{VERSION 6 0 "SUN SPARC SOLARIS" "6.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 
0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 
0 0 1 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 
{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 
0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "Using the general theory f
or fixed-period straining we find an asymptotic approximation" }}
{PARA 0 "" 0 "" {TEXT -1 14 "to the problem" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "diff(y(t),t$2)+omega(epsilon*
t)^2*y(t)=epsilon*f(y(t),diff(y(t),t))" "6#/,&-%%diffG6$-%\"yG6#%\"tG-
%\"$G6$F+\"\"#\"\"\"*&-%&omegaG6#*&%(epsilonGF0F+F0F/-F)6#F+F0F0*&F6F0
-%\"fG6$-F)6#F+-F&6$-F)6#F+F+F0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 85 "where y(0)=a  and y'(0)=b for a,b constan
t. The assumed expansion for the solution is" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "y(t,epsilon)=Y[0](t[1],t[2])+
epsilon*Y[1](t[1],t[2])+O(epsilon^2)" "6#/-%\"yG6$%\"tG%(epsilonG,(-&%
\"YG6#\"\"!6$&F'6#\"\"\"&F'6#\"\"#F2*&F(F2-&F,6#F26$&F'6#F2&F'6#F5F2F2
-%\"OG6#*$F(F5F2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 25 "where the strained times " }{XPPEDIT 18 0 "t[1]" "6#&%\"t
G6#\"\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "t[2]" "6#&%\"tG6#\"\"#
" }{TEXT -1 13 " are given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {XPPEDIT 18 0 "t[1]=int(omega(s),s=0..t)" "6#/&%\"tG6#
\"\"\"-%$intG6$-%&omegaG6#%\"sG/F.;\"\"!F%" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "t[2]=epsilon*t" "6#/&%\"tG6#
\"\"#*&%(epsilonG\"\"\"F%F*" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 69 "The leading solution to the O(1) problem has th
e amplitude-phase form" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {XPPEDIT 18 0 "Y[0](t[1],t[2]) =A(t[2])*cos(t[1]-phi(t[2])" "6#/-
&%\"YG6#\"\"!6$&%\"tG6#\"\"\"&F+6#\"\"#*&-%\"AG6#&F+6#F0F--%$cosG6#,&&
F+6#F-F--%$phiG6#&F+6#F0!\"\"F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "O(epsilon)" "6#-%\"O
G6#%(epsilonG" }{TEXT -1 11 " problem is" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "diff(diff(Y[1],t[1]),t[1])+Y[1] \+
= F[1](Y[0])" "6#/,&-%%diffG6$-F&6$&%\"YG6#\"\"\"&%\"tG6#F-&F/6#F-F-&F
+6#F-F--&%\"FG6#F-6#&F+6#\"\"!" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "F[1](Y[0])" "6#-&%
\"FG6#\"\"\"6#&%\"YG6#\"\"!" }{TEXT -1 38 " is determined by manual ca
lculations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "F[1]=-2/omega(t[2])*diff(diff
(Y[0],t[1]),t[2])-diff(omega(t[2]),t[2])/omega(t[2])^2*diff(Y[0],t[1])
-f(Y[0],omega*diff(Y[0],t[1]))/omega(t[2])^2" "6#/&%\"FG6#\"\"\",(*(\"
\"#F'-%&omegaG6#&%\"tG6#F*!\"\"-%%diffG6$-F36$&%\"YG6#\"\"!&F/6#F'&F/6
#F*F'F1*(-F36$-F,6#&F/6#F*&F/6#F*F'*$-F,6#&F/6#F*F*F1-F36$&F86#F:&F/6#
F'F'F1*&-%\"fG6$&F86#F:*&F,F'-F36$&F86#F:&F/6#F'F'F'*$-F,6#&F/6#F*F*F1
F1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "For
 a given a given " }{XPPEDIT 18 0 "omega" "6#%&omegaG" }{TEXT -1 67 " \+
and function f the following expressions derive and then solve the" }}
{PARA 0 "" 0 "" {TEXT -1 20 "equations for A and " }{XPPEDIT 18 0 "phi
" "6#%$phiG" }{TEXT -1 22 ". Simply define f and " }{XPPEDIT 18 0 "ome
ga" "6#%&omegaG" }{TEXT -1 21 " and then hit return." }}{PARA 0 "" 0 "
" {TEXT -1 29 "Initial conditions for A and " }{XPPEDIT 18 0 "phi" "6#
%$phiG" }{TEXT -1 36 " must be determined manually but are" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "A(0)^2=a^2+b^2/
omega(0)^2" "6#/*$-%\"AG6#\"\"!\"\"#,&*$%\"aGF)\"\"\"*&%\"bGF)*$-%&ome
gaG6#F(F)!\"\"F-" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "tan(phi(0))=b/a/omega(0)" "6#/-%$tanG6#-%$phiG6#\"\"!*(
%\"bG\"\"\"%\"aG!\"\"-%&omegaG6#F*F/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "f:=(y,dy)->-diff(d(T2),T2
)/d(T2)*dy;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$%\"yG%#dyG6\"
6$%)operatorG%&arrowGF),$*(-%%diffG6$-%\"dG6#%#T2GF5\"\"\"F2!\"\"9%F6F
7F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "omega:=T2->1/sqr
t(d(T2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&omegaGf*6#%#T2G6\"6$%)
operatorG%&arrowGF(*&\"\"\"F--%%sqrtG6#-%\"dG6#9$!\"\"F(F(F(" }}}
{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 122 "F1:=Y->-2/omega(T2)*diff(diff(Y,T1),T2)-diff(omega(T
2),T2)/omega(T2)^2*diff(Y,T1)+1/omega(T2)^2*f(Y,omega(T2)*diff(Y,T1));
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#F1Gf*6#%\"YG6\"6$%)operatorG%&a
rrowGF(,(*(\"\"#\"\"\"-%&omegaG6#%#T2G!\"\"-%%diffG6$-F66$9$%#T1GF3F/F
4*(-F66$F0F3F/F0!\"#F8F/F4*&F0F?-%\"fG6$F:*&F0F/F8F/F/F/F(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "XF1:=expand(subs(T1=psi+phi(
T2),eval(F1(A(T2)*cos(T1-phi(T2))))));" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$XF1G,(**\"\"#\"\"\"-%\"dG6#%#T2G#F(F'-%%diffG6$-%\"AGF+F,F(-%
$sinG6#%$psiGF(F(*,F'F(F)F-F1F(-%$cosGF5F(-F/6$-%$phiGF+F,F(!\"\"*&F-F
(**F)#F>F'-F/6$F)F,F(F1F(F3F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 8 "To have " }{XPPEDIT 18 0 "Y[1]" "6#
&%\"YG6#\"\"\"" }{TEXT -1 12 " bounded in " }{XPPEDIT 18 0 "t[1]" "6#&
%\"tG6#\"\"\"" }{TEXT -1 12 " we require " }{XPPEDIT 18 0 "F[1]" "6#&%
\"FG6#\"\"\"" }{TEXT -1 21 " to be orthogonal to " }{XPPEDIT 18 0 "sin
(psi)" "6#-%$sinG6#%$psiG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "cos(psi
)" "6#-%$cosG6#%$psiG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 57 "
Thus the following functions (consistency conditions) of " }{XPPEDIT 
18 0 "t[2]" "6#&%\"tG6#\"\"#" }{TEXT -1 13 " must vanish:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "Phi[
1]:=expand(simplify(eval(int(XF1*sin(psi),psi=0..2*Pi))));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>&%$PhiG6#\"\"\",&**\"\"#F'-%\"dG6#%#T2G#F'F
*-%%diffG6$-%\"AGF-F.F'%#PiGF'F'*&F/F'**F+#!\"\"F*-F16$F+F.F'F3F'F5F'F
'F'" }}}{EXCHG {PARA 12 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 54 "Phi[2]:=simplify(eval(int(XF1*cos(psi),psi=0..2*
Pi)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%$PhiG6#\"\"#,$*,F'\"\"\"-
%\"dG6#%#T2G#F*F'-%\"AGF-F*-%%diffG6$-%$phiGF-F.F*%#PiGF*!\"\"" }}}
{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 79 "Here we solve the differential equations for the slowly varying
 functions using" }}{PARA 0 "" 0 "" {TEXT -1 14 "A(0)=AIC  and " }
{XPPEDIT 18 0 "phi" "6#%$phiG" }{TEXT -1 22 "(0)=phiIC for notation" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 49 "subs(AIC=A(0),dsolve(\{Phi[1]=0,A(0)=AIC\},A(T2)));" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#/-%\"AG6#%#T2G*(-F%6#\"\"!\"\"\"-%\"dGF*#F,\"\"
%-F.F&#!\"\"F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "subs(phiI
C=phi(0),dsolve(\{Phi[2]=0,phi(0)=phiIC\},phi(T2)));" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/-%$phiG6#%#T2G-F%6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 29 "DE:=diff(y(t),t)+p(t)*y(t)=0;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#DEG/,&-%%diffG6$-%\"yG6#%\"tGF-\"\"\"*&-%\"pGF,F.F*F
.F.\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(DE,y(t))
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"tG*&%$_C1G\"\"\"-%$ex
pG6#-%$IntG6$,$-%\"pGF&!\"\"F'F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 85 "ODE:=omega(t)*diff(d(t),t)/d(t)*A(t)+2*omega(t)*diff(
A(t),t)+diff(omega(t),t)*A(t)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
$ODEG/,&*&#\"\"\"\"\"#F)*(-%\"dG6#%\"tG#!\"$F*-%%diffG6$F,F/F)-%\"AGF.
F)F)F)*(F*F)F,#!\"\"F*-F36$F5F/F)F)\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "dsolve(ODE,A(t));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
/-%\"AG6#%\"tG*&%$_C1G\"\"\"-%\"dGF&#!\"\"\"\"%" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 36 "eval(int(diff(d(t),t)/d(t),t=0..T));" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#-%$intG6$*&-%\"dG6#%\"tG!\"\"-%%diffG6$F'F*
\"\"\"/F*;\"\"!%\"TG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}}{MARK "9 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 
2 33 1 1 }