl[13]\});" }}}
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