FACULTY OF SCIENCE
UNIVERSITY OF COPENHAGEN
First year project
Christoffer Østfeldt May 28th 1989 & Karsten Møller Dideriksen May 18th 1989
Bouncing Ball
Chaotic mechanics, project no. 2012-27
Jörg Helge Müller, Jean-Baptiste Sylvain Béguin
Submitted March 23rd 2012. 19 Pages plus 5 pages appendices.
Resumé
Denne rapport omhandler et simpelt deterministisk system, bestående af en kugle og en
vibrerende højttaler, hvori kaos opstår for bestemte parametre. Vi forventer at se bifurkationer,
altså flerperiodiske oscillationer af kuglen, når systemets kontrolparametre varieres. Vi undersøger
vha. forskellige eksperimentelle tilgange forskellige aspekter af sammenstødet mellem kugle og
underlag. Vi sammenholder vores observerede størrelser med en numerisk simulering.
Det tilstræbes at rapporten er forståelig for førsteårsstuderende på fysikstudiet.
Abstract
The report deals with a simple deterministic system, consisting of a ball and a vibrating surface,
wherein chaotic behaviour is observed for certain parameters. We expect to see bifurcations, i.e.
multiple periodic oscillations of the ball, when the control parameters of the system is varied. We
investigate different aspects of the collision between ball and surface using different experimental
techniques. We will compare our observed results with a numeric simulation.
It is our goal that this report is readable by any first year physics student.
Figure 1: The setup
2
CONTENTS
LIST OF PICTURES
Contents
1
Introduction to chaos
4
2
The bouncing ball
7
3
The experiment
9
4
Numeric Simulation
14
5
Data analysis - varied frequency
16
6
Conclusion
18
7
References
19
A N-stable periods
20
B N oscillations per impact
21
C Amplitude scan
22
D Simulation - constant amplitude
23
E Amplitude in frequency span of the experiment
24
List of pictures
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
The setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The logistic map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The bifurcation diagram of the logistic map . . . . . . . . . . . . . . . . . . .
Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Amplitude as a function of frequency and digital value . . . . . . . . . . . .
Measuring the coefficient of restitution . . . . . . . . . . . . . . . . . . . . .
Acceleration of loudspeaker membrane showing clear impact disturbances
A simulation of how the ball bounces on the loudspeaker. . . . . . . . . . .
Attractors in the phase space . . . . . . . . . . . . . . . . . . . . . . . . . . .
Experimental data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Cobwebs of n-periodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Minimum amplitude for stable 1-periodicity over N oscillations . . . . . . .
Varied amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Simulation with constant amplitude . . . . . . . . . . . . . . . . . . . . . . .
The amplitude in the frequency span of the experiment . . . . . . . . . . . .
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1
1
INTRODUCTION TO CHAOS
Introduction to chaos
This chapter will contain an introduction to relevant concepts in chaos, and the theoretical aspects
we need to accurately describe our observations of and predictions for the system. We will not be
very rigorous, and a deeper introduction to chaos is beyond the scope of this report, but we will
recommend the magnificent book [1] "Nonliniear Dynamics and Chaos" by S. Strogatz for a deeper
introduction to chaos theory.
1.1
What is chaos?
Chaos, in physics, is the study of certain non-linear dynamic systems. The observed systems
are deterministic, but extremely sensitive to the initial conditions. This means that slight, even
unmeasurable, differences in initial conditions may create very different results, an effect popularly
named the butterfly effect. Even though the systems are deterministic, i.e. no random factors influence
the system, prediction of the systems behaviour over long time scales is impossible.
In [1] pp. 323, which we have based our theoretic approach on, the following working definition1
is used:
Chaos is aperiodic long-term behaviour in a deterministic system that exhibits sensitive
dependence on initial conditions
1.2
Phase space
An important tool in solving the nonlinear equations is the visualization of the system in a so called
phase space.
The phase space is space which has as its axes as the parameters needed to uniquely describe
any possible the state of a system. This might for instance be the position and velocity of a jumping
ball affected only by gravity and impact with for instance the ground. For any possible state of this
simple system we can fully describe the state as a combination of position and velocity.
When the system develops in time it will follow a trajectory in the phase space, called a phase
plot. It follows directly from the fact that the described system are deterministic that for any
given combination of conditions, only one possible future is possible, and thereby only one phase
plot. Hence trajectories in the phase space can never intersect itself. That would mean that at the
intersection the system could go two ways randomly.
It follows that if n parameters are needed to describe the state of the system, the phase space
will be n-dimensional. It is also obvious that for a periodic solution to exist we need at least
two parameters to make a closed loop in the phase space that doesn’t intersect itself2 , while for
periodic motions with periodicity larger than one we need at least three parameters to make a closed
circuit that intersects some plane in the phase space multiple times, before returning to the initial
conditions.
1.3
Attractors
When dynamic systems over timer converge to a certain state (periodic or stable) we say that the
point to which they converge is an attractor. This might, for instance, be the state of a dampened
pendulum with velocity and displacement for equilibrium equal to zero, the stable oscillations of
a loudspeaker driven by an amplifier with a constant frequency and amplitude and so on. More
mathematically speaking the attractor is the set of values the system converges to and may hence be
points, curves, planes etc. in the phase space. Three conditions are listed in [1] pp. 324:
We define an attractor to be a closed set A with the following properties:
1. A is an invariant set: Any trajectory x(t) that starts in A stays in A for all time.
2. A attracts an open set of initial conditions: there is an open set U containing A such
that if x(0)∈ U, then the distance from x(t) to A tends to zero as t → ∞ (...).
3. A is minimal: There is no proper subset of A that satisfies conditions 1 and 2.
1 It
is a working definition because there exists no universally accepted definition of chaos.
the one parameter is a circle - like the phase of any rotational, undampened motion. Then closed loops can exist
for a 1-dimensional system.
2 Unless
4
1.4
The logistic map
1
INTRODUCTION TO CHAOS
In chaotic systems a special kind of attractors exist. Because of their somewhat peculiar nature they
are called strange attractors. Strange attractors are the trajectories in the phase space for a given set
of parameters in a chaotic system, once any initial transient has died out. However because of the
nature of chaotic systems we require several things: That the trajectory in the phase space does
not return to the same place within an finite time, that it does not intersect itself and, because the
physical system only exists for certain values, that the attractors stays within finite boundaries. This
means that even though strange attractors are trajectories - curves in the phase space - they will look
like planes, because any loop on the trajectory will be infinitesimally close to another loop. This is
also a sign of chaos - if we choose to infinitesimally close points on the "attractor plane" we may get
wildly varying results.
When different loops of a trajectory are very close to each other, it is difficult to determine where
on the trajectory the system is. Consider two systems with initial states on two different loops of
a trajectory very close to each other. As time passes their states will begin to differ significantly3 .
Meaning that the determination of the initial condition has great importance for predictability of a
chaotic system. This is what is meant by sensitive dependency on initial conditions in the definition
of chaos.
As well as determination of initial conditions is difficult in the chaotic regime, the system is also
vulnerable to disturbances. A minor disturbance can cause the system to jump to another close-by
loop of the trajectory.
Attractors are closely linked to fixed points. These are points or curves in the phase space that, if
a system assumes this exact combination of values, it will stay forever. However there exist stable
and unstable fixed points. The stable ones are the ones that system converge to and the unstable are
the ones for which even the smallest perturbation away from the point will "blow up" and send the
system to a state far from the unstable point. It is possible for a point to be stable if approached
from one direction and unstable from the other.
1.4
The logistic map
A central concept in chaos theory is the logistic or quadratic map. This is the plot of the difference
equation
xn+1 = λxn (1 − xn ) = −λxn2 + λxn
(1)
where λ is a control parameter between 0 and 4. If plotted, as in figure 2, we will see a parabola with
its top point in ( xn , xn+1 ) = ( 21 , λ4 ). This means that we can change the intersection of xn+1 = xn (red
line) and equation (1) (blue line) by selection of λ. For 0 < λ ≤ 1 only the origin is a fixed point.
This is a trivial and uninteresting solution. For 1 < λ ≤ 3 one fixed point exists at the intersection
because xn+1 = xn . For 3 ≤ λ multiple fixed points may exist simultaneously.
As shown in figure 2a on the following page, xn+1 converges to the intersection when doing a
lot of iterations (for λ > 1). The intersection is a fixed point. However, when the intersection moves
to the down-slope, eventually (at λ = 3) xn+1 will converge differently. Now it will only stay in
the intersection if it assumes the exact value of the intersection, making it an unstable fixed point.
Otherwise it will jump back and forth around the intersection until stabilizing in two new fixed
points (see figure 2(b)). When this happens we talk about a bifurcation. For the described logistic
map the first bifurcation happens when λ = 3 and we have a 2-period stable state.
We define n-periodicity to mean that a system assumes a value n times before returning to the
original state. For instance that it impacts the loudspeaker in our system n times, so that x = 0,
where x is the balls height above the speaker.
One may find xn+1 by drawing a horizontal line from xn to the line xn+1 = xn (this point is
directly above xn ) and from there a vertical line to the plot of the parabola. This is known as the
graphical method for finding the next iterate. It is shown in figure 2a on the next page.
Using a MatLab simulation we have found the stable points for the logistic map for different
values of λ. The plot is shown in figure 3.
3 See
[1] p. 320 for a clarification of why this happens.
5
1.4
The logistic map
1
(a) The logistic map and graphical method
INTRODUCTION TO CHAOS
(b) λ=3. 2-period stable. Zoomed in
Figure 2: The logistic map
Figure 3: The bifurcation diagram is drawn by choosing an arbitrary initial value of 0 < x < 1 and doing
200 iterations on the logistic map. Then the next 40 values are plotted as final x. This is done in steps of
∆λ = 0.0001 on the logistic map with 2.8 ≤ λ ≤ 4.
6
1.5
1.5
Feigenbaum’s delta
2
THE BOUNCING BALL
Feigenbaum’s delta
Feigenbaums delta or Feigenbaums constant is an import constant in chaos theory. It is the limit of the
function
λ m +1 − λ m
δ = lim
= 4.669...
(2)
m → ∞ λ m +2 − λ m +1
where λn is the value of λ for the n’th bifurcation. It has been shown to be true for many different
kinds of chaotic systems, not just the logistic map for which it was originally proven.
According to [1] p. 355 the values of the first five bifurcations in the logistic map are:
λ1 = 3, λ2 = 3.499..., λ3 = 3.5409..., λ4 = 3.5644..., λ5 = 3.568759...
(3)
From that the first three deltas are δ1 = 4.7218, δ2 = 4.6819, δ3 = 4.6593. So even the first deltas are
close to the limit value.
That Feigenbaum’s delta is the limit value of the relation of distance between consecutive
bifurcations, implicate that the more bifurcations the system has gone through, the smaller the
absolute difference between two consecutive values will be. As each bifurcation also means twice
as many stable points we have n → ∞ while λ → 3.5699464 . Chaos arises when the periodicity is
infinite, i.e. when λ = λ∞ 5 . This period doubling within a finite range of control parameters is
known as the doubling route to chaos. The concept is clearly visible in figure 3 on the preceding page.
2
The bouncing ball
The system we deal with is a simple deterministic system. To model the system in a simple way,
several assumptions must be made.
First of all we assume no movement except in the vertical direction, and only along a straight
line. I.e. we will assume that there is no sideways movement. This in turn means that we will
consider the ball to be a point like particle, with no vertical size.
Secondly we assume all impacts to be instantaneous. This means that we can consider the
impacts to be inertial in the reference frame of the moving loudspeaker.
2.1
Dynamics of the system
The description of the loudspeakers position and velocity are then the simple and well known
equations for a harmonic oscillation:
xt (t) = A sin(ωt + φ0 )
(4)
vt (t) = Aω cos(ωt + φ0 )
(5)
where ωt + φ0 is the phase of its movement.
Since we assume no air resistance6 the formula for calculating the position of the steel ball is:
1
xb (t) = xb,0 + vb↑ t − gt2
2
(6)
vb = vb↑ − gt
(7)
and the velocity
Hence the time between impacts will depend on vb↑ and the difference in the loudspeaker position,
∆x, according to following:
q
vb↑ + v2b↑ − 2∆x · g
∆t =
(8)
g
4
[1] pp 355.
happens when λ > λ∞ will not be covered in this report
6 Not a wild assumption - the speed is nowhere large enough to account for forces larger than 1/1000 of the gravitational
forces acting on the ball - that’s smaller than our uncertainty on g.
5 What
7
2.2
Periodicity
2
THE BOUNCING BALL
Upon impact the steel ball will reverse its velocity relative to the loudspeaker, however, since the
collision is inelastic, the velocity will smaller by a factor of Cr 7 , the the coefficient of restitution. This
means the new upward velocity of the steel ball, vb↑ , in the lab system will be
vb↑ − vt = −Cr · (vb↓ − vt )
vb↑ = Cr · (vt − vb↓ ) + vt
2.2
(9)
(10)
Periodicity
The next realization to make is that there exists stable states for the periodic movement of the ball.
The simplest of such will be a single period in which the ball hits the membrane in the same phase
of its movement for every impact it makes. In this case the loudspeaker adds just as much kinetic
energy to the ball as is lost in the in collision. If we combine (10) and (5) we get:
vb↑ = Cr · ( Aω cos(φ) − vb↓ ) + Aω cos(φ)
(11)
where we are only interested in the phase of the impact and not the time. As we are looking at a
single period movement the ball will hit the membrane in the same position as it left. Hence we
may write vb↑ = −vb↓ . Therefore (11) is reduced to:
vb↑ =
1 + Cr
· A · ω · cos φ
1 − Cr
(12)
We demand that the fly time for the ball equals any full number of period of the membrane motion,
hence ∆t = nf = 2πn
ω and ∆y = 0. From (8) we see that the fly time dependency of vb↑ is given by:
∆t =
2vb↑
g
(13)
Given this relation we can determine a phase of impact for which the ball movement is periodically
stable:
cos φ∗ =
1 − Cr πgn
·
1 + Cr ω 2 A
(14)
This equation has two solutions of a phase φ∗ 8 which are both fixed points. However one is stable
and the other one unstable. To realize this one must consider that the single period movement
depends only on the velocity of the membrane and not its position since the ball hits in the same
position for the next impact. An oscillating surface has the same upward speed in two different
phases of its movement. One just after the beginning of the upward movement and one just before
ending the upward movement.
Now consider the last mentioned; in this situation the membrane is slowing down which means
that should the ball hit the membrane in a phase of its movement that is a little higher than the
phase found in (14) it will gain less speed on the next fly and therefore the next impact will be in an
earlier phase. And vice versa for an impact a little lower. Hence this is the stable fixed point.
Examining the other solution to (14) will show the opposite relation for an impact close to this
fixed point and thereby sending the impact away from this phase. Hence this is the unstable fixed
point.
Furthermore (14) shows that there exists a range of parameters ω, A for which it is not possible
πg
−Cr
to have a single period state. I.e. when 1 < 11+
Cr · ω 2 A
Approaching this condition the two fixed points will melt together and disappear. For instance, if
one varies only the amplitude we can introduce the critical amplitude:
Acrit =
1 − Cr πg
·
1 + Cr ω 2
An analysis of the critical amplitude is given in Appendix B.
7 This
8 the
is a real, physical ball, so 0 < Cr < 1
∗ denotes a fixed or stable point
8
(15)
2.3
2.3
Dimensions of the phase space
3
THE EXPERIMENT
Dimensions of the phase space
To fully describe the state of the system we need specification of three quantities. First of all
there is the current phase the membrane motion which is somewhat proportional to time since
0 ≤ φ(t) = ωt ≤ 2π. This quantity holds information about the membrane position and velocity
(see (4) and (5)). Secondly since the motion of the ball is non-linear we cannot express its position
by a single quantity (for a linear case i.e. time), but we will need to regard them as two quantities.
For simplicity reason we express the position relative to the membrane, x b . This way an impact is
always when x b = 0. When the ball is in the air, the velocity is given by (7) and combining (4) and
(6) we get:
1
x b (t) = vb↑ t − gt2 − A sin(φ(t))
2
(16)
In chapter 4 we will show examples of attractors in the phase space.
2.4
Our system and the logistic map
Now, one might ask, what has this system to do with the logistic map? Well, we cannot give a
quantitative reason why these two system are connected, but a qualitative connection is clearly visible
when considering the stable states of the two systems.
In the logistic map the trajectories converge towards the intersection between the line xn+1 = xn
because xn slightly above the intersection will give xn+1 sligthly below, but closer to the intersection,
and vice versa. This is the same mechanism described for convergence to the stable periods in our
system in subsection 2.2. Therefore we may learn a lot about the anticipated dynamics of our system
from the logistic map.
3
3.1
The experiment
Setup
In order to investigate chaos in a physical system we use a setup which is basically a steel ball
bouncing on glass. For the glass to be oscillating up and down it is glued to a loudspeaker membrane.
The glass is concave which helps to keep the ball in position (i.e. in the centre of the membrane).
To measure each impact of the ball we use a piezoelectric device that gives us an electric pulse for
each impact. The signal is low pass filtered by a small trigger circuit set to give only one signal from
each impact.
As we have shown in chapter 1 the task is to slowly change a control parameter, λ, to see at what
condition the behaviour starts to bifurcate and eventually goes into chaos. Since the coefficient of
restitution is difficult to control (as we would need to change ball during the experiment) equation
ω
and amplitude A.
(14) leaves us only two parameters, namely frequency 2π
To gain maximum control of these parameters a microcomputer is used. The microcomputer sends
out a reference voltage and an electronic 12 bit sine signal. A DAC unit then translates the electronic
signal into an analogue quantity of the reference voltage. All together this circuit enables us to
control the amplitude of the loudspeaker membrane (i.e. by controlling the reference voltage)
and the frequency of its movement (i.e. by changing the period of the sine signal send from the
microcomputer). Both are digitally controlled by connecting the microcomputer to a PC.
The DAC output signal needs amplification by a 15V amplifier and is then able to set the membrane
in a oscillating motion.
When the microcomputer receives a signal of an impact it sends information of the current time
and phase to the PC.
9
3.2
Limitations
3
1
2
3
4
5
6
7
8
9
THE EXPERIMENT
Loudspeaker membrane
Glass with piezo element
Ball
Trigger circuit
Microcomputer
DAC unit
Amplifier
Power supply
Accelerometer
Figure 4: Setup seen from above
3.2
Limitations
There are two main challenges of our system: One dimensional movement control and membrane
amplitude control. One dimensional movement is important for the simplicity of the system. As
long as the ball is moving solely vertically we do not have to take account of any horizontal velocity.
Simple observations told us that the ball moved in horisontal circles no larger than 1 mm, that
quickly died out. When measuring each impact with a piezoelectric we get a time stamp which
enables us to determine the time interval between impacts. This does not tell us where the ball is
going. Should there be any horizontal motion it would inflict the relation between the motion of the
ball and that of the membrane making it difficult for the system to settle down in stable periodicity.
We discovered when examining the amplitude of the membrane motion that the resulting
amplitude for given voltage has a strong dependency on the frequency due to loudspeaker resonance
(see section 3.4). This makes it extremely difficult to regulate only the frequency without changing
the amplitude as well. To compensate we would need to investigate thoroughly the relation between
the two. This has proven complicated, and we have conducted the experiment without compensation.
One last limitation to mention is that with the setup we are unable to determine any loss of
energy when the ball is in the air. This might cause some contradiction between the experiment and
simulation, but should not prevent us from seeing bifurcations.
3.3
Choice of parameters
The parameters and system components has been chosen through a trial and error process. We have
found the best results with these parameters:
1. A small and dense ball, i.e. a steel ball
2. We coated the glass surface with tape, to give a lower Cr , meaning faster convergence to the
stable periods
3. Frequency in the area where the amplitude is most linear, without going into the area with
falling amplitude, as this might make a bifurcation disappear (see section 3.4)
4. Amplitude to get bifurcations in that frequency range
All experiments have been conducted so that the ball was in period 1 orbit over 1 oscillation
at the start frequency before the frequency sweep was started. This way the start height, velocity
and phase was never an issue. This approach was chosen because we did not want to look at
sticking solutions, where all oscillations of the ball died out and it simply rested on the loudspeaker,
or jumps over more than one oscillation of the loudspeaker. It is also clear from the formulas in
section 2.1 that v0 , φ0 etc. does influence the stable periods. We note also that [2] concludes no such
dependency.
The setup is placed on a concrete floor to avoid it from standing on something vibrating.
10
3.4
Measuring the amplitude of the speaker
3
(a) Sweeped frequency at amplitude value 1700
THE EXPERIMENT
(b) Sweeped amplitude at 21Hz
Figure 5: Study of the loudspeaker motion shows a significant rise in amplitude in the resonance area and no
significant fluctuations in amplitude when changing the digital value sent from the PC.
3.4
Measuring the amplitude of the speaker
An important parameter for our system is the amplitude of the speaker. We want to find this value
experimentally as a function of the digital amplitude value send to the microcomputer.
We know that the position of the speaker is governed by the general formula for a harmonic
oscillator. This differentiated two times will give the acceleration of the loudspeaker, which can be
measured with an accelerometer connected to a PC through a microcomputer (Arduino UnoR3). If
we assume sinusoidal movement9 and take the maximum acceleration measured we know this will
be the acceleration of the time where sin(ωt) = 1.
∂2 x ( t )
= a(t)
∂t2
∂2
A sin(ωt) = − Aω 2 sin(ωt)
∂t2
amax A = 2 2 4π f
(17)
(18)
(19)
The accelerometer used is capable of measuring accelerations up to 3.5g with a linearity of 0.6% Full
scale, meaning 0.6% of Vre f 10 . The output is a voltage where 0g will be 0.5 · Vre f . The voltage is
read with the 10 bit ADC of the Arduino to produce a value between 0 and 1023. We have calibrated
the accelerometer by simple turning it 0, 90 and 180 degrees from vertical and reading the values.
This gives us digital read out values for +1, 0 and −1 times g, which can then be interpolated to
give a value of acceleration in terms of g.
We can now measure the amplitude of the loudspeaker with automatic data collection and
handling by sweeping both through frequencies for a set amplitude or through amplitudes with a
fixed frequency. For constant amplitude value see figure 5a and for sweeped amplitude see figure
5b.
For the sweeped frequency we see a clear resonant frequency of the loudspeaker around 19Hz.
This means that the frequency spectrum where we perform our experiment has a large rise in
amplitude. This is a non-linearity in our system that we do not like at all.
As a function of the digital amplitude value sent, we see that amplitude is very close to being
linear within our uncertainty.
9 We
don’t know if this is actually true, but the output signal from the DAC is almost perfectly sinusoidal.
to the datasheet. We have tried to confirm this and have not found any large deviations from the declared
accuracy
10 According
11
3.5
Measuring the coefficient of restitution
3
Uncertainties were calculated using the law of error propagation:
s
2 2
−2amax
1
σA =
σ
+
σ
a
4π 2 f 2
4π 2 f 3 f
THE EXPERIMENT
(20)
We know from the amplitude vs. frequency scan in figure 5a that our experiment is done with an
increase in amplitude as well as frequency. Therefore we made a finer scan of the amplitude vs.
frequency in the area and fitted a function that would give us a good amplitude for our simulation.
This is shown in figure 15 on page 24.
Even though the accelerometer is light, it does have a mass which may effect the oscillations of
the loudspeaker, and we have therefore conducted the entire experiment with it mounted.
We have noted a possible error in our setup. The reference voltage for our microcomputer and
DAC have been the 5 V supply from a standard USB connection. This seems however, to be quite
dependent on the how much current is drawn, whether the PC i charging, plugged in but fully
charged or running on battery etc. This is an unfortunate error, discovered to late to correct it. It
could have been solved by using a dedicated, high quality power supply instead. The error will
affect the amplitude most severely.
3.5
Measuring the coefficient of restitution
The coefficient of restitution, Cr , was measured by letting the ball drop from a small height onto the
stationary loudspeaker and measuring the time between the impacts. Assuming only gravitational
forces, the time between an impact and the next is then only determined by the velocity with which
the ball jumps. Then we can measure Cr by simply finding the ratio between two consecutive jumps
time (see equation (13)).
The time series were recorded with a digital oscilloscope that records one data point for each
20µs. Considering then the width of one impact peak in the (v,t)-plot of the oscilloscope we
determined one oscillation of the piezoelectric crystal to be approximately 260µs. A good estimate
of the uncertainty on the impact time will then be 2602+20 µs. When considering a difference the
uncertainty will then be 280µs because each time value is free to move in both directions.
Using the error propagations law on our formula for the coefficient of restitution we get:
Cr =
∆ti
∆ti−1
s
(21)
1
·σ
∆ti−1 ∆ti
σi =
2
+
−∆ti
·σ
(∆ti−1 )2 ∆ti−1
2
(22)
where σi denotes the error of the individual measurement of Cr .
Each time we let go of the ball it jumps approximately seven times which gives five individual
measurements of Cr,i . A graphical presentation is given in figure 6b.
By weighing according to error, σ12 , we find an estimate of Cr for one run, Ĉr . The deviation from
this mean is calculated by:
i
s
σĈr =
n
1
(C − Ĉr )2
∑
n − 1 i=1 r,i
(23)
where n − 1 is necessary because one degree of freedom is lost when the mean was found.
Impact data from 8 runs generates a series of Ĉr . Again weighing according to error - now with
σĈr as error - we find an overall estimate of Cr . The uncertainty for the overall value is set to the
deviation of σĈr ’s from the overall estimate according to (23).
We determine the coefficient of restitution to be:
Cr = 0.685 ± 0.009
(24)
Without the tape-coat of the glass surface the Cr of our system was approximately 0.97, giving us a
very slow convergence to the stable periods.
We note a dependency in measurements of the coefficient of restitution upon the impact velocity
(the last Cr ’s of a run are lower than the first). However, since our model deals with this as a
constant, we have chosen to neglect this. It should also be noted that the impact velocities found in
our experimental setup seems to be higher than where we starts to see this deviation.
12
3.6
Measuring the phase shift
3
THE EXPERIMENT
(b) Cr estimates
(a) Piezo element output
Figure 6: Data from one run, measuring the coefficient of restitution of our ball. (a) shows the piezo element
output from seven impacts on fixed loudspeaker. (b) shows Cr estimates from each pair of consecutive time
intervals between impacts plotted in the order of the impacts. One sees lower Cr for the last impacts.
3.6
Measuring the phase shift
When a sinusoidal voltage signal is fed to an amplifier/speaker system, one does generally not
expect the input signal and the physical oscillations of the loudspeaker to be in phase. Furthermore
we cannot rule out a frequency dependence on the size of the phase shift.
Therefore we have tried to measure this phase shift. We recorded the acceleration with the
same accelerometer as we used to measure the amplitude, but this time with a digital oscilloscope,
yielding a higher sample rate. By comparing the acceleration of the loudspeaker with and without
a bouncing ball we could estimate the impact time/phase by finding a sharp discontinuity in
the acceleration signal. This of course all depends on whether the loudspeaker and the measured
acceleration is in phase or not. We see no reason for this not the be the case.
By comparing the measured impact phase with the phase recorded by the microcomputer we
can then find the phase shift between the microcomputer and loudspeaker.
The measured phase of the impact is 1.03 ± 0.02 rad at 17.7 Hz. The microcomputer output for
the same parameters is a phase of 0.72 ± 0.03 rad. This gives a phase shift of 0.31 ± 0.04 rad. We
have not been able to detect a frequency dependent phase shift within our frequency range, and
have therefore chosen to regard it as a constant.
Figure 7: Acceleration of loudspeaker membrane showing clear impact disturbances
13
4
4
NUMERIC SIMULATION
Numeric Simulation
To be able to investigate the behaviour of the system more thoroughly we have written a MatLab
program to do numeric simulation. From the experiment we get a measurement of the time interval
between each impact, and by doing the necessary equipment studies we are able to get a reliable
determination of where in the phase of the membrane motion there is an impact. What happens
between impacts, our experiment cannot tell us.
Furthermore, a simulation gives an opportunity to test whether the behaviour we see in the
experiment can be explained by the dynamics stated in chapter 2 or if there are other circumstances
that should be taken into account.
The simulation is driven by the equations from section 2.1. It calculates the quantities xb , vb , xt , vt
over time with a time step of 0.0008s. When xb < xt it takes a step backwards and calculates with
half the previous time step. It repeats this precision method several times until the distance between
the two is very small and registers this as an impact.
It repeats the whole process a thousand times - hereby having a thousand impacts - and plots
the time interval between the last 10. The programme then changes the frequency in order to sweep
a spectrum of frequencies and performs the calculations once more for the new frequency.
(a) period-1
(b) period-2
(c) period-4
Figure 8: A simulation of how the ball bounces on the loudspeaker.
Here we discovered the importance of continuation in the simulation. Of course when starting
the simulation, the ball and membrane is set in a chosen position. Starting in a period-1 set of
parameters (amplitude and frequency) the ball quickly finds the stable period-1 state. However, if
every time the frequency changes the ball and membrane position is reset then periodicity becomes
randomly unstable in the sweep of frequencies. After correcting the programme so the ball and
membrane position is kept when changing the frequency, the randomly instability disappears.
This realization shows the strength of the attractors in the system. The period-1 attractor is
strong, meaning that almost any initial condition will find it. When approaching more-period
attractors there is a smaller interval of initial conditions which allows for the system to find the
more-period attractor. There is a more significant tendency to stabilize in the period-1 motion
jumping over two oscillations of the membrane or entering a sticky solution - which the simulation
allows for.
4.1
Attractor studies
As mentioned in section 2.3 the state of the system can be fully described in a three dimensional
phase space. The numeric simulation described above can be used to graphically express the
attractor. This is done by plotting x b , vb , φ for each time step between the last 100 impacts.
From figure 9 we get an idea of how the attractor changes with the frequency. From (a) to (b)
the curve sort of splits now making two loops but still being one curve. From (b) to (c) we see that
the distance between the two loops grows until it "splits" again in (d). Same thing happens again
and eventually we have (f) where we see extremely close loops. This bifurcating behaviour is highly
comparable to the behaviour of the logistic map (see Appendix A).
Two straight lines, representing discontinuities can be seen for each loop on the trajectories. One
is vertical and represent nothing else than the time when the phase of the loudspeakers reaches 2π
14
4.1
Attractor studies
4
NUMERIC SIMULATION
(a) period-1
(b) period-2 right after bifurcation
(c) period-2 for larger λ
(d) period-4
(e) period-8
(f) chaotic regime
Figure 9: Attractors in the phase space drawn using the simulation and showing states between the last 100
impacts. First axis is the distance between loudspeaker and ball, second axis is the velocity of the ball and third
axis is the phase of the loudspeaker motion.
15
5
DATA ANALYSIS - VARIED FREQUENCY
and returns to zero. The other it along the velocity-axis, i.e., when x=0. It represents impacts, were
velocities are reversed. It goes from negative to positive. This also gives the direction of the flow in
the phase space.
5
Data analysis - varied frequency
To analyse the route to chaos for our system we want to plot experiment data in a way that shows
the bifurcations. The plot will be much like the one in figure 3. For the logistic map we have a
control parameter λ which is gradually changed in the bifurcation diagram. In chapter 2 we have
seen that the control parameters of the ideal system configuration are Cr , frequency and amplitude.
Unfortunately as we have seen in section 3.4 we cannot change the frequency alone, and studies have
shown that if we only change the amplitude we will only see the first bifurcation (see Appendix C).
On the first axis of the plotted bifurcation diagram in figure 10 is frequency, but the reader should
regard this as a combination of all three control parameters. Amplitude according to figure 15 and a
small variation in Cr with the impact speed.
The logistic map is one dimensional and therefore the second axis of the diagram is an easy
choice. However, the map of our system is two dimensional. Because of the choice of position
coordinate an impact is always when x b = 0 and leaves two dimensions for a final state indication,
i.e. phase and velocity. Both should be showing bifurcations for the same set of parameters. The
microcomputer sends information about the phase of each impact making it directly readable. The
velocity of the ball just before an impact is closely related to the fly time until the next impact (see
equation (8)). Therefore the time interval between impacts, ∆t, will function as an expression of
velocity.
Our experimental results, plotted together with our simulation, is shown in figure 10 on the
following page. The experiment was run with a frequency shift of 0.002Hz and 100 impacts pr.
frequency step. We have plotted the last 40.
16
5.1
Discussion of the experimental results
5
(a) phase plot
DATA ANALYSIS - VARIED FREQUENCY
(b) phase plot with phase correction
(c) ∆t-plot
Figure 10: Experimental data showing three bifurcations. Blue: Experimental data. Green: Simulation. Red:
One oscillation of the loudspeaker
5.1
Discussion of the experimental results
We see a clear correlation between our measured and simulated results. They both show bifurcations
at approximately the same frequencies. The phase plot assumes almost exactly the same shape as
the measured results, when correcting for the phase shift found in section 3.6 on page 13.
We note that the measured results seem to converge to points further from each other than
those found in the simulation. This may for example be due to the rise in amplitude, described in
section 3.4 on page 11, which may not have been reproduced correctly in our simulation, a different
Cr etc.
We also note that the last bifurcations occur at higher frequencies in the simulation. This may be
due to the same reasons as the width of the branches.
On close inspection the experimental data seems to indicate two-periodicity before the first
bifurcation. This is due to a to low number of impacts pr. frequency. When the frequency is changed
the ball is disturbed in its trajectory and converges only slowly to the stable one-periodic motion.
We also see this after the bifurcation where the impacts converge only slowly to the new stable
points. These ghosts of stable states before/after the bifurcation is commonly found in systems like
ours - the attracting strength of a stable points gets weaker when close to the bifurcation.11
11
[1] p. 99.
17
5.2
5.2
Our experiment and Feigenbaum’s delta
6
CONCLUSION
Our experiment and Feigenbaum’s delta
Since we have seen three bifurcations we can make an estimate of Feigenbaum’s delta. Note however,
that since the amplitude was varied as well as the frequency, it is not the true ratio of λ that we plug
into our formula. We have
λ n +1 − λ n
δ = lim
(25)
n → ∞ λ n +2 − λ n +1
and uncertainty is then given by
s
σδ =
∂δ
σ
∂λ1 ∂λ1
2
+
∂δ
σ
∂λ2 λ2
2
+
∂δ
σ
∂λ3 λ3
2
(26)
We read the three bifurcations to occur at: λ1 = 17.82 ± 0.02Hz, λ2 = 18.08 ± 0.005Hz and
λ3 = 18.14 ± 0.01Hz. When plugged in we get:
δ = 4.3 ± 0.9
(27)
We note that the theoretical value of δ, 4.699... and the first delta of the logistic map δ1 = 4.7218 falls
within the uncertainty of our measured δ. We also note that the value found in our experiment
(4.3) is listed as the most commonly found for our experiment in the article "Chaotic dynamics of a
bouncing ball" [3] p. 943.
6
Conclusion
We have investigated the dynamics of a simple deterministic system, showing clear signs of chaotic
behaviour. We have measured, using different experimental techniques, different aspects of our
setup, including amplitude, phase shift and the coefficient of restitution, and conducted automatic
data collection and handling. The focus of the project have shifted gradually, and as new problems
and intriguing questions arose, we have investigated the things we found most fruitful.
From these studies we have come to thorough knowledge of our experimental setup which have
enabled us to simulate the experiment in detail and explain to some extent when results were not as
expected.
We have found an experimental estimate of Feigenbaum’s constant, 4.3 ± 0.9. The value found is
a little lower than the value of the logistic map, but within the uncertainty. However, the uncertainty
of our experimental estimate is large and we have therefore not been able to give a good estimate
of Feigenbaum’s constant with our setup. Things to consider for a better estimate would be a
better control of parameters, e.g. keeping the amplitude constant while changing frequency, and
improving the setup in order to see a fourth bifurcation.
We have seen the expected bifurcations, indicating the doubling route to chaos, as a function of
changed frequency (and amplitude), and shown a clear correlation to our numerical simulations of
the system. From this correlation we have made it clear that the chaotic behaviour of the experiment
is - to a large extent - explained by the dynamic equations in chapter 2. Hence we can consider it a
deterministic system with chaotic behaviour.
18
7
7
REFERENCES
References
[1] Steven H. Strogatz. Nonlinear Dynamics and Chaos. Perseus Publishing, 1994.
[2] Ivan Iakupov, Therkel Andreas Zøllner Olesen, and Amalie Christensen. Bouncing ball undersøgelse af et kaotisk system - first year project. 2009.
[3] N.B. Tufillaro and A.M. Albano. Chaotic dynamics of a bouncing ball. American Journal of
Physics, 10:939–944, 1986.
19
A
A
N-STABLE PERIODS
N-stable periods
(a) 4 points
(b) 8 points
(c) 16 points
(d) 32 points
Figure 11: Cobwebs of n-periodicity
This figure shows the logistic map for the 4-, 8-, 16- and 32-stable periodicities. The graphs are
made by doing ten thousand iterations and drawing the last hundred values with connecting lines.
20
B
B
N OSCILLATIONS PER IMPACT
Minimum amplitude for stable 1-periodicity over N oscillations
With large enough amplitudes it is possible to have stable 1-periodicity where the surface goes
through more than one oscillation per impact. This is of course also true for 2-, 4- etc. periodicities.
The smallest amplitude needed for a given frequencies and n oscillations of the surface is given by
the formula12
A( f , n) =
(1 − cr ) gn
4(1 + cr ) f 2 π
(28)
Figure 12: Minimum amplitude for stable 1-periodicity over N oscillations
The plot shows the minimum amplitude for stable periodicities over n oscillations of the
loudspeaker for n = 1, 2..10, with n = 1 being the one closest to the x-axis and n = 10 the one
furthest away. As seen from the graph the parameter zone of our experiment ( f ∈ [18.2; 19.2]Hz, A ∈
[0.72; 0.82]mm) allows for stable periodicities over one and two oscillations of the loudspeaker. One
can easily hear when the ball goes into stable motion over two oscillations. By laying a hand on
the loudspeaker membrane the dampening effect quickly sets the ball back in a motion over one
oscillation.
12 Deducted
from equation 14
21
C
C
AMPLITUDE SCAN
Data analysis - varied amplitude
We have also tried changing the amplitude instead of the frequency, to see wheter we could achieve
bifurcations here as well. A plot is seen in figure 13. We have only been able so see one bifurcation
as a funciton of varied amplitude. Therefore we have chosen to focus on the varied frequency. We
note however, that the resemblance to the plot in figure 10 on page 17.
Figure 13: Experimental results - varied amplitude. 17.3Hz
22
D
D
SIMULATION - CONSTANT AMPLITUDE
Simulation - constant amplitude
Figure 14: Simulation with constant amplitude
We ran our simulation with a constant amplitude to see what kind of values for λ and δ it
would give us. It gave us five clear values for λ: 17.475 Hz, 18.465 Hz, 18.635 Hz, 18.6737 Hz and
18.6805 Hz,which again gave us three different o values of δ: 5.8235, 4.3928 and 5.6912. These values
vary quite a bit from the expect value of 4.699.
23
E
E
AMPLITUDE IN FREQUENCY SPAN OF THE EXPERIMENT
Amplitude in frequency span of the experiment
Figure 15: The amplitude in the frequency span of the experiment. Amplitude value 3000.
To get a better starting point for our numeric simulation, we ran a fine scan of the amplitude
in the frequency span of the experiment, and fitted a quadratic funtion to it. The amplitude as a
frunction of frequency is of course not quadratic on a larger scale, but it it’s a quite nice fit for the
relevant frequencies.
24