Chaos
Dan Brunski
Dustin Combs
Sung Chou
Daniel White
With special thanks to:
Dr. Matthew Johnson, Dr. Joel Keay,
Ethan Brown, Derrick Toth, Alexander
Brunner, Tyler Hardman, Brittany
Pendelton, Shi-Hau Tang
Outline
• Motivation and History
• Characteristics of Chaos
• The SHO (R.O.M.P.)
• The Pasco Setup
• The Lorenzian Waterwheel
• Feedback, Mapping, and Feigenbaum
• Conclusions
Motivation
Chaos theory offers ordered models for seemingly disorderly systems, such as:
• Weather patterns
• Turbulent Flow
• Population dynamics
• Stock Market Behavior
• Traffic Flow
Pre-Lorenz History
•
The qualitative idea of small changes sometimes having large effects has been
present since ancient times
•
Henry Poincaré recognizes this chaos in a three-body problem of celestial
mechanics in 1890
•
Poincaré conjectures that small changes could commonly result in large differences
in meteorology
Modern version of Three Body Problem
What is it all about?
• A dissipative (non-conservative) system couples somehow to the
environment or to an other system, because it loses energy
• The coupling is described by some parameters (e.g. the friction constant
for the damped oscillator)
• The whole system can be described by its phase-flux (in the phase-space)
which depends on the coupling parameters
• The question is now: are there any critical parameters for which the
phase-flux changes considerably?
• We study now the long term behaviour of various systems among differing
initial conditions
Sensitivity to Initial Conditions
• First noted by Edward
Lorenz, 1961
• Changing initial value by
very small amount
produces drastically
different results
The Strange Attractor of the turbulent flow equations. Each color represents varying ICs
by 10-5 in the x coordinate.
Non-Linearity
• Most physical relationships are not linear and aperiodic
• Usually these equations are approximated to be linear
– Ohm’s Law, Newton’s Law of Gravitation, Friction
Nonlinear diffraction patterns of
alkali metal vapors.
The Damped & Driven SHO
•
This motion is determined by the nonlinear
equation
Driven here with F0
x  rx  sin x  F cos t
•
x = oscillating variable (θ)
•
r = damping coefficient
•
F0 = driving force strength
•
ω = driving angular frequency
•
t = dimensionless time
•
Motion is periodic for some values of F0, but
chaotic for others
Damped here with r
Random Oscillating Magnetic Pendulum
(R.O.M.P.)
Demonstration of Chaos
– Non-linear equation of motion
http://www.physics.upenn.edu/courses/gladney/mathphys/subsection3_2_5.html
Where b, C are amplitudes of damping and the
driving force, respecitively
http://www.thinkgeek.com/geektoys/cubegoodies/6758/
Random Oscillating Magnetic Pendulum
(R.O.M.P.)
Right: Potential
energy diagram
of nine repelling
magents
Video displaying chaotic motion of
R.O.M.P. with nine repelling
magnets.
Potential energy diagram showing
magnetic repelling peaks in a
gravitational bowl
http://www.4physics.com:8080/phy_demo/ROMP/ROMP.html
Random Oscillating Magnetic Pendulum
(R.O.M.P.)
• Sensitivity to initial conditions
Colors signify the final state of the
pendulum given an initial value.
A plot shows three close initial values yield
three wildly varying results
http://www.inf.ethz.ch/personal/muellren/pendulum/index.html
Lorenzian Water Wheel
Sketch and description
• Clockwise and counterclockwise rotation possible
• Constant water influx
• Holes in bottom of cups empty at steady rate
• As certain cups fill, others empty
Lorenz attractor
Attractor: A subset of the phase-space, which can not be left under
the dynamic of the system.
• In 1963 the meteorogolist Edward Lorenz formulated s set of
equations, which were an idealization of a hydrodynamic system in
order to make a long term weather forecast
• He derived his equations from the Navier-Stokes equations, the
basic equation to describe the motion of fluid substances
• The result were the three following coupled differential equations,
and the solution of these is called the Lorenz attractor:
Lorenz attractor and the waterwheel
Lorenzian waterwheel
• Fortunately the theoretical description of the Lorenzian Waterwheel
leads to the Lorenz attractor (maybe because both systems are
hydrodynamic)
• The equations of the Lorenz attractor can be solved numerically, the
solution shows that the behaviour is very sensitive to initial conditions
initial points differ only by 10-5 in the x-coordinate, a = 28, b = 10, c=8/3
The PASCO Pendulum
•
Weight attached to rotating disc
•
Springs attached to either side of
disc in pulley fashion
•
One spring is driven by sinusoidal
force
•
Sensors take angular position,
angular velocity and driving
frequency data
PASCO Chaos Setup
• Driven, double-spring oscillator
• Necessary two-minima potential
• Variable:
– Driving Amplitude
– Driving Frequency
– Magnetic Damping
– Spring Tension
The magnetic damping measurement
The measurement of the amplitude
Mapping the Potential
I.
Let the weight rotate all the way around once, without driving force
II.
Take angular position vs. angular velocity data for the run
III. Potential energy is defined by the equation U  c 
1 2
I
2
Potential vs Position
0
-450
-400
-350
-300
-250
-200
-150
-100
-50
0
50
-20
Potential vs Position
-(omega)^2
-40
-60
-80
-100
-120
(theta)
Two “wells” represent equilibrium points. In the lexicon of chaos theory, these are
“strange attractors”.
Mapping the Potential
We notice that the potential curve is highly dependent on the position of the
driving arm
(Left and Right refer to directions when facing the apparatus)
0
-20
-40
-60
-80
Driving Arm Down
Driving Arm Up
-100
-120
-140
-160
-180
-500
-400
-300
Right Well
-200
-100
Left Well
0
Chaos Data
• Data Studio Generates:
– Driving Frequency Measured with photogate
– Phase Plot - Angular Position
vs. Angular Velocity (Above)
5
– Poincare Diagram - Slices of
Phase Plot taken periodically
(Below)
2.5
-350
-300
-250
-200
-150
-100
-50
-2.5
-5
-7.5
Chaos Data
• Data Studio Generates
– Driving Frequency Measured with photogate
– Phase Plot - Angular Position
vs. Angular Velocity (Above)
5
– Poincare Diagram - Slices of
Phase Plot taken periodically
(Below)
– For a movie of this data, see
chaos-mechanical.wmv in
the AdvLab-II\Chaos\2008S\
Folder
2.5
-350
-300
-250
-200
-150
-100
-50
-2.5
-5
-7.5
Chaos Data
Below Chaotic Region
Frequency ~0.65 Hz
Chaotic Region
Frequency ~0.80 Hz
Above Chaotic Region
Frequency ~1.00 Hz
Chaotic Regions
• Chaotic Regions Dependent on:
• Driving Frequency, Driving Amplitude, Magnetic Damping
• Larger Amplitude – Larger Region
• More Damping – Higher Amplitudes, and narrower range of Frequency
• Hysteresis – Dependent on direction of approach
•4.7 Volts
* Damping distance of 0.3 cm yielded no chaotic points
Probing Lower Boundary
Frequency ~ 0.67 Hz
Left Well
Frequency plotted:
1000*f-900
Left Well
Frequency plotted:
Right Well
1000*f-400
Frequency ~ 0.80 Hz
The Chaotic Circuit
R = 47 kΩ
C = 0.1 μF
How it works
• Using Kirchoff’s Law
dx
 x
dt
dV
V2   RC 1  x
dt
R
 R
dV2
RC
  V2   V0  V3
dt
 Rv 
 R0 
V1   RC
V3  V1  D ( x)
R
 R
x    x  x  D ( x)   V0
 Rv 
 R0 
Mapping
• You call xn+1=f(xn) mapping
• With f(α,xn) you can form a difference equation where x is in
[0,1] and α is a model-dependent parameter
• A famous example is the logistic equation:
• The function f(α, xn) generates a set of xn, this set is said to be a map
Logistic Map
Concepts of Chaos Theory
We end up at the same point, no
matter where we start
Chaos and Stability
Feigenbaum‘s number
Pitchfork Bifurcation and Chaos
xn+1 = xn
α = 3.1
α ==44
to solve the iteration graphically
easier and to get a better overview,
we draw the 450 line in the plot
Alexander Brunner
Chaos and Stability
Bifurcation Diagram
Δα is the range in
which the program
varies α
Initial x is equal
to x0 ,the value
with which the
iteration starts
Signifies how often the
program should execute
the logistic map and tells it
how many points it should
calculate for one α
Convergence
Feigenbaum‘s number
Concepts of Chaos Theory
let Dan = an - an-1 be the width between successive
period doublings
Dan+1
n
1
2
3
4
5
an
Da
dn
3.0
3.449490
3.544090
3.564407
3.568759
0.449490
0.094600
0.020317
0.004352
4.7515
4.6562
4.6684
3.5699456
4.6692
limn®¥ dn » 4.669202 is called the Feigenbaum´s number d
Alexander Brunner
Chaos and Stability
Feigenbaum‘s Number
Concepts of Chaos Theory
• The limit δ is a universal property when the function f (α,x) has a
quadratic maximum
Facts
• It is also true for two-dimensional maps
• The result has been confirmed for several cases
• Feigenbaum's constant can be used to predict when chaos will
arise in such systems before it ever occurs .
(First found by Mitchell Feigenbaum in the 1970s)
Alexander Brunner
Chaos and Stability
Lyapunov Exponents
Concepts of Chaos Theory
1
2
Alexander Brunner
Chaos and Stability
Lyapunov Exponents
‹
Application to the Logistic Map
• As we found out, a > 0 means chaos and a < 0 indicates nonchaotic behaviour
a
a<0
Alexander Brunner
Chaos and Stability
Conclusions
• R.O.M.P.
– Simplest way to demonstrate chaotic behavior
• Pasco Chaos Generator
– Exhibits chaos in regions shown by phase plot
– Increased driving amplitude expands chaotic frequency range
– Increased damping
• Requires larger driving amplitude for chaos
• Shifts chaotic region to lower frequency
• Logistical Mapping
– We can characterize a system by determining Lyapunov Exponents,
which allow the mapping of chaotic and non-chaotic regions
• Future study
– Examine hysteresis in detail
– Refine phase plot by taking more data points
Useful Viewgraphs
From Thornton:
• Poincure through with side-by-side of 3-Space. (p. 168)
• Two point Poincure (p. 167)
Sources
• General Information
–
–
–
–
–
http://en.wikipedia.org/wiki/Lorenz_attractor
http://www.imho.com/grae/chaos/chaos.html
http://www.adver-net.com/mmonarch.jpg
http://www.gap-system.org/~history/Mathematicians/Poincare.html
Thornton, Steven T. and Jerry B. Marion. Classical Dynamics of Particles and Systems.,
Chapter 4: Nonlinear Oscillations and Chaos
• R.O.M.P.
– http://www.thinkgeek.com/geektoys/cubegoodies/6758/
– http://www.4physics.com:8080/phy_demo/ROMP/ROMP.html
– http://www.inf.ethz.ch/personal/muellren/pendulum/index.html
• Mapping and Lyapunov Exponents
– Theoretische Physik I: Mechanik by Matthias Bartelmann, Kapitel 14: Strabilitaet und Chaos
– http://de.wikipedia.org/wiki/Hauptseite