Transcript Lecture 50
Nonlinear Oscillations; Chaos
Mainly from Marion, Chapter 4
• Confine general discussion to oscillations &
oscillators. Most oscillators seen in undergrad
mechanics are Linear Oscillators: Obey Hooke’s
“Law”, linear restoring force: F(x)= -kx
Potential energy V(x)= (½)kx2
• Real world: Most oscillators are, in fact, nonlinear!
– Techniques for solving linear problems might or might not
be useful for nonlinear ones!
– Often, rather than having a general technique for nonlinear
problem solution, technique may be problem dependent.
– Often need numerical techniques to solve diff. eqtns.
• Many techniques exist for solving (or at least
approximately solving) nonlinear differential
equations.
• Often, nonlinear systems reveal a rich &
beautiful physics that is simply not there in
linear systems. (e.g. Chaos).
• The related areas of nonlinear mechanics and
Chaos are very modern & are (in some places)
hot topics of current research!
• I am not an expert!
Nonlinear Oscillations
• Consider a general 1 d driven, damped oscillator for which
Newton’s 2nd Law equation of motion is: (v = dx/dt)
m(d2x/dt2) + f(v)+ g(x) = h(t) (1)
– f(v), g(x),h(t) problem dependent!
– If f(v) contains powers of v higher than linear & / or
g(x) contains powers of x higher than linear, (1) is a
non-linear differential equation.
– Complete, general solutions are not always available!
– Sometimes special treatment adapted for a problem is
needed.
– Can learn a lot by considering deviations from linearity.
– Sometimes examining the phase (v-x) diagrams is useful.
• A Brief History & Terminology: The Early 1800’s:
Laplace’s idea: N’s 2nd Law. If at t = 0 positions & velocities of all
particles in universe are known, & if force laws governing particle
interactions are known, then we can know the exact future of the
universe by integrating N’s 2nd Law equations.
A Deterministic view of nature. Recently: In the past 35 years
or so, researchers in many (different) areas of science have realized
that knowing the laws of nature is not enough. Much of nature is
chaotic! Deterministic Chaos: Motion of a system whose time
evolution depends sensitively initial conditions.
Deterministic: For given initial conditions, N’s 2nd Law equations
give the exact future of the system. Chaos: Only slight changes in
initial conditions can result in drastic changes in the system motion.
Random: There is no correlation between the system’s present state
& it’s immediate past state. Chaos & randomness are different!
• Chaos: Measurements on the system at given time might not
allow the future to be predicted with certainty, even if the force
laws are known exactly! (Experimental uncertainties in the initial
conditions). Deterministic Chaos: Always associated with a
system nonlinearity. Nonlinearity: Necessary for chaos, but not
sufficient! All
chaotic systems are nonlinear but not all
nonlinear systems are chaotic.
– Chaos occurs when the system depends very sensitively on
its previous state. Even a tiny change in initial conditions
can completely change the system motion.
• Chaotic Systems: Can only be solved numerically. Only with
the availability of modern computers has it become possible to study
these phenomena. No simple, general rules for when the system will be
chaotic or not. Chaotic phenomena in the real world: Irregular heartbeats,
Planet motion in the solar system, Electrical circuits, Weather patterns, …
• Chaotic Systems:
– Henri Poincaré: In the late 1800’s. The first to
recognize chaos (in celestial mechanics).
– Real breakthroughs in understanding came in the
1970’s (with computers).
– Chaos study is widespread. We will only give a
brief, elementary introduction.
– Many textbooks and popular texts exist! See
References to Goldstein’s Ch. 11 or Marion’s Ch. 4.
– This is a very popular & popularized topic even in
the mainstream, popular press.
Chaos in a Pendulum
• Use the damped, driven pendulum to introduce
Chaos concepts.
• Pendulum: The nonlinearity has been known for hundreds
of years. Chaotic behavior only known (& explored) recently.
Some pendulum motions which are known to be chaotic:
– Forced oscillating support:
• Double pendulum:
• Coupled pendula:
• Magnetic pendulum:
• We aren’t going to analyze these!
• Consider the ordinary pendulum, but
add a driving torque (Nd cos(ωdt)) &
a damping term (-b(dθ/dt))
• The Equation of Motion is obtained
by equating the total torque around the
pivot point with the moment of
inertia the angular acceleration.
N = I(d2θ/dt2)
= - b(dθ/dt) -mgsinθ + Nd cos(ωdt)
• Equation of motion:
N = I(d2θ/dt2) = - b(dθ/dt) - mgsinθ + Nd cos(ωdt)
Divide by I = m2
(d2θ/dt2) = - b(m2)-1(dθ/dt) - g()-1sinθ
+ Nd(m2)-1cos(ωdt)
• Go to dimensionless variables (for ease of numerical solution):
Divide eqtn by natural frequency squared: (ω0)2 (g/)
• Define dimensionless variables: Time: t ω0t (g/)½t
Driving frequency: ω (ωd/ω0) (/g)½ωd
Variable: x θ
Damping constant: c b(m2ω0)-1 b(mg)-1
Driving force strength: F Nd(m2ω0)-1 Nd(mg)-1
• To solve this numerically, its first convert this 2nd order differential
equation to two 1st order diff. eqtns!
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 tat
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.
• 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 = (2 /ω) is 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!