{VERSION 4 0 "IBM INTEL NT" "4.0" }
{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 Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 
257 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 
0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 
0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }
{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 
262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 
{CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 
0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 
1 16 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }}
{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\");" }}}{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;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "We provide a solution exa
mple for the case where the oscillator initinally has sufficient energ
y 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;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "sol:=dsolve(\{DE,IC\},\{x(t)
,y(t)\},numeric,output=listprocedure):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "X:=subs(sol,x(t)): Y:=subs(sol,y(t)):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot('X(t)',t=0..100,numpoints=500,
title=\"Trajectory\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 143 "The ph
ase-space diagram illustrates how the loss of mechanical energy leads \+
to the particle winding up in the x<0 well after five oscillations:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "plot(['X(t)','Y(t)',t=0..5
0],numpoints=500,title=\"Phase-space diagram\");" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 261 11 "Exercise 1:" }}{PARA 0 "" 0 "" {TEXT -1 122 "Chan
ge the initial conditions and observe whether the particle gets trappe
d 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 a
dd a periodic driving force (amplitude 1 and circular frequency 1) and
 start the oscillator from the unstable equilibrium point:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "t:='t': nu:=1: DF:=sin(nu*t);" }}}
{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;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 59 "IC:=x(0)=0,y(0)=0; #start at the unstable equili
brium point" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "sol:=dsolve(
\{DE,IC\},\{x(t),y(t)\},numeric,output=listprocedure):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "X:=subs(sol,x(t)): Y:=subs(sol,y(t)
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "plot('X(t)',t=0..100,
numpoints=500,title=\"Trajectory\");" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 72 "plot(['X(t)','Y(t)',t=0..50],numpoints=500,title=\"Ph
ase-space Diagram\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Now we \+
want the Poincare section: we accumulate a list of points by strobosco
pically sampling with the driving force period:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "tau:=2*Pi/nu;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 75 "sol:=dsolve(\{DE,IC\},\{x(t),y(t)\},numeric,method=ls
ode,output=listprocedure):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "X:=subs(sol,x(t)): Y:=subs(sol,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=point,symbol=box,symbolsize=4,axes=box
ed,view=[-4..4,-4..4],title=\"Poincare Section\");" }}}{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
 (asymptotically 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-sid
e well with a positive velocity." }}{PARA 0 "" 0 "" {TEXT -1 110 "The \+
few scattered point must have something to do with the transient behav
iour before 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 "Remove the first " }{TEXT 258 1 "M" }{TEXT -1 111 " points from
 the Poincare section, and confirm that the stroboscopic phase-space d
iagram with strobe frequency " }{TEXT 19 2 "nu" }{TEXT -1 80 " results
 in points which cluster around a single location in this case. Try fo
r " }{TEXT 259 1 "M" }{TEXT -1 167 "=10,20,30,40 , and observe the Poi
ncare section both on the large scale (-4 < x < 4, -4 < x' < 4), as we
ll 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 double the frequency of the driving force. For the same amplitu
de as used before (1), we get a limit cycle inside one of the two well
s. However, when we increase the amplitude rather interesting motion o
ccurs:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "nu:=2: F0:=2.5: D
F:=F0*sin(nu*t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "DE:=dif
f(x(t),t)=y(t),diff(y(t),t)=-gamma*y(t)+eta*x(t)-delta*x(t)^3+DF;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "IC:=x(0)=0,y(0)=0; #start at
 the unstable equilibrium point" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 62 "sol:=dsolve(\{DE,IC\},\{x(t),y(t)\},numeric,output=listprocedu
re):" }}}{EXCHG {PARA 0 "> " 0 "" {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,title=\"Trajectory\");" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "plot(['X(t)','Y(t)',t=0..50],numpoi
nts=1000,title=\"Phase-space Diagram\");" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 208 "The phase-space diagram is displayed with some small err
ors occasionally. Increasing the numpoints variable leads to more accu
racy, 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;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "t:='t': sol:=dsolve(\{DE,IC\},\{x(t
),y(t)\},numeric,method=lsode,output=listprocedure):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 37 "X:=subs(sol,x(t)): Y:=subs(sol,y(t)):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "LL:=[[X(0),Y(0)]]: for i fr
om 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=point,symbol=
box,symbolsize=4,axes=boxed,view=[-5..5,-8..4],title=\"Poincare Sectio
n\");" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 268 "The structure that appe
ars in the Poincare section in this case can be proven to be a complic
ated curve, namely a fractal. This leads to the name 'strange attracto
r' for this oscillator, and is an indication that the system is chaoti
c (sensitive to intial conditions)." }}{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 freq
uency), and observe what happens to the Poincare section in these case
s. Pick a frequency, and increase systematically the amplitude. Explor
e for which amplitude values F0 the Poincare diagram has a few discret
e accumulation points, and for which values a multitude of points is o
bserved (indicative of a strange attractor)." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "0 0 0
" 18 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }