Transcript Document 7738332

Chaos in a Pendulum Section 4.6
• To introduce chaos concepts, use the damped,
driven pendulum. This is a prototype of a
nonlinear oscillator which can display chaos.
The Pendulum
– The nonlinearity has been known for hundreds of
years.
– Thoroughly discussed (for the undamped case!) in
Sect. 4.4.
– Chaotic behavior in pendula has only been
discovered (& explored) recently (20-30 years).
• Some pendulum systems for which the motion has
been found to be chaotic:
Support undergoing forced
sinusoidal oscillations 
Double pendulum 
Coupled pendula 
Magnetic pendulum 
• We aren’t going to analyze these!
• Instead, as a prototype for
pendula which exhibit chaos,
go back to the ordinary plane
pendulum we’ve already seen,
but add a sinusoidal driving
force and a damping term 
(sinusoidal)

• Equation of motion (rotational version of Newton’s 2nd Law):
Total torque about the axis = (moment of inertia)  (angular acceleration)
Assume a damping torque proportional to the angular
velocity & sinusoidal driving torque:
N = I(d2θ/dt2)
= - b(dθ/dt) - mgsinθ + Nd cos(ωdt)
Divide by I = m2 & define
B  (b/I), (ω0)2  (g/), D  (Nd/I)
ω0  Natural frequency for small θ
of (“simple”) pendulum
 (d2θ/dt2) =
-B(dθ/dt) – (ω0)2 sinθ + D cos(ωdt)
• For ease of numerical solution, go to dimensionless
variables.
– Divide the equation of motion by the natural frequency
squared: (ω0)2  (g/),
Define the dimensionless variables:
Time: t  ω0t  (g/)½t
Oscillating variable: x  θ
Driving frequency: ω  (ωd/ω0)  (/g)½ωd
Damping constant: c  b/(m2ω0)  b/(mg)
Driving force (torque) strength:
F  Nd/(m2ω0)  Nd/(mg)
• Also note that:
x = (dx/dt) = (dθ/dt)(dt/dt) = (dθ/dt)(ω0)-1 = θ(ω0)-1
x = (d2x/dt2) = (d/dt)[(dθ/dt)(ω0)-1] = (d2θ/dt2)(ω0)-2 = θ(ω0)-2
• So, the equation of motion finally becomes:
x + cx + sin(x) = F cos(ωt)
• A nonlinear driven oscillator equation! The
author used numerical methods to solve for x(t) for
various values of the parameters c, F, ω
• To solve this numerically, its first convert this 2nd order differential
equation to two 1st order differential equations!
x + cx + sin(x) = F cos(ωt). DEFINE: y  (dx/dt) = x
(angular velocity), z  ωt  (dy/dt) = - cy – sin(x) + F cos(z)
• Results are shown in the rather complicated figure (next page), which
we’ll now look at in detail! For c = 0.05, ω = 0.7, results are shown
for (driving torque strength) F = 0.4, 0.5, 0.6, 0.7, 0.8, 0.9
• Bottom line of the results from the figure:
1. The motion is periodic for F = 0.4, 0.5, 0.8, 0.9
2. The motion is chaotic for F = 0.6, 0.7, 1.0
• This indicates the richness of the results which can come from
nonlinear dynamics! This is surprising only if you think
linearly! Thinking linearly, one would expect the solution for F
= 0.6 to not be much different from that for F = 0.5, etc.
• Sideways view!
• Left figure shows y  (dx/dt) (angular velocity) vs time tat
steady state (transient effects have died out).
F = 0.8 
F = 0.4 
~ simple harmonic motion
F = 0.9 
F = 0.5 
periodic, but not very
“simple”!
F = 0.6 
F = 1.0 
F = 0.7 
F = 0.6, 0.7, 1.0 are
VERY different from
the others: CHAOS!
F = 0.8, 0.9 are
~ similar to F = 0.5.
• Middle figure shows x – (dx/dt) phase space plots for the
same cases (periodic, so only -π < x < π is needed).
F = 0.4 
~ ellipse, as expected for
simple harmonic motion
F = 0.5 
Much more complicated!
2 complete revolutions
& 2 oscillations!
F = 0.6
& F = 0.7 
Entire phase plane is accessed.
A SIGN OF CHAOS!
F = 0.8 
Periodic again.
One complete
revolution +
oscillation.
F = 0.9 
2 different
revolutions
in one cycle
(“period
doubling”).
F = 1.0: The entire phase
plane is accessed again!
A SIGN OF CHAOS!
• Right column: “Poincaré Sections”: Need lots of further
explanation!
F = 0.4 
F = 0.8 
F = 0.5 
F = 0.9 
F = 0.6 
F = 1.0 
F = 0.7 
Poincaré Sections
• Poincaré Sections: Poincaré invented a technique to
“simplify” representations of complicated phase space
diagrams, such as we’ve just seen.
• They are essentially 2d representations of 3d phase
space diagram plots. In our case, the 3d are:
y [= (dx/dt) = (dθ/dt)] vs x (= θ) vs z (= ωt). Left column of
the first figure (angular velocity y vs. t) = the projection of
this plot onto a y-z plane, showing points corresponding to
various x. Middle column of the first figure = the projection
onto a y-x plane, showing points belonging to various z.
• The figure on the next page shows a 3d phase space diagram,
intersected by a set of y-x planes, perpendicular to the z axis &
at equal z intervals.
Poincaré Sections
Explanation follows!
• Poincaré Section Plot: Or, simply, Poincaré
Section  The sequence of points formed by the
intersection of the phase path
with these parallel planes in
phase space, projected onto
one of the planes. The phase
path pierces the planes as a
function of angular speed [y = (dθ/dt)], time (z  ωt) &
phase angle (x = θ). The points of intersection are labeled A1,
A2, A3, etc. The resulting set of points {Ai}forms a PATTERN
when projected onto one of the planes. Sometimes, the pattern
is regular & recognizable, sometimes irregular. Irregularity of
the pattern can be a sign of chaos.
• Poincaré realized that
1. Simple curves generated like this represent
regular motion with possibly analytic solutions,
such as the regular curves for F = 0.4 & 0.5 in the
driven pendulum problem.
2. Many complicated, irregular, curves
represent CHAOS!
• Poincaré Section: Effectively reduces an N
dimensional diagram to N-1 dimensions for graphical
analysis. Can help to visualize motion in phase space
& determine if chaos is present or not.
• For the driven, damped
F = 0.4
pendulum, the regularity of the
0.5
motion is due to the forcing period.
A description of the motion
0.6
depends on x (θ), y (dθ/dt) & z (t).
Complete description requires 3d
0.7
phase space diagrams. All values of z are
included in middle column of the figure
 Choose to take Poincaré Sections only for z = 2nπ (frequency
= that of driving force).
• Right column of figure = Poincaré Section for the same system as
in left 2 columns. F = 0.4: Simple periodic motion. The system
always comes back to same phase point (x,y). For simple harmonic
motion, (F = 0.4) all projected points are either the same or fall on a
smooth curve.  Poincaré Section = one point, as shown.
• F = 0.5 Poincaré Section F = 0.4
has 3 points, because of more
0.5
complex motion.
0.6
• In general, the number
of points n in the
0.7
Poincaré Section shows
that the motion is periodic
with a period different than the period of the driving
force. In general this period is T = T0(n/m), where
T0 = (2π/ω) is the period of the driving force & m =
integer. (m = 3 for F = 0.5)
• F = 0.8: The Poincaré F = 0.8
Section again has only 0.9
1 point (“simple”,
1.0
regular motion.)
• F = 0.9: 2 points (more
complex motion). T = T0(n/m), m = 2.
• F = 0.6, 0.7, 1.0: CHAOTIC MOTION & new
period T  . The Poincaré Section is rich
in structure!
• Recall from earlier discussion:
• ATTRACTOR  A set of points (or one point) in
phase space towards which a system motion
converges when damping is present. When there is an
attractor, the regions traversed in phase space are bounded.
• For Chaotic Motion, trajectories which are very near each
other in phase space are diverging from one another. However,
they must eventually return to the attractor.
• Attractors in chaotic motion  “Strange”
Attractors or Chaotic Attractors.
• Because Strange Attractors are bounded in
phase space, they must fold back into the nearby
phase space regions.
 Strange Attractors create intricate patterns,
as seen in the Poincaré Sections of the example
we’ve discussed. Because of the uniqueness of the
solutions to the Newton’s 2nd Law differential
equations, the trajectories must still be such that no
one trajectory crosses another.
• It is also known, that some of these Strange or
Chaotic Attractors are FRACTALS!