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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 65 "The Duffing equation - forced and
 damped oscillations (continued)" }}{PARA 0 "" 0 "" {TEXT -1 89 "Adapt
ed from material written by Peter Young at the University of Californi
a, Santa Cruz " }}{PARA 0 "" 0 "" {TEXT -1 6 "See:  " }{URLLINK 17 "ht
tp://bartok.ucsc.edu/peter/115/duffing.nb.pdf" 4 "http://bartok.ucsc.e
du/peter/115/duffing.nb.pdf" "" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Pete
r Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version
:  2.4.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " }{TEXT 0 7 "desolv
e" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }
{TEXT 267 7 "DEsol.m" }{TEXT -1 32 " is required by this worksheet. " 
}}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by
 a command similar to the one that follows, where the file path gives \+
its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\
\\Maple/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 45 "load root-finding procedures etc. includi
ng: " }{TEXT 0 7 "findmax" }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-
file " }{TEXT 267 7 "roots.m" }{TEXT -1 38 " contains the code for the
 procedures " }{TEXT 0 7 "findmax" }{TEXT -1 25 " used in this workshe
et. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple sess
ion by a command similar to the one that follows, where the file path \+
gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \+
\"K:\\\\Maple/procdrs/roots.m\";" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 65 "A periodic solution with period 4
 times that of the driving force" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "d^2*x/(d*t^2)+1/10;
" "6#,&*(%\"dG\"\"#%\"xG\"\"\"*&F%F(*$%\"tGF&F(!\"\"F(*&F(F(\"#5F,F(" 
}{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dt-x+x^3 = A*cos(omega*t);" "6#/,(*
&%#dxG\"\"\"%#dtG!\"\"F'%\"xGF)*$F*\"\"$F'*&%\"AGF'-%$cosG6#*&%&omegaG
F'%\"tGF'F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 7 "We set " }{XPPEDIT 18 0 "A = 0" "6#/%\"AG\"\"!" 
}{TEXT -1 1 "." }{XPPEDIT 18 0 "338=169/500" "6#/\"$Q$*&\"$p\"\"\"\"\"
$+&!\"\"" }{TEXT -1 42 " to increase the driving force a bit more." }}
{PARA 0 "" 0 "" {TEXT -1 64 "As before, we start with the particle off
 at rest at the origin." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 101 "First we perform a quick calculation with relativel
y low accuracy to search for a periodic solution. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 242 "A := 169/50
0: omega := 7/5:\nde := diff(x(t),t$2)+0.1*diff(x(t),t)-x(t)+x(t)^3=A*
cos(omega*t):\nic := x(0)=0,D(x)(0)=0:\nsol := dsolve(\{de,ic\},x(t),n
umeric,maxfun=0,output=listprocedure):\nxproc := subs(sol,x(t));\nupro
c := subs(sol,diff(x(t),t));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 173 "omega := 7/5:\nT := evalf(2
*Pi/omega);\ntmax := 200*T;\ntmin := 196*T;\nplot([xproc(t),uproc(t),t
=tmin..tmax],color=COLOR(RGB,.5,0,1),\n                    labels=[`x`
,`dx/dt`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG$\"+2&*)z[%!\"*" 
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urve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
18 "Obtain values for " }{XPPEDIT 18 0 "x(t)" "6#-%\"xG6#%\"tG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "`x '`(t)" "6#-%$x~'G6#%\"tG" }
{TEXT -1 18 " at some value of " }{TEXT 263 1 "t" }{TEXT -1 33 " where
 the solution is periodic. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "x0 := evalf(xproc(tmax+2*T));\nu0 :
= evalf(uproc(tmax+2*T));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G$!+
px1i6!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#u0G$\"+5,s)4&!#5" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "Now use t
hese values as the starting values for a more accurate solution. " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
277 "A := 169/500: omega := 7/5:\nde := diff(x(t),t$2)+0.1*diff(x(t),t
)-x(t)+x(t)^3=A*cos(omega*t):\nic := x(0)=x0,D(x)(0)=u0:\nsol := dsolv
e(\{de,ic\},x(t),type=numeric,method=dverk78,abserr=1e-11,\n          \+
         relerr=1e-11,maxfun=0,output=listprocedure):\nxproc := subs(s
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0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "evalf[
12](findmax('xproc(t)',t = \{118.5,119,119.5\}));\nevalf[12](findmax('
xproc(t)',t = \{136.5,137,137.5\}));\nevalf(evalf[12](%[1]-%%[1]));" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"-P;AQ)=\"!\"*$!-q\"Go(zA!#7" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"-n?9!zO\"!\"*$!-\\t'\\'zA!#7" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V?>&z\"!\")" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 41 "The period is approximate
ly  17.95192043." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 22 "This is approximately " }{TEXT 261 10 "four times" }
{TEXT -1 34 " the period of the driving force  " }{XPPEDIT 18 0 "8*Pi/
omega;" "6#*(\"\")\"\"\"%#PiGF%%&omegaG!\"\"" }{TEXT -1 1 " " }{TEXT 
264 1 "~" }{TEXT -1 14 " 17.95195802. " }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "evalf(evalf[12](8*Pi/1.4
));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+-e>&z\"!\")" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "The period has " }
{TEXT 261 13 "doubled again" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "The calculation can also be mad
e using the procedure " }{TEXT 0 7 "desolve" }{TEXT -1 2 ". " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 283 "A
 := 169/500: omega := 7/5:\nT := evalf(2*Pi/omega);\ntmax := 32*T;\nde
 := diff(x(t),t$2)+1/10*diff(x(t),t)-x(t)+x(t)^3=A*cos(omega*t):\nic :
= x(0)=x0,D(x)(0)=u0:\n(fn,dfn) := desolve(\{de,ic\},x(t),t=0..145,typ
e=numeric,method=rk78,\n                      tolerance=1e-11,output=p
roc_pair):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG$\"+2&*)z[%!\"*" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tmaxG$\"+Um:O9!\"(" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "T := ev
alf(2*Pi/omega):\ntmin := 20*T;\ntmax := 32*T;\nplot('fn'(t),t=tmin..t
max,x=-1.4..0,color=red);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tminG$
\"+9!zf(*)!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tmaxG$\"+Um:O9!\"
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0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 131 "evalf[12](findmax('fn'(t),t = \{118.5,119,119.5
\}));\nevalf[12](findmax('fn'(t),t = \{136.5,137,137.5\}));\nevalf(eva
lf[12](%[1]-%%[1]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"-N;AQ)=\"
!\"*$!-n\"Go(zA!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$$\"-n?9!zO\"!
\"*$!-]t'\\'zA!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+V?>&z\"!\")" 
}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" 
}{TEXT -1 93 ": The next section continues working with the numerical \+
solution constructed in this section." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "
" 0 "" {TEXT -1 59 "Reconstructing the solution and its symmetrical co
unterpart" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "
" 0 "" {TEXT 261 4 "Note" }{TEXT -1 97 ": This section continues worki
ng with the numerical solution constructed in the previous section." }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "We can r
ecover the periodic part of the motion easily as follows." }}{PARA 0 "
" 0 "" {TEXT -1 105 "First note that the 23 times the period of the dr
iving force brings into the periodic part of the motion." }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "evalf
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}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Now cal
culate the values of the solution and its derivative at this value of \+
" }{TEXT 265 1 "t" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "evalf[15](fn(230/7*Pi));\nev
alf[15](dfn(230/7*Pi));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0CmEI-)[7
!#9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!0i'ecE[9D!#:" }}}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "Using these values a
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}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 257 "A := 169/500: omega := 7/5:\nde := diff(x(t),t$2)+1/10*diff(x(t
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=-.251448265658662;\n(gn,dgn) := desolve(\{de,ic\},t=0..20,type=numeri
c,\n          method=rk78,tolerance=1e-11,output=proc_pair):" }}{PARA 
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ic := x(0)=.230342171313424,D(x)(0)=.515771390225484e-1;\n(hn,dhn) := \+
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$$\"+)4.P#eF*$\"+O9jL;F-7$$\"+ar/$)eF*$\"+AH:b$*F-7$$\"+V+WDgF*$\"+o\"
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$p]BC)F*$\"+$eg\\I&F*7$$\"+^WbI))F*$\"+s5TtcF*7$$\"+`9(4X*F*$\"+OWx-fF
*7$$\"+$4D(35Fap$\"+3#fa'fF*7$$\"+xP,s5Fap$\"+5#\\_$eF*7$$\"+%*>IH6Fap
$\"+%>xt^&F*7$$\"+o&)Q#=\"Fap$\"+\"[R#**\\F*7$$\"+)eZ#H7Fap$\"+Mfs!G%F
*7$$\"+.h\"zE\"Fap$\"+h5#[P$F*7$$\"+/[n(H\"Fap$\"+\"oKvD#F*7$$\"+vM'\\
J\"Fap$\"+6[/85F*7$$\"+'3_(=8Fap$!+Rh0%)HF-7$$\"+>\\o38Fap$!+;#f+h\"F*
7$$\"+inx&G\"Fap$!+*GU'HGF*7$$\"+s9)3D\"Fap$!+V:ALRF*7$$\"+NLT07Fap$!+
%\\6-)[F*7$$\"+ic7^6Fap$!+!Rxak&F*7$$\"+4T%p3\"Fap$!+;)o:C'F*7$$\"+<#f
u,\"Fap$!+eHZKmF*7$$\"+4-,[%*F*$!+G^ZMoF*7$$\"+aq\")3()F*$!+_^*4(oF*7$
$\"+1,J+!)F*$!+\\-LunF*7$$\"+%Q/vI(F*$!+Md#>d'F*7$$\"+w$Q-k'F*$!+\"Qn]
G'F*7$$\"+DGH1gF*$!+7%e?$fF*7$$\"+g;v)R&F*$!+ErD=bF*7$$\"+0vNP[F*$!+^$
oN1&F*7$$\"+FX\"fK%F*$!+'[fvd%F*7$$\"+/#\\t'QF*$!+@XJnSF*7$$\"+%*40nMF
*$!+R?ZUNF*7$$\"+\"olF7$F*$!+YXk-IF*7$$\"+Z<\"f$GF*$!+[Yr]CF*7$$\"+STj
2EF*$!+SA0*)=F*7$$\"+S+uVBF*$!+zXb%Q)F-7$$\"+(fjVG#F*$\"+G*G5D#F-7$$\"
+CYFpCF*$\"+e&[CW\"F*7$$\"+Nzn@HF*$\"+gq5OEF*7$$\"+[+j3KF*$\"+;>EbJF*7
$$\"+0')QYNF*$\"+![)HhOF*7$$\"+q?[LRF*$\"+&\\F1:%F*7$$\"+=V/oVF*$\"+2&
G)=YF*7$$\"+G6+\")[F*$\"+?c\\)3&F*7$$\"+JzfTaF*$\"+@4J>bF*7$$\"+&f#=Xg
F*$\"+moZ+fF*7$$\"+gl#eo'F*$\"+()fd=iF*7$$\"+Rf\"oK(F*$\"+Sl0\\kF*7$$
\"+_Tv')zF*$\"+r;c!f'F*7$$\"+Nqub')F*$\"+(o3^i'F*7$$\"+^_/A$*F*$\"+gh?
MlF*7$$\"+7B&>+\"Fap$\"+Xd$pF'F*7$$\"+$eHz1\"Fap$\"+L!4y$eF*7$$\"+&3/
\"G6Fap$\"+$QB\"3_F*7$$\"+U:T!=\"Fap$\"+%fv7R%F*-%+AXESLABELSG6$%\"xG%
*u~=~dx/dtG-%&COLORG6&%$RGBG$\"\"&!\"\"\"\"!\"\"\"-%%VIEWG6$%(DEFAULTG
Fedn" 1 2 0 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1
" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Not
e" }{TEXT -1 216 ": It is quite likely that the numerical solutions ca
lculated by different methods will initially diverge due to errors but
, with luck, any method should eventually hit one or other of the two \+
closed periodic schemes." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 17 "Poincar\351 sections" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 74 "A useful way of ana
lyzing chaotic motion is to look at what is called the " }{TEXT 261 
16 "Poincar\351 section" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
225 "Rather than considering the phase space trajectory for all times,
 which gives a continuous curve, the Poincar\351 section is just the d
iscrete set of points on the trajectory of at every period of the driv
ing force, that is, at " }{XPPEDIT 18 0 "t = T,2*T,3*T,` . . . `;" "6&
/%\"tG%\"TG*&\"\"#\"\"\"F%F(*&\"\"$F(F%F(%(~.~.~.~G" }{TEXT -1 14 ", e
tc., where " }{XPPEDIT 18 0 "T = 2*Pi/omega" "6#/%\"TG*(\"\"#\"\"\"%#P
iGF'%&omegaG!\"\"" }{TEXT -1 15 " is the period." }}{PARA 0 "" 0 "" 
{TEXT -1 57 " Note that, at each of these points, the forcing function
" }{TEXT 262 1 " " }{TEXT -1 20 "has the same value. " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "For a periodic orbit \+
the Poincar\351 section is a single point, when the period has doubled
 it consists of two points, and so on." }}{PARA 0 "" 0 "" {TEXT -1 19 
"With the parameter " }{TEXT 266 1 "A" }{TEXT -1 63 ", which determine
s the strength of the driving force, set to 0." }{XPPEDIT 18 0 "338=16
9/500" "6#/\"$Q$*&\"$p\"\"\"\"\"$+&!\"\"" }{TEXT -1 94 ", we have seen
 in the previous sections that the motion is eventually periodic with \+
period 4. " }}{PARA 0 "" 0 "" {TEXT -1 84 "In the following computatio
n an initial 280 periods bypass the transient behaviour.\n" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 435 "A := 169/500: omega := 7/5:\nT := \+
evalf(2*Pi/omega);\nnumpts := 300:\ntmax := numpts*T;\nde := diff(x(t)
,t$2)+1/10*diff(x(t),t)-x(t)+x(t)^3=A*cos(omega*t):\nic := x(0)=0,D(x)
(0)=0:\nsol := dsolve(\{de,ic\},x(t),type=numeric,range=0..tmax,maxfun
=0):\npts := [seq(subs(\{op(2..3,sol(i*T))\},[x(t),diff(x(t),t)]),\n  \+
            i=numpts-20..numpts)]:\nplot(pts,style=point,color=COLOR(R
GB,.3,0,.6),symbol=circle,\n               labels=[`x`,`dx/dt`]);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG$\"+2&*)z[%!\"*" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%%tmaxG$\"+_oRY8!\"'" }}{PARA 13 "" 1 "" 
{GLPLOT2D 375 375 375 {PLOTDATA 2 "6(-%'CURVESG6#777$$!3mslpe(e4@\"!#<
$\"3wbDaDrs#y%!#=7$$!3yT,@!)=\")[7F*$!3t+L#)G^59DF-7$$!3A/j.X[Og6F*$\"
3#f$4xP$fY3&F-7$$!3GC7x!*G,!H\"F*$!3%QZM?(zcY:F-7$$!3\\:&RRAf4@\"F*$\"
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NTG-%&COLORG6&%$RGBG$\"\"$!\"\"\"\"!$\"\"'F_s-%%VIEWG6$%(DEFAULTGFfs" 
1 5 4 1 10 0 2 6 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }
{XPPEDIT 18 0 "A=0" "6#/%\"AG\"\"!" }{TEXT -1 1 "." }{XPPEDIT 18 0 "33
98 = 1699/5000;" "6#/\"%)R$*&\"%*p\"\"\"\"\"%+]!\"\"" }{TEXT -1 118 ",
 the period is multiplied by 8.  \nIn the following computation an ini
tial 980 periods bypass the transient behaviour." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 433 "A := 1699/5
000: omega := 7/5:\nT := evalf(2*Pi/omega);\nnumpts := 1000:\ntmax := \+
numpts*T;\nde := diff(x(t),t$2)+1/10*diff(x(t),t)-x(t)+x(t)^3=A*cos(om
ega*t):\nic := x(0)=0,D(x)(0)=0:\nsol := dsolve(\{de,ic\},x(t),numeric
,range=0..tmax,maxfun=0):\npts := [seq(subs(\{op(2..3,sol(i*T))\},[x(t
),diff(x(t),t)]),\n              i=numpts-20..numpts)]:\nplot(pts,styl
e=point,color=COLOR(RGB,.3,0,.6),symbol=circle,\n               labels
=[`x`,`dx/dt`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG$\"+2&*)z[%!
\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tmaxG$\"+2&*)z[%!\"'" }}
{PARA 13 "" 1 "" {GLPLOT2D 374 374 374 {PLOTDATA 2 "6(-%'CURVESG6#777$
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!$\"\"'F_s-%%VIEWG6$%(DEFAULTGFfs" 1 5 4 1 10 0 2 6 1 4 2 1.000000 
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{PARA 0 "" 0 "" {TEXT -1 64 "We may construct a phase trajectory of th
is 8 cycle as follows. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 319 "A := 1699/5000: omega := 7/5:\nT :
= evalf(2*Pi/omega);\nde := diff(x(t),t$2)+1/10*diff(x(t),t)-x(t)+x(t)
^3=A*cos(omega*t):\nic := x(0)=0,D(x)(0)=0:\nsol := dsolve(\{de,ic\},x
(t),numeric,range=0..5000,maxfun=0):\nplots[odeplot](sol,[x(t),diff(x(
t),t)],4950..5000,refine=1,\n        color=COLOR(RGB,.5,0,1),labels=[`
x`,`dx/dt`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG$\"+2&*)z[%!\"*
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 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Wh
en " }{XPPEDIT 18 0 "A=0" "6#/%\"AG\"\"!" }{TEXT -1 77 ".35 we are in \+
the chaotic regime, and the Poincare section shows many points." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
359 "T := evalf(2*Pi/1.4);\nnumpts := 1000:\ntmax := 1000*T;\nde := di
ff(x(t),t$2)+0.1*diff(x(t),t)-x(t)+x(t)^3=0.35*cos(1.4*t):\nic := x(0)
=0,D(x)(0)=0:\nsol := dsolve(\{de,ic\},x(t),numeric,range=0..tmax,maxf
un=0):\npts := [seq(subs(\{op(2..3,sol(i*T))\},[x(t),diff(x(t),t)]),i=
0..numpts)]:\nplot(pts,style=point,color=COLOR(RGB,.3,0,.6),symbol=poi
nt,labels=[`x`,`dx/dt`]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"TG$\"
+2&*)z[%!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tmaxG$\"+2&*)z[%!\"
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iagram is, not surprisingly, called a strange attractor. It is the lim
iting set of points to which the trajectory tends (after the initial t
ransient) every period of the driving force. Notice the that the struc
ture is complicated but not random. One can blow up regions of it and \+
see the same patterns appearing on finer scales. The strange attractor
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