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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 37 "Maple workbook:  Differ
ence equations" }}{PARA 0 "" 0 "" {TEXT -1 25 "RL Dewar,  7 April 1999
\n\n" }{TEXT 264 6 "Notes:" }{TEXT -1 59 " \n* For ease of reading, th
e execution group input displays" }{TEXT -1 26 " (lines beginning with
 >) " }{TEXT -1 99 " have been toggled from Maple input notation to St
andard Math using the left-most button (x) in the" }{TEXT -1 16 " exec
ution group" }{TEXT -1 130 " toolbar that appears immediately above th
e Maple desktop window when inputting Maple commands (replacing the te
xt input toolbar)." }}{PARA 0 "" 0 "" {TEXT -1 163 "   To see the Mapl
e commands in their normal, line editor form, put the cursor in a line
 beginning with > and click on the x button in the execution group too
lbar." }}{PARA 0 "" 0 "" {TEXT -1 59 "\n* For further discussion of th
is problem, see Chapter 5.4." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 257 17 
"Problem 4.7.2 (a)" }{TEXT -1 43 "\nAction for piecewise-linear trial \+
function" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Statement of part (a)
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 32 376 "Using the harmonic os
cillator Lagrangian $L = \\half m(\\dot\{x\}^2 -\n\\omega_0^2 x^2)$\n
\n\\noindent(a) calculate an approximate action integral $S$ using \nt
he piecewise-linear trial function\n\\begin\{displaymath\}\n\011x(t) =
\n\011\\frac\{1\}\{\\Delta t\}\\left[(t_\{n+1\}-t)x_n + (t-t_n)x_\{n+1
\}\\right]\n\\end\{displaymath\}\nfor $t$ in each range $t_n \\equiv n
\\Delta t < t < t_\{n+1\} \\equiv \n(n+1)\\Delta t$." }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 36 "Calculation of action integral from " }
{XPPEDIT 18 0 "t[n];" "6#&%\"tG6#%\"nG" }{TEXT -1 4 " to " }{XPPEDIT 
18 0 "t[n+1];" "6#&%\"tG6#,&%\"nG\"\"\"\"\"\"F(" }}{PARA 0 "" 0 "" 
{TEXT -1 45 "Define harmonic oscillator Lagrangian (using " }{TEXT 
258 1 "w" }{TEXT -1 12 " instead of " }{XPPEDIT 18 0 "omega[0];" "6#&%
&omegaG6#\"\"!" }{TEXT -1 2 "):" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 
19 1 "restart;\nL := t -> (m/2)*(diff(x(t),t)^2 - w^2*x(t)^2);" "6#C$%
(restartG>%\"LGR6#%\"tG7\"6$%)operatorG%&arrowG6\"*(%\"mG\"\"\"\"\"#!
\"\",&*$-%%diffG6$-%\"xG6#F)F)\"\"#F1*&%\"wG\"\"#-F:6#F)\"\"#F3F1F.F.F
." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LGR6#%\"tG6\"6$%)operatorG%&a
rrowGF(,$*&%\"mG\"\"\",&*$)-%%diffG6$-%\"xG6#9$F9\"\"#\"\"\"F/*&)%\"wG
F:F;)F6F:F;!\"\"F/#F/F:F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 29 "Define \+
linear trial function " }{TEXT 259 1 "x" }{TEXT -1 1 "(" }{TEXT 260 1 
"t" }{TEXT -1 14 ") on interval " }{XPPEDIT 18 0 "t[n] = n*h;" "6#/&%
\"tG6#%\"nG*&F'\"\"\"%\"hGF)" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "t[n+1
] = (n+1)*h;" "6#/&%\"tG6#,&%\"nG\"\"\"\"\"\"F)*&,&F(F)\"\"\"F)F)%\"hG
F)" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "x := t -> (x[n+1]*(t-n*h
) + x[n]*((n+1)*h - t))/h;" "6#>%\"xGR6#%\"tG7\"6$%)operatorG%&arrowG6
\"*&,&*&&F$6#,&%\"nG\"\"\"\"\"\"F4F4,&F'F4*&F3F4%\"hGF4!\"\"F4F4*&&F$6
#F3F4,&*&,&F3F4\"\"\"F4F4F8F4F4F'F9F4F4F4F8F9F,F,F," }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"xGR6#%\"tG6\"6$%)operatorG%&arrowGF(*&,&*&&F$6#,&
%\"nG\"\"\"F3F3F3,&9$F3*&F2F3%\"hGF3!\"\"F3F3*&&F$6#F2F3,&*&F1F3F7\"\"
\"F3F5F8F3F3F>F7!\"\"F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 19 "where we \+
have used " }{TEXT 261 1 "h" }{TEXT -1 12 " instead of " }{XPPEDIT 18 
0 "Delta*t;" "6#*&%&DeltaG\"\"\"%\"tGF%" }{TEXT -1 22 ".\nNow define f
unction " }{XPPEDIT 18 0 "S(x[n],x[n+1])" "6#-%\"SG6$&%\"xG6#%\"nG&F'6
#,&F)\"\"\"\"\"\"F-" }{TEXT -1 53 "  as action integral of trial funct
ion over interval " }{XPPEDIT 18 0 "t[n];" "6#&%\"tG6#%\"nG" }{TEXT 
-1 4 " to " }{XPPEDIT 18 0 "t[n+1];" "6#&%\"tG6#,&%\"nG\"\"\"\"\"\"F(
" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "S := unapply(simplify(int(
L(t), t=n*h..(n+1)*h)),\nx[n],x[n+1]);" "6#>%\"SG-%(unapplyG6%-%)simpl
ifyG6#-%$intG6$-%\"LG6#%\"tG/F1;*&%\"nG\"\"\"%\"hGF6*&,&F5F6\"\"\"F6F6
F7F6&%\"xG6#F5&F<6#,&F5F6\"\"\"F6" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>
%\"SGR6$%#y1G%#y2G6\"6$%)operatorG%&arrowGF),$*&*&%\"mG\"\"\",.*()%\"w
G\"\"#\"\"\")%\"hGF6F7)9$F6F7F1*$F:F7!\"$*&9%F1F;F1\"\"'*$)F?F6F7F=*(F
4F7F8F7FBF7F1**F4F7F8F7F?F7F;F7F1F1F7F9!\"\"#!\"\"F@F)F)F)" }}}{PARA 
0 "" 0 "" {TEXT -1 9 "Simplify " }{XPPEDIT 18 0 "S(x[n],x[n+1]);" "6#-
%\"SG6$&%\"xG6#%\"nG&F'6#,&F)\"\"\"\"\"\"F-" }{TEXT -1 30 " by collect
ing like powers of " }{TEXT 262 1 "h" }{TEXT -1 5 " and " }{TEXT 263 
2 "w " }{TEXT -1 1 ":" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "colle
ct(S(x[n],x[n+1])/m, [h,w]);" "6#-%(collectG6$*&-%\"SG6$&%\"xG6#%\"nG&
F+6#,&F-\"\"\"\"\"\"F1F1%\"mG!\"\"7$%\"hG%\"wG" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#,&*(,(*$)&%\"xG6#%\"nG\"\"#\"\"\"#!\"\"\"\"'*$)&F)6#,&F
+\"\"\"F6F6F,F-F.*&F3F6F(F6F.F6)%\"wGF,F-%\"hGF6F6*&,(F&#F6F,F7F/F1F=F
-F:!\"\"F6" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 50 "Problem 4.7.2 (b
)\nDiscrete Euler-Lagrange equation" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 21 "Statement of part (b)" }}{PARA 0 "" 0 "" {TEXT 32 342 "\\noinde
nt(b) Show that the approximate action integral from $t_\{-N\}$ \nto $
t_N$ ($N \\geq 1$ being an arbitrary integer) is stationary for $-N < \+
\nn < N$ if $x_n$ obeys the second-order difference equation\n\\begin
\{displaymath\}\n\011x_\{n-1\} - 2x_n + x_\{n+1\}\n\011= -\\frac\{\\om
ega_0^2(\\Delta t)^2\}\{6\}\n\011\\left(x_\{n-1\} + 4x_n + x_\{n+1\}\\
right)\n\\end\{displaymath\}" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 13 "
Variation of " }{XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }}{PARA 0 "" 
0 "" {TEXT -1 45 "The two terms in the total action containing " }
{XPPEDIT 18 0 "x[n];" "6#&%\"xG6#%\"nG" }{TEXT -1 6 "  are " }
{XPPEDIT 18 0 "S(x[n-1],x[n]);" "6#-%\"SG6$&%\"xG6#,&%\"nG\"\"\"\"\"\"
!\"\"&F'6#F*" }{TEXT -1 7 "  and  " }{XPPEDIT 18 0 "S(x[n],x[n+1])" "6
#-%\"SG6$&%\"xG6#%\"nG&F'6#,&F)\"\"\"\"\"\"F-" }{TEXT -1 62 ". The gra
dient of the action with respect to the vector \{..., " }{XPPEDIT 18 
0 "x[n-1],x[n],x[n+1];" "6%&%\"xG6#,&%\"nG\"\"\"\"\"\"!\"\"&F$6#F'&F$6
#,&F'F(\"\"\"F(" }{TEXT -1 15 " ,...\}  is thus" }}{EXCHG {PARA 0 "> \+
" 0 "" {XPPEDIT 19 1 "gradS_n := diff(S(x[n-1],x[n]) + S(x[n],x[n+1]),
 x[n]);" "6#>%(gradS_nG-%%diffG6$,&-%\"SG6$&%\"xG6#,&%\"nG\"\"\"\"\"\"
!\"\"&F-6#F0F1-F*6$&F-6#F0&F-6#,&F0F1\"\"\"F1F1&F-6#F0" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%(gradS_nG,&*&*&%\"mG\"\"\",*&%\"xG6#,&%\"nGF)!\"
\"F)\"\"'&F,6#F/!\"'*()%\"wG\"\"#\"\"\")%\"hGF8F9F2F)F8*(F6F9F:F9F+F)F
)F)F9F;!\"\"#F0F1*&*&F(F9,*F5F8F2F4&F,6#,&F/F)F)F)F1*(F6F9F:F9FBF)F)F)
F9F;F=F>" }}}{PARA 0 "" 0 "" {TEXT -1 107 "The finite-difference appro
ximation to the equation of motion is obtained by setting this gradien
t to zero:" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "collect(gradS_n/
m = 0,[h,w,m]);" "6#-%(collectG6$/*&%(gradS_nG\"\"\"%\"mG!\"\"\"\"!7%%
\"hG%\"wGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(,(&%\"xG6#%\"nG#!
\"#\"\"$&F(6#,&F*\"\"\"!\"\"F1#F2\"\"'&F(6#,&F*F1F1F1F3F1)%\"wG\"\"#\"
\"\"%\"hGF1F1*&,(F.F2F'F:F5F2F;F<!\"\"F1\"\"!" }}}}}{SECT 1 {PARA 3 "
" 0 "" {TEXT -1 57 "Problem 4.7.2 (c)\nDiscrete Hamiltonian equation o
f motion" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 21 "Statement of part (c)
" }}{PARA 0 "" 0 "" {TEXT 32 612 "\\noindent(c) Using the approximate \+
action integral \n$\\bar\{S\}(x_n,x_\{n+1\})$ evaluated over the range
 $t_n < t < t_\{n+1\}$, \nas a type 1 generating function, $F_1(q,Q) =
 -\\bar\{S\}(x_n,x_\{n+1\})$, \nwith $q=x_n$ and $Q=x_\{n+1\}$, find t
he linear canonical transformation \nfrom $(x_n,p_n \\equiv p)$ to $(x
_\{n+1\},p_\{n+1\} \\equiv P)$.  Represent \nthis linear transformatio
n as a matrix and show that the determinant \nis unity.  (It would be \+
advisable to use Maple or Mathematica for the \nalgebra.)\n\nThis prov
ides a discrete-time dynamical system approximation to the \ntrue cont
inuous-time system for the harmonic oscillator." }}}{SECT 0 {PARA 4 "
" 0 "" {TEXT -1 36 "Action as Type-1 generating function" }}{PARA 0 "
" 0 "" {TEXT -1 124 "Solve the Type-1 canonical transformation (CT) eq
uations to give the new canonical variables explicitly in terms of the
 old:" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "CT := solve(\{p+diff(
S(q,Q),q) = 0, P-diff(S(q,Q),Q) = 0\},\{P, Q\});" "6#>%#CTG-%&solveG6$
<$/,&%\"pG\"\"\"-%%diffG6$-%\"SG6$%\"qG%\"QGF3F,\"\"!/,&%\"PGF,-F.6$-F
16$F3F4F4!\"\"F5<$F8F4" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#CTG<$/%\"
PG,$*&,***%\"mG\"\"\")%\"wG\"\"#\"\"\"%\"hGF-%\"qGF-!#7%\"pG\"#7*(F.F1
)F2F0F1F5F-!\"%**)F/\"\"%F1)F2\"\"$F1F,F1F3F1F-F1,&\"\"'F-*&F.F1F8F1F-
!\"\"#F-F0/%\"QG,$*&,(*&F5F1F2F1!\"$**F,F1F.F1F8F1F3F1F-*&F,F1F3F1FJF1
*&F,\"\"\"F?\"\"\"FB!\"#" }}}{PARA 0 "" 0 "" {TEXT -1 111 "Define the \+
Jacobian matrix of the transformation, which, given that the CT in thi
s case is linear, is constant:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 111 "M[1,1]:=diff(eval(Q,CT),q);\nM[1,2]:=diff(eval(Q,CT),p);\nM[2,1
]:=diff(eval(P,CT),q);\nM[2,2]:=diff(eval(P,CT),p);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>&%\"MG6$\"\"\"F',$*&,&*(%\"mGF')%\"wG\"\"#\"\"\")%\"
hGF/F0F'F,!\"$F0*&F,\"\"\",&\"\"'F'*&F-F0F1F0F'\"\"\"!\"\"!\"#" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"MG6$\"\"\"\"\"#,$*&%\"hG\"\"\"*&%
\"mG\"\"\",&\"\"'F'*&)%\"wGF(F,)F+F(F,F'\"\"\"!\"\"F1" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>&%\"MG6$\"\"#\"\"\",$*&,&*(%\"mGF()%\"wGF'\"\"\"%
\"hGF(!#7*()F/\"\"%F0)F1\"\"$F0F-F0F(F0,&\"\"'F(*&F.F0)F1F'F0F(!\"\"#F
(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"MG6$\"\"#F',$*&,&\"#7\"\"
\"*&)%\"wGF'\"\"\")%\"hGF'F0!\"%F0,&\"\"'F,F-F,!\"\"#F,F'" }}}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "J := matrix([[M[1,1], M[1,2]], [M[2,1
], M[2,2]]]):" "6#>%\"JG-%'matrixG6#7$7$&%\"MG6$\"\"\"\"\"\"&F+6$\"\"
\"\"\"#7$&F+6$\"\"#\"\"\"&F+6$\"\"#\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 
32 "Load the linear algebra package:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "with(linalg):" }}{PARA 7 "" 1 "" {TEXT -1 32 "Warning
, new definition for norm" }}{PARA 7 "" 1 "" {TEXT -1 33 "Warning, new
 definition for trace" }}}{PARA 0 "" 0 "" {TEXT -1 76 "Calculate the d
eterminant of the Jacobian matrix to verify that it is unity:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "det(J);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#\"\"\"" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 50 "Probl
em 4.7.2 (d)\nIteration of area-preserving map" }}{SECT 1 {PARA 4 "" 
0 "" {TEXT -1 21 "Statement of part (d)" }}{PARA 0 "" 0 "" {TEXT 32 
250 "\\noindent(d) Iterate the map obtained in (c) 100 times taking th
e \ninitial point as $x_0 = 1$, $p_0 = 0$ and plot the result. Use uni
ts \nsuch that $m = \\omega_0 = 1$ and use three timesteps: $\\Delta t
 = \n0.1$, $\\Delta t = 3.45$ and $\\Delta t = 3.47$." }}}{SECT 0 
{PARA 4 "" 0 "" {TEXT -1 9 "Iteration" }}{PARA 0 "" 0 "" {TEXT -1 27 "
First define the procedure " }{XPPEDIT 18 0 "plot(x0,p0,n);" "6#-%%plo
tG6%%#x0G%#p0G%\"nG" }{TEXT -1 26 " , which iterates the map " }
{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 13 " times given " }{XPPEDIT 
18 0 "x0,p0;" "6$%#x0G%#p0G" }{TEXT -1 97 "  as initial conditions, an
d returns a list containing the initial point in phase space, and the \+
" }{XPPEDIT 18 0 "n;" "6#%\"nG" }{TEXT -1 10 " iterates:" }}{EXCHG 
{PARA 0 "> " 0 "" {XPPEDIT 19 1 "orbit := proc(x0,p0,n)\n  local i,x,p
;\n  x[0] := x0; p[0]:=p0;\n  for i from 1 to n do\n    x[i] := M[1,1]
*x[i-1] + M[1,2]*p[i-1];\n    p[i] := M[2,1]*x[i-1] + M[2,2]*p[i-1];\n
  od;\n  RETURN([[x[j],p[j]]$j=0..n])\nend:" "6#>%&orbitGR6%%#x0G%#p0G
%\"nG7%%\"iG%\"xG%\"pG6\"F.C&>&F,6#\"\"!F'>&F-6#F3F(?(F+\"\"\"\"\"\"F)
%%trueGC$>&F,6#F+,&*&&%\"MG6$\"\"\"\"\"\"F9&F,6#,&F+F9\"\"\"!\"\"F9F9*
&&FB6$\"\"\"\"\"#F9&F-6#,&F+F9\"\"\"FJF9F9>&F-6#F+,&*&&FB6$\"\"#\"\"\"
F9&F,6#,&F+F9\"\"\"FJF9F9*&&FB6$\"\"#\"\"#F9&F-6#,&F+F9\"\"\"FJF9F9-%'
RETURNG6#7#-%\"$G6$7$&F,6#%\"jG&F-6#F^p/F^p;F3F)F.F.F." }}}{PARA 0 "" 
0 "" {TEXT -1 71 "As an example, iterate the map 10 times, starting fr
om the point [1,0]:" }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 1 "m := 1;
 w := 1; h := 0.1;\norbit(1,0,10);" "6#C&>%\"mG\"\"\">%\"wG\"\"\">%\"h
G$\"\"\"!\"\"-%&orbitG6%\"\"\"\"\"!\"#5" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"mG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"\"" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#7-7$\"\"\"\"\"!7$$\"+%>$3]**!#5$!+&fT](**!#67$$\"+:6$
3!)*F*$!+v)\\])>F*7$$\"++Qt`&*F*$!+?\"yF&HF*7$$\"+6\"e7@*F*$!+:x-\"*QF
*7$$\"+2J#ox)F*$!+w<V!z%F*7$$\"+3fwa#)F*$!+F7,UcF*7$$\"+,%)H]wF*$!+VWE
PkF*7$$\"+._XppF*$!+C@DorF*7$$\"+<M.>iF*$!+blnFyF*7$$\"+vZ_1aF*$!+lW&*
3%)F*" }}}{PARA 0 "" 0 "" {TEXT -1 37 "Now plot 100 iterates using tim
estep " }{XPPEDIT 18 0 "h = .1" "6#/%\"hG$\"\"\"!\"\"" }{TEXT -1 90 " \+
to show the orbit in phase space is essentially a circle, as in the co
ntinuous-time case:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 167 "m :=
 1; w := 1; h := 0.1;\nplot(orbit(1,0,100), x=-1..1, style=point,symbo
l=circle,\nlabels=[x,p],axesfont=[SYMBOL,12],labelfont=[TIMES,ITALIC,1
4],\nscaling=CONSTRAINED);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"
\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"\"" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#>%\"hG$\"\"\"!\"\"" }}{PARA 13 "" 1 "" {INLPLOT "6)-
%'CURVESG6$7aq7$$\"\"\"\"\"!F*7$$\"1+++%>$3]**!#;$!1+++&fT](**!#<7$$\"
1+++:6$3!)*F.$!1+++v)\\])>F.7$$\"1++++Qt`&*F.$!1+++?\"yF&HF.7$$\"1+++6
\"e7@*F.$!1+++:x-\"*QF.7$$\"1+++2J#ox)F.$!1+++w<V!z%F.7$$\"1+++3fwa#)F
.$!1+++F7,UcF.7$$\"1+++,%)H]wF.$!1+++VWEPkF.7$$\"1+++._XppF.$!1+++C@Do
rF.7$$\"1*****pTL!>iF.$!1+++blnFyF.7$$\"1+++vZ_1aF.$!1+++lW&*3%)F.7$$
\"1+++W3/SXF.$!1+++YFG1*)F.7$$\"1+++Y?BGOF.$!1+++\"R'p9$*F.7$$\"1+++(H
,-o#F.$!1+++f!=,j*F.7$$\"1+++JJT0<F.$!1+++!y)R\\)*F.7$$\"1,+++@*f8(F1$
!1+++(R\\.(**F.7$$!1+++\"H)Q`GF1$!1+++;Cw\"***F.7$$!1+++G#G9G\"F.$!1++
+jSU8**F.7$$!1+++mYskAF.$!1+++=k6O(*F.7$$!1+++S:TDKF.$!1+++4'4;Y*F.7$$
!1+++mz*Q:%F.$!1+++MTk#4*F.7$$!1+++GX\"4/&F.$!1+++5N!Hj)F.7$$!1+++@egx
eF.$!1+++$\\xp3)F.7$$!1+++`)=cl'F.$!1******fiJguF.7$$!1+++Qk=ntF.$!1++
+'*f<fnF.7$$!1+++TZ?0!)F.$!1+++Pkb!*fF.7$$!1+++&>/Lc)F.$!1+++!*48i^F.7
$$!1+++\"48f.*F.$!1+++E,<#G%F.7$$!1*****R?8$=%*F.$!1+++7)e%fLF.7$$!1++
+'*oo1(*F.$!1+++2)3KS#F.7$$!1+++F_:)*)*F.$!1+++,n'HU\"F.7$$!1+++-n!3**
*F.$!1+++]g=&G%F17$$!1+++Yjr$)**F.$\"1+++sa2-dF17$$!1+++X\\&p()*F.$\"1
+++76Cj:F.7$$!1+++P$)er'*F.$\"1+++w#o1a#F.7$$!1+++qnmp$*F.$\"1+++K5t#
\\$F.7$$!1+++OW?u*)F.$\"1+++$fC*4WF.7$$!1,++z$\\\"*[)F.$\"1+++$G#4$G&F
.7$$!1+++zSM>zF.$\"1+++cp^.hF.7$$!1+++3rZqsF.$\"1*****\\,3I'oF.7$$!1++
+Qj-\\lF.$\"1+++)=$)Rb(F.7$$!1+++uU>idF.$\"1+++=Uap\")F.7$$!1+++Wh$y\"
\\F.$\"1+++Rda.()F.7$$!1+++![\"QCSF.$\"1******>ml]\"*F.7$$!1+++j*\\24$
F.$\"1+++$>8k]*F.7$$!1+++tBEE@F.$\"1+++5QEn(*F.7$$!1+++QvaS6F.$\"1+++1
VgI**F.7$$!1+++^?YM9F1$\"1+++$*R![***F.7$$\"1******H?(3b)F1$\"1+++W>Af
**F.7$$\"1+++nR3X=F.$\"1+++'Q8U#)*F.7$$\"1+++\"fgm\"GF.$\"1+++fh7\"f*F
.7$$\"1+++wu6gPF.$\"1+++dsGi#*F.7$$\"1+++^d.mYF.$\"1+++'fz4%))F.7$$\"1
+++18PDbF.$\"1+++W#49L)F.7$$\"1+++%3X&HjF.$\"1+++DMmQxF.7$$\"1+++G(G02
(F.$\"1+++N(f'oqF.7$$\"1+++JZ#4u(F.$\"1+++iq3GjF.7$$\"1+++(GSSL)F.$\"1
+++6)QV_&F.7$$\"1+++5T&R%))F.$\"1+++#4Ram%F.7$$\"1+++Ybdl#*F.$\"1+++5E
'*fPF.7$$\"1+++&R&p%f*F.$\"1+++j!\\p\"GF.7$$\"1+++Oz-G)*F.$\"1+++'*G\"
e%=F.7$$\"1+++EPCj**F.$\"1+++MJ\\i&)F17$$\"1+++dG**)***F.$!1+++b^i=9F1
7$$\"1+++J%=\\$**F.$!1+++!3e&Q6F.7$$\"1+++F,mr(*F.$!1+++3q)Q7#F.7$$\"1
+++7y%3^*F.$!1+++0C,)3$F.7$$\"1+++p_3b\"*F.$!1+++f!48-%F.7$$\"1+++/U#z
q)F.$!1+++L&fW\"\\F.7$$\"1+++)yGQ<)F.$!1+++#=Z&edF.7$$\"1+++)3J\"evF.$
!1+++w^9XlF.7$$\"1+++Py(p'oF.$!1+++A1SmsF.7$$\"1+++m!ps5'F.$!1+++oH6:z
F.7$$\"1+++B#*e'G&F.$!1*****H)e![[)F.7$$\"1+++^988WF.$!1+++<>zp*)F.7$$
\"1+++%z:c\\$F.$!1+++!GH_O*F.7$$\"1+++U@?VDF.$!1+++y,<n'*F.7$$\"1+++&y
)Rl:F.$!1+++D-gs)*F.7$$\"1+++p[n>dF1$!1+++R&o%z**F.7$$!1+++)oR<F%F1$!1
+++:#3n)**F.7$$!1+++$z]?U\"F.$!1+++xpC%*)*F.7$$!1+++r2t-CF.$!1+++**y+.
(*F.7$$!1+++>MUfLF.$!1+++!>+\\T*F.7$$!1+++Ixd#G%F.$!1+++L,!G.*F.7$$!1+
++Av(H;&F.$!1+++rB_g&)F.7$$!1+++WM$=*fF.$!1+++B=y-!)F.7$$!1+++I2(3w'F.
$!1+++:m9ltF.7$$!1+++.=TiuF.$!1+++*[#)Rl'F.7$$!1+++CHX*3)F.$!1*****HD*
QweF.7$$!1+++;TtN')F.$!1+++,*H,/&F.7$$!1+++c;!e4*F.$!1+++8Jb`TF.7$$!1+
++BD1l%*F.$!1+++/*4bA$F.7$$!1+++a-$)R(*F.$!1+++l_ElAF.7$$!1,++V<O<**F.
$!1+++mcS#G\"F.7$$!1+++HY)e***F.$!1+++![Lu'GF17$$!1+++!*\\hu**F.$\"1++
+Fj\"y6(F17$$!1+++o^w`)*F.$\"1+++S1?.<F.7$$!1+++];aM'*F.$\"1+++\")fhxE
F.7$$!1+++CI8>$*F.$\"1+++:(*HDOF.7$$!1,++G\")o5*)F.$\"1+++s2zOXF.7$$!1
+++:YG8%)F.$\"1+++4%*)HS&F.-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%'SYMBOLG6
#%'CIRCLEG-%&STYLEG6#%&POINTG-%+AXESLABELSG6%%\"xG%\"pG-%%FONTG6%%&TIM
ESG%'ITALICG\"#9-%*AXESTICKSG6%%(DEFAULTGFh\\m-F`\\m6$Fc[m\"#7-%(SCALI
NGG6#%,CONSTRAINEDG-%%VIEWG6$;$Fa[mF*F(Fh\\m" 2 375 281 281 5 4 1 0 2 
9 1 4 1 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 1 0 0 0 0 0 1 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 37 "Now plot 1
00 iterates using timestep " }{XPPEDIT 18 0 "h = 3.45;" "6#/%\"hG$\"$X
$!\"#" }{TEXT -1 47 " to show the orbit is a very flattened ellipse:" 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "m := 1; w := 1; h := 3.45
;\nplot(orbit(1,0,100), x=-1..1, style=point,symbol=circle,\nlabels=[x
,p],axesfont=[SYMBOL,12],labelfont=[TIMES,ITALIC,14]);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"mG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"wG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"$X$!\"#" }}
{PARA 13 "" 1 "" {INLPLOT "6(-%'CURVESG6$7aq7$$\"\"\"\"\"!F*7$$!1+++K$
Qb%**!#;$!1+++?wj%R*!#=7$$\"1,++Wlu#y*F.$\"1+++sXpo=!#<7$$!1+++>yR8&*F
.$!1+++P4dxFF77$$\"1+++#*fUS\"*F.$\"1+++tI>cOF77$$!1+++'f$*ym)F.$!1+++
O2*\\\\%F77$$\"1+++8w%45)F.$\"1+++hu#[G&F77$$!1+++2MwXuF.$!1+++$3+r,'F
77$$\"1+++juZ4nF.$\"1+++WC$Qo'F77$$!1+++`'4,!fF.$!1,++uAwxsF77$$\"1+++
\")eZE]F.$\"1+++!H?Cz(F77$$!1+++d?4)4%F.$!1+++m1?A#)F77$$\"1+++[/2DJF.
$\"1,++!4ACc)F77$$!1+++H&4!=@F.$!1+++e(y$4))F77$$\"1+++W&yy3\"F.$\"1**
***\\t!Qg*)F77$$!1++++B!)*e%F1$!1,++dKy8!*F77$$!1+++`V#e'**F7$\"1,++YY
+p*)F77$$\"1+++&y2#G?F.$!1+++XE`E))F77$$!1+++?7uPIF.$\"1+++2\">ze)F77$
$\"1+++ll=9SF.$!1+++\"4jdD)F77$$!1+++^!3p%\\F.$\"1+++]DoLyF77$$\"1+++m
huDeF.$!1+++E\\FEtF77$$!1+++6$G6k'F.$\"1+++sq1RnF77$$\"1+++rI<%Q(F.$!1
+++_]XygF77$$!1+++cpyY!)F.$\"1+++tWj^`F77$$\"1+++aDv@')F.$!1+++1@_mXF7
7$$!1*****z92G5*F.$\"1+++d'p;t$F77$$\"1+++K4r%[*F.$!1+++=1<cGF77$$!1++
+\"3/Lw*F.$\"1+++;7c\\>F77$$\"1+++m?bN**F.$!1+++Vlr@5F77$$!1+++0(y&***
*F.$\"1,++!H*Hu#)!#>7$$\"1+++)f'oa**F.$\"1*****4A=8d)F17$$!1+++DZO,)*F
.$!1+++s/n(y\"F77$$\"1+++>JGT&*F.$\"1+++>st)p#F77$$!1+++wYFx\"*F.$!1++
+?%3/e$F77$$\"1+++'H/Lr)F.$\"1+++>13BWF77$$!1++++dUa\")F.$!1+++\\^d<_F
77$$\"1******zjs1vF.$\"1+++6\"Q_&fF77$$!1+++$Hhsx'F.$!1+++MY.GmF77$$\"
1+++;g(R(fF.$\"1+++$RO'GsF77$$!1,++*>?c5&F.$!1+++M9]]xF77$$\"1+++lAl\"
=%F.$\"1+++Ha%z=)F77$$!1+++zj87KF.$!1+++#f.i`)F77$$\"1+++KGj2AF.$\"1++
+zD[\"z)F77$$!1+++7I3z6F.$!1+++1=+^*)F77$$\"1+++qA!pP\"F7$\"1+++OP-8!*
F77$$\"1+++fK-_!*F7$!1+++1G(o(*)F77$$!1+++9^BQ>F.$\"1,++%yUH%))F77$$\"
1+++**f:]HF.$!1+++![#p7')F77$$!1+++,G%*HRF.$\"1+++e)H')G)F77$$\"1+++cL
#p'[F.$!1+++:ZGuyF77$$!1+++f<*3v&F.$\"1+++P-<utF77$$\"1+++J&>Ad'F.$!1+
++PQt$z'F77$$!1+++'\\g>K(F.$\"1+++ByHRhF77$$\"1+++E#[>*zF.$!1+++I0*zT&
F77$$!1+++\"*\\)[d)F.$\"1+++w'owj%F77$$\"1+++\\7Uk!*F.$!1+++*yJo!QF77$
$!1+++4[Ab%*F.$\"1+++?'HX$HF77$$\"1+++2*QIu*F.$!1+++wNEI?F77$$!1+++w&G
Z#**F.$\"1+++DK)Q5\"F77$$\"1+++#z9$)***F.$!1+++&3!za;F17$$!1+++Hg*H'**
F.$!1+++0mFZxF17$$\"1+++\"*p:>)*F.$\"1+++Qd\\1<F77$$!1+++?WOo&*F.$!1++
+&4w'>EF77$$\"1+++L+N8#*F.$\"1+++m?K/NF77$$!1+++l2)zv)F.$!1+++$z(z]VF7
7$$\"1+++]m@2#)F.$\"1+++!=$))\\^F77$$!1+++%zcqc(F.$!1+++AV(G*eF77$$\"1
+++LSZWoF.$\"1+++6$y;d'F77$$!1+++y*Qt/'F.$!1+++*Q,*yrF77$$\"1+++*GMV=&
F.$\"1+++![Hzq(F77$$!1+++.,'[E%F.$!1+++L-+`\")F77$$\"1+++Q;$*)H$F.$\"1
,++)yl#4&)F77$$!1+++4,2(H#F.$!1+++$eXGx)F77$$\"1+++E\")=q7F.$\"1+++='o
3%*)F77$$!1+++5$3ZH#F7$!1+++JZ]6!*F77$$!1+++u$fu8)F7$\"1+++HX)R)*)F77$
$\"1+++9\"*4[=F.$!1+++rxge))F77$$!1+++z@KiGF.$\"1+++.,uO')F77$$\"1+++x
yOXQF.$!1+++#=)z?$)F77$$!1+++X&Gly%F.$\"1*****HNBU\"zF77$$\"1+++\\Fbvc
F.$!1+++kTW@uF77$$!1+++Spv-lF.$\"1+++H\"Gy%oF77$$\"1+++M48fsF.$!1+++nK
i*>'F77$$!1+++`gVOzF.$\"1+++S+*Q[&F77$$\"1+++j[HF&)F.$!1+++bWU3ZF77$$!
1+++N:FD!*F.$\"1+++SJn\")QF77$$\"1+++S>%\\U*F.$!1+++Z8k7IF77$$!1+++MF&
>s*F.$\"1+++g[z5@F77$$\"1+++x(oI\"**F.$!1+++\"*o&f=\"F77$$!1+++p$3i***
F.$\"1+++M3,#[#F17$$\"1*****f#fYq**F.$\"1+++I@eApF17$$!1+++V=7O)*F.$!1
+++-s<D;F77$$\"1+++T%RYf*F.$\"1+++>URSDF77$$!1+++F!\\'[#*F.$!1+++:/%zU
$F77$$\"1+++W#>>!))F.$\"1+++U$[\"yUF77$$!1,++@gJf#)F.$!1+++^sv\"3&F77$
$\"1+++t&\\ni(F.$\"1+++lR,IeF77$$!1+++E+66pF.$!1+++B#oZ^'F77$$\"1+++jB
>?hF.$\"1+++a9cGrF77$$!1+++C:hi_F.$!1+++:!3Zm(F7-%'COLOURG6&%$RGBG$\"#
5!\"\"F*F*-%'SYMBOLG6#%'CIRCLEG-%&STYLEG6#%&POINTG-%+AXESLABELSG6%%\"x
G%\"pG-%%FONTG6%%&TIMESG%'ITALICG\"#9-%*AXESTICKSG6%%(DEFAULTGFj\\m-Fb
\\m6$Fe[m\"#7-%%VIEWG6$;$Fc[mF*F(Fj\\m" 2 375 281 281 5 4 1 0 2 9 1 4 
2 1.000000 45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 
0 0 0 0 17368 0 0 0 0 0 0 }}}{PARA 0 "" 0 "" {TEXT -1 37 "Now plot 100
 iterates using timestep " }{XPPEDIT 18 0 "h = 3.47;" "6#/%\"hG$\"$Z$!
\"#" }{TEXT -1 73 " to show the orbit is unstable, diverging radially \+
along a diagonal line:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 161 "m
 := 1; w := 1; h := 3.47;\nplot(orbit(1,0,100), x=-160..160,y=-10..10,
 style=point,symbol=circle,\nlabels=[x,p],axesfont=[SYMBOL,12],labelfo
nt=[TIMES,ITALIC,14]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"hG$\"$Z$!\"#" }}{PARA 13 "" 1 "" {INLPLOT "6(-%'CUR
VESG6$7aq7$$\"\"\"\"\"!F*7$$!1+++rqE-5!#:$\"1+++p!oL$R!#=7$$\"1+++j&y!
45F.$!1+++)eqX)yF17$$!1+++h`Y?5F.$\"1+++(G_r=\"!#<7$$\"1+++$4zk.\"F.$!
1+++L-B\"f\"F<7$$!1+++pB>d5F.$\"1+++UI_-?F<7$$\"1+++/\"*p#3\"F.$!1+++v
b*GU#F<7$$!1+++^\\686F.$\"1+++fQDaGF<7$$\"1+++>yd[6F.$!1+++OPb)H$F<7$$
!1+++,&[#*=\"F.$\"1+++C(4yv$F<7$$\"1+++09JN7F.$!1+++`T5MUF<7$$!1+++(Qv
pG\"F.$\"1+++7mfHZF<7$$\"1+++)puWM\"F.$!1+++KP`Y_F<7$$!1+++[+239F.$\"1
+++\"Qfsy&F<7$$\"1+++(y\\!y9F.$!1+++'GDUN'F<7$$!1+++87ta:F.$\"1******
\\@+]pF<7$$\"1+++6?YQ;F.$!1+++48HxvF<7$$!1+++H=iH<F.$\"1+++#)p$*Q#)F<7
$$\"1+++(*RiG=F.$!1+++k\"Rz$*)F<7$$!1+++0u\"f$>F.$\"1+++JsYx'*F<7$$\"1
+++Q&))>0#F.$!1+++Ju3Y5!#;7$$!1+++yOOx@F.$\"1+++Z7<H6F\\r7$$\"1+++%H6E
J#F.$!1+++')[P<7F\\r7$$!1+++=YMeCF.$\"1+++v#)468F\\r7$$\"1+++DWA:EF.$!
1+++qjw59F\\r7$$!1+++I?'Ry#F.$\"1+++\"3Jo^\"F\\r7$$\"1+++:DKlHF.$!1+++
>LxH;F\\r7$$!1+++$>G,;$F.$\"1+++!=0,v\"F\\r7$$\"1+++SBEpLF.$!1+++mAPy=
F\\r7$$!1++++Kn$f$F.$\"1+++eh::?F\\r7$$\"1+++%GyV$QF.$!1+++aq2h@F\\r7$
$!1+++%)*oC4%F.$\"1+++!e'z;BF\\r7$$\"1+++>b6pVF.$!1+++!z?I[#F\\r7$$!1+
++VAdlYF.$\"1+++pL]gEF\\r7$$\"1+++ML=$)\\F.$!1+++[!\\+&GF\\r7$$!1+++%)
))QB`F.$\"1+++as^_IF\\r7$$\"1+++O9t(o&F.$!1+++1g#)oKF\\r7$$!1+++pH'y2'
F.$\"1+++zg&**\\$F\\r7$$\"1+++&R_b\\'F.$!1+++\\a&pu$F\\r7$$!1+++wNpUpF
.$\"1+++ZS%4,%F\\r7$$\"1+++:RJ@uF.$!1+++M)=JH%F\\r7$$!1+++XNeLzF.$\"1+
++J#fZf%F\\r7$$\"1+++o^#=[)F.$!1+++=HB<\\F\\r7$$!1*******eC&o!*F.$\"1+
++P?+i_F\\r7$$\"1+++))>M'p*F.$!1+++?)H1j&F\\r7$$!1+++)R7o.\"!#9$\"1+++
ywyCgF\\r7$$\"1+++^;p36Fhy$!1+++xFEYkF\\r7$$!1+++qyf&=\"Fhy$\"1+++Yh'p
*oF\\r7$$\"1+++e(zyE\"Fhy$!1+++R8%*ytF\\r7$$!1+++#R5fN\"Fhy$\"1+++*ptV
*yF\\r7$$\"1+++;*)3]9Fhy$!1+++p-gX%)F\\r7$$!1+++]B%3b\"Fhy$\"1+++)Q?^.
*F\\r7$$\"1+++Bvie;Fhy$!1+++Dqgl'*F\\r7$$!1+++[J$Rx\"Fhy$\"1+++!*=*R.
\"F.7$$\"1+++P?G(*=Fhy$!1+++V8616F.7$$!1+++sMBH?Fhy$\"1+++igC$=\"F.7$$
\"1+++SdQq@Fhy$!1+++'yXdE\"F.7$$!1+++X)y8K#Fhy$\"1+++#e%)RN\"F.7$$\"1+
++6u*G[#Fhy$!1+++NDO[9F.7$$!1+++'ytcl#Fhy$\"1+++qvI\\:F.7$$\"1+++j8\\S
GFhy$!1+++&Qxsl\"F.7$$!1+++J\")=QIFhy$\"1+++I:ws<F.7$$\"1+++u/m\\KFhy$
!1+++GOG'*=F.7$$!1+++Rs'eZ$Fhy$\"1+++XPSG?F.7$$\"1+++!3Myr$Fhy$!1+++K4
sp@F.7$$!1+++3\"em(RFhy$\"1+++Uf(3K#F.7$$\"1+++nG^`UFhy$!1+++LTb#[#F.7
$$!1+++eOl\\XFhy$\"1+++w&)[bEF.7$$\"1+++CKUm[Fhy$!1+++\"Qj/%GF.7$$!1++
+VyD0_Fhy$\"1+++]sJQIF.7$$\"1+++RQpnbFhy$!1+++\"GZ*\\KF.7$$!1+++\\XPbf
Fhy$\"1+++MIJwMF.7$$\"1+++sx0qjFhy$!1+++$)3W=PF.7$$!1,++TPi8oFhy$\"1++
+q'Gu(RF.7$$\"1,++ZO3)G(Fhy$!1+++%o]WD%F.7$$!1+++e()e&z(Fhy$\"1+++\")H
w]XF.7$$\"1+++u,WQ$)Fhy$!1+++$34x'[F.7$$!1+++h#*4>*)Fhy$\"1+++ogs1_F.7
$$\"1+++4))>S&*Fhy$!1+++)3^$pbF.7$$!1+++(\\b/-\"!#8$\"1+++O$Gs&fF.7$$
\"1+++.!=:4\"Fcdl$!1+++.l6sjF.7$$!1+++2'Hv;\"Fcdl$\"1+++]n*e\"oF.7$$\"
1+++c\\$)[7Fcdl$!1+++S7e!H(F.7$$!1++++F!eL\"Fcdl$\"1+++fAK)z(F.7$$\"1+
++jr#)G9Fcdl$!1+++y>UT$)F.7$$!1+++J,LG:Fcdl$\"1*****z)GMA*)F.7$$\"1+++
kFwM;Fcdl$!1+++n*=Pa*F.7$$!1+++Vwg[<Fcdl$\"1+++in$3-\"Fhy7$$\"1+++e4Qq
=Fcdl$!1+++Y-$>4\"Fhy7$$!1+++X[j+?Fcdl$\"1+++'puz;\"Fhy7$$\"1+++%*)f*R
@Fcdl$!1+++5\\J\\7Fhy7$$!1+++Dy)*)G#Fcdl$\"1+++&p>jL\"Fhy7$$\"1+++_VR[
CFcdl$!1+++VNQH9Fhy7$$!1+++YA!*=EFcdl$\"1+++>%G*G:Fhy7$$\"1+++9YG,GFcd
l$!1+++tcSN;Fhy7$$!1+++/%oj*HFcdl$\"1+++*3)H\\<Fhy7$$\"1+++a\"Q]?$Fcdl
$!1+++q?7r=Fhy7$$!1+++-+CGMFcdl$\"1+++%)*H9+#Fhy7$$\"1+++wf)pm$Fcdl$!1
+++mE\"39#Fhy7$$!1+++&eeB#RFcdl$\"1+++'4-**G#Fhy7$$\"1+++Cd^&>%Fcdl$!1
+++nUP\\CFhy-%'COLOURG6&%$RGBG$\"#5!\"\"F*F*-%'SYMBOLG6#%'CIRCLEG-%&ST
YLEG6#%&POINTG-%+AXESLABELSG6%%\"xG%\"pG-%%FONTG6%%&TIMESG%'ITALICG\"#
9-%*AXESTICKSG6%%(DEFAULTGF\\]m-Fd\\m6$Fg[m\"#7-%%VIEWG6$;$!$g\"F*$\"$
g\"F*;$!#5F*$Fd[mF*" 2 375 281 281 5 4 1 0 2 9 1 4 2 1.000000 
45.000000 45.000000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 3 0 
0 0 0 0 0 }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 59 "Problem 4.7.2 (bon
us 1)\nComparison with difference equation" }}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 33 "Statement of first bonus question" }}{PARA 0 "" 0 "" 
{TEXT 32 492 "For a bonus mark verify one of the solutions by showing \+
that the \nsecond-order difference equation derived in (b) above (when
 arranged \nto give $x_\{n+1\}$ in terms of $x_n$ and $x_\{n-1\}$) giv
es the same \nsequence of $x$-values as the discrete-time dynamical sy
stem in (c) if \ntwo successive values of $x$ given by the dynamical s
ystem are used to \nstart the second-order difference equation.  I.e. \+
the discrete time \n``Lagrangian'' and ``Hamiltonian'' discriptions ar
e dynamically \nequivalent." }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 31 "S
olution of difference equation" }}{PARA 0 "" 0 "" {TEXT -1 72 "First g
et coefficients for forward iteration of the difference equation:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "unassign('m','w','h');\nx[n+
1] := solve(gradS_n,x[n+1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"x
G6#,&%\"nG\"\"\"F)F),$*&,*&F%6#,&F(F)!\"\"F)\"\"'&F%6#F(!#7*()%\"wG\"
\"#\"\"\")%\"hGF8F9F2F)\"\"%*(F6F9F:F9F-F)F)F9,&F1F)*&F6F9F:F9F)!\"\"F
0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "c1 := diff(x[n+1],x[n]
); c2 := diff(x[n+1],x[n-1]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c1
G,$*&,&!#7\"\"\"*&)%\"wG\"\"#\"\"\")%\"hGF-F.\"\"%F.,&\"\"'F)F*F)!\"\"
!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#c2G!\"\"" }}}{PARA 0 "" 0 
"" {TEXT -1 101 "Now define a procedure for forward iteration of the d
ifference equation using the above coefficients:" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 162 "orbitEL := proc (x0, x1, n)\n  local i, x;\n \+
 x[0] := x0; x[1] := x1;\n  for i from 2 to n do\n    x[i] := c1*x[i-1
]+c2*x[i-2];\n  od;\n  RETURN([x[j] $ j = 0..n])\nend;" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%(orbitELGR6%%#x0G%#x1G%\"nG6$%\"iG%\"xG6\"F-C&>&
8%6#\"\"!9$>&F16#\"\"\"9%?(8$\"\"#F89&%%trueG>&F16#F;,&*&%#c1GF8&F16#,
&F;F8!\"\"F8F8F8*&%#c2GF8&F16#,&F;F8!\"#F8F8F8-%'RETURNG6#7#-%\"$G6$&F
16#%\"jG/FX;F3F=F-F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 58 "First iterate \+
the phase-space difference equation 10 times" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 55 "m := 1; w := 1; h := 0.1;\nphase_orbit := orbit(1,
0,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"wG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
\"hG$\"\"\"!\"\"" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%,phase_orbitG7-7
$\"\"\"\"\"!7$$\"+%>$3]**!#5$!+&fT](**!#67$$\"+:6$3!)*F,$!+v)\\])>F,7$
$\"++Qt`&*F,$!+?\"yF&HF,7$$\"+6\"e7@*F,$!+:x-\"*QF,7$$\"+2J#ox)F,$!+w<
V!z%F,7$$\"+3fwa#)F,$!+F7,UcF,7$$\"+,%)H]wF,$!+VWEPkF,7$$\"+._XppF,$!+
C@DorF,7$$\"+<M.>iF,$!+blnFyF,7$$\"+vZ_1aF,$!+lW&*3%)F," }}}{PARA 0 "
" 0 "" {TEXT -1 23 "then use the first two " }{TEXT 266 1 "x" }{TEXT 
-1 104 " values of the phase-space points to start the iteration of th
e configuration-space difference equation:" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 62 "config_orbit := orbitEL(phase_orbit[1,1],phase_orbi
t[2,1],10);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-config_orbitG7-\"\"
\"$\"+%>$3]**!#5$\"*7J3!)*!\"*$\"+1Qt`&*F)$\"*7e7@*F,$\"+9J#ox)F)$\"*
\"fwa#)F,$\"+'R)H]wF)$\"*>b%ppF,$\"+%RL!>iF)$\"*uClS&F," }}}{PARA 0 "
" 0 "" {TEXT -1 55 "Comparing phase_orbit and config_orbit we see that
 the " }{TEXT 265 1 "x" }{TEXT -1 49 " values are the same up to the 7
th decimal place." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 54 "Problem 4.
7.2 (bonus 2)\nEigenvalues of Jacobian matrix" }}{SECT 1 {PARA 4 "" 0 
"" {TEXT -1 21 "Statement of part (d)" }}{PARA 0 "" 0 "" {TEXT 32 132 
"For another bonus mark calculate the eigenvalues for the three \nvalu
es of $\\Delta t$ given above. (Again, use Maple or Mathematica.)" }}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 26 "Calculation of eigenvalues" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "m := 1; w := 1; h := 0.1;\ne
igenvalues(J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"wG\"\"\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"hG$\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,
&$\"+&>$3]**!#5\"\"\"%\"IG$!+K/?z**!#6,&F$F'F($\"+K/?z**F+" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "m := 1; w := 1; h := 3.45;\neigenva
lues(J);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"wG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"hG$\"$X$!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&$!+N$Qb%**!#5\"
\"\"%\"IG$!+W9CU5F&,&F$F'F($\"+W9CU5F&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "m := 1; w := 1; h := 3.47;\neigenvalues(J);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%\"wG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG$\"$Z$!\"#
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+b7kp5!\"*$!+u)G*[$*!#5" }}}
{PARA 0 "" 0 "" {TEXT -1 111 "Thus we see that the Jacobian matrices c
orresponding to the two stable orbits have complex eigenvalues (on the
 " }{TEXT -1 91 "unit circle), while that for the unstable orbit has r
eal (mutually reciprocal) eigenvalues." }}}}}{MARK "1" 0 }{VIEWOPTS 1 
1 0 1 1 1803 }