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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 18 "Duffing oscillator" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 106 "This non
linear oscillator is an example of a system which becomes chaotic when
 driven by a periodic force." }}{PARA 0 "" 0 "" {TEXT -1 47 "The oscil
lator itself is given by the equation:" }}{PARA 0 "" 0 "" {TEXT -1 29 
"x\"(t)=-d*x(t)^3+e*x(t)-g*v(t)" }}{PARA 0 "" 0 "" {TEXT -1 187 "The c
ubic term -d*x(t)^3 provides a nonlinear restoring force at large x, w
hile the linear term pushes away from the origin. In addition, there i
s the usual velocity-proportional damping." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "The potential for this oscillat
or has a double-well structure. At " }{TEXT 256 1 "x" }{TEXT -1 209 "=
0 there is an unstable equilibrium point, and given some damping the p
article has to fall into one side of the well or the other if it appro
aches the equilibrium point with just enough energy to move over it." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 265 "The ho
mogeneous problem (non-driven oscillator) has no surprises in it. Give
n an initial condition there is a unique phase-space trajectory that l
eads to the particle winding up at the bottom of one of the two wells \+
after the mechanical energy is converted to heat." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 266 "When the oscillator is d
riven by a periodic force the system can reach a limit cycle, where as
 much mechanical energy is lost per cycle as is dumped into the system
 by the crank. Chaos appears as a result of the two wells connected by
 the unstable equilibrium point." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "restart; unprotect(gamma):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 32 "gamma:=1/10: eta:=1: delta:=1/4:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 64 "plot(-1/2*eta*x^2+delta*x^4/4,x=-3..3,tit
le=\"Potential Energy\");" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 
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45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "
DE:=diff(x(t),t)=y(t),diff(y(t),t)=-gamma*y(t)+eta*x(t)-delta*x(t)^3;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG6$/-%%diffG6$-%\"xG6#%\"tGF-
-%\"yGF,/-F(6$F.F-,(F.#!\"\"\"#5F*\"\"\"*&#F7\"\"%F7*$)F*\"\"$F7F7F5" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "We provide a solution example \+
for the case where the oscillator initinally has sufficient energy to \+
move across the unstable equilibrium point at the origin." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "IC:=x(0)=3,y(0)=10;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%#ICG6$/-%\"xG6#\"\"!\"\"$/-%\"yGF)\"#5" }}}
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{MPLTEXT 1 0 37 "X:=subs(sol,x(t)): Y:=subs(sol,y(t)):" }}}{EXCHG 
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particle gets trapped in the left-hand-side or right-hand-side well." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 
11 "Driven case" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 136 "Now we add a periodic driving force (amplitude 1 and cir
cular frequency 1) and start the oscillator from the unstable equilibr
ium point:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "t:='t': nu:=1
: DF:=sin(nu*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DFG-%$sinG6#%\"
tG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "DE:=diff(x(t),t)=y(t)
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" {XPPMATH 20 "6#>%#DEG6$/-%%diffG6$-%\"xG6#%\"tGF--%\"yGF,/-F(6$F.F-,
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "IC:=x(0)=0,y(0)=0; #start at
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55 "plot('X(t)',t=0..100,numpoints=500,title=\"Trajectory\");" }}
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1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 28 "What does this plot tell us?" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 336 "For a simple limit cycle (asympto
tically periodic motion with frequency of the driving force) we expect
 a single point: we catch the system at the same time during the cycle
. Thus, we expect the Poincare points to cluster at a single point. In
 our case it happens to be somewhere inside the left-hand-side well wi
th a positive velocity." }}{PARA 0 "" 0 "" {TEXT -1 110 "The few scatt
ered point must have something to do with the transient behaviour befo
re the limit cycle sets in." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT 257 11 "Exercise 2:" }}{PARA 0 "" 0 "" {TEXT -1 17 "Re
move the first " }{TEXT 258 1 "M" }{TEXT -1 111 " points from the Poin
care section, and confirm that the stroboscopic phase-space diagram wi
th strobe frequency " }{TEXT 19 2 "nu" }{TEXT -1 80 " results in point
s which cluster around a single location in this case. Try for " }
{TEXT 259 1 "M" }{TEXT -1 167 "=10,20,30,40 , and observe the Poincare
 section both on the large scale (-4 < x < 4, -4 < x' < 4), as well as
 on a fine scale (the vicinity of the accumulation point)." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 216 "Now we d
ouble the frequency of the driving force. For the same amplitude as us
ed before (1), we get a limit cycle inside one of the two wells. Howev
er, when we increase the amplitude rather interesting motion occurs:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "nu:=2: F0:=2.5: DF:=F0*si
n(nu*t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DFG,$-%$sinG6#,$%\"tG\"
\"#$\"#D!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "DE:=diff(x
(t),t)=y(t),diff(y(t),t)=-gamma*y(t)+eta*x(t)-delta*x(t)^3+DF;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#DEG6$/-%%diffG6$-%\"xG6#%\"tGF--%\"
yGF,/-F(6$F.F-,*F.#!\"\"\"#5F*\"\"\"*&#F7\"\"%F7*$)F*\"\"$F7F7F5*&$\"#
DF5F7-%$sinG6#,$F-\"\"#F7F7" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
59 "IC:=x(0)=0,y(0)=0; #start at the unstable equilibrium point" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ICG6$/-%\"xG6#\"\"!F*/-%\"yGF)F*" }
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{MPLTEXT 1 0 37 "X:=subs(sol,x(t)): Y:=subs(sol,y(t)):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot('X(t)',t=0..100,numpoints=2000
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s displayed with some small errors occasionally. Increasing the numpoi
nts variable leads to more accuracy, but then the lines do not connect
 in same places, but are shown in dot form." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 54 "The Poincare section should be \+
rather interesting now:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "
tau:=2*Pi/nu;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$tauG%#PiG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "t:='t': sol:=dsolve(\{DE,IC
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "X:=subs(sol,x(t)): Y:=subs(s
ol,y(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "LL:=[[X(0),Y(0
)]]: for i from 1 to 1000 do: t_i:=i*tau; LL:=[op(LL),[X(t_i),Y(t_i)]]
: od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "plot(LL,style=poi
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2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 268 "The structure that appears in the Poincare section in \+
this case can be proven to be a complicated curve, namely a fractal. T
his leads to the name 'strange attractor' for this oscillator, and is \+
an indication that the system is chaotic (sensitive to intial conditio
ns)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 11 "
Exercise 3:" }}{PARA 0 "" 0 "" {TEXT -1 386 "Change the parameters of \+
the driving force (amplitude and frequency), and observe what happens \+
to the Poincare section in these cases. Pick a frequency, and increase
 systematically the amplitude. Explore for which amplitude values F0 t
he Poincare diagram has a few discrete accumulation points, and for wh
ich values a multitude of points is observed (indicative of a strange \+
attractor)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}}{MARK "0 9 0" 43 }{VIEWOPTS 1 1 0 1 1 1803 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }