Chaos

1
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
2
Outline

• Motivation and History
• Characteristics of Chaos
• The SHO (R.O.M.P.)
• The Pasco Setup
• The Lorenzian Waterwheel
• Feedback, Mapping, and Feigenbaum
• Conclusions

3
Motivation

• Chaos theory offers ordered models for seemingly
  disorderly systems, such as
• Weather patterns
• Turbulent Flow
• Population dynamics
• Stock Market Behavior
• Traffic Flow

4
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
5
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

6
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.
7
Non-Linearity

• Most physical relationships are not linear and
  aperiodic
• Usually these equations are approximated to be
  linear
• Ohms Law, Newtons Law of Gravitation, Friction

Nonlinear diffraction patterns of alkali metal
vapors.
8
The Damped Driven SHO

• This motion is determined by the nonlinear
  equation
• x oscillating variable (?)
• r damping coefficient
• F0 driving force strength
• ? driving angular frequency
• t dimensionless time
• Motion is periodic for some values of F0, but
  chaotic for others

Driven here with F0
Damped here with r
9
Random Oscillating Magnetic Pendulum (R.O.M.P.)
Demonstration of Chaos

• Non-linear equation of motion

http//www.physics.upenn.edu/courses/gladney/mathp
hys/subsection3_2_5.html
Where b, C are amplitudes of damping and the
driving force, respecitively
http//www.thinkgeek.com/geektoys/cubegoodies/6758
/
10
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.com8080/phy_demo/ROMP/ROMP.ht
ml
11
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
12
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

13
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

14
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, c8/3
15
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

16
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
17
Mapping the Potential

1. Let the weight rotate all the way around once,
   without driving force
2. Take angular position vs. angular velocity data
   for the run
3. Potential energy is defined by the equation

Two wells represent equilibrium points. In the
lexicon of chaos theory, these are strange
attractors.
18
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)
Left Well
Right Well
19
Chaos Data

• Data Studio Generates
• Driving Frequency - Measured with photogate
• Phase Plot - Angular Position vs. Angular
  Velocity (Above)
• Poincare Diagram - Slices of Phase Plot taken
  periodically (Below)

20
Chaos Data

• Data Studio Generates
• Driving Frequency - Measured with photogate
• Phase Plot - Angular Position vs. Angular
  Velocity (Above)
• 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

21
Chaos Data
Below Chaotic Region Frequency 0.65 Hz
Chaotic Region Frequency 0.80 Hz
Above Chaotic Region Frequency 1.00 Hz
22
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
23
Probing Lower Boundary
Frequency 0.67 Hz
Left Well
Frequency plotted 1000f-900
Left Well
Frequency plotted 1000f-400
Right Well
Frequency 0.80 Hz
24
The Chaotic Circuit
R 47 kO C 0.1 µF
25
How it works

• Using Kirchoffs Law

26
Mapping

• You call xn1f(xn) mapping
• With f(a,xn) you can form a difference equation
  where x is in 0,1 and a is a model-dependent
  parameter
• A famous example is the logistic equation

• The function f(a, xn) generates a set of xn, this
  set is said to be a map

27
Logistic Map
Concepts of Chaos Theory
We end up at the same point, no matter where we
start
Chaos and Stability
28
Pitchfork Bifurcation and Chaos
Feigenbaums number
xn1 xn
a 3.1
4
a 4
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
29
Bifurcation Diagram
?a is the range in which the program varies a
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 a
30
Convergence
31
Feigenbaums number
Concepts of Chaos Theory
let Dan an - an-1 be the width between
successive period doublings
Dan1
n
an
Da
dn
1
3.0
2
3.449490
0.449490
4.7515
3
3.544090
0.094600
4.6562
4
3.564407
0.020317
4.6684
5
3.568759
0.004352
3.5699456
4.6692
limn dn 4.669202 is called the Feigenbaums
number d
Alexander Brunner
Chaos and Stability
32
Feigenbaums Number
Concepts of Chaos Theory

• The limit d is a universal property when the
  function f (a,x) has a quadratic maximum
• 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)

Facts
Alexander Brunner
Chaos and Stability
33
Lyapunov Exponents
Concepts of Chaos Theory
1
2
Alexander Brunner
Chaos and Stability
34
Lyapunov Exponents

35
Application to the Logistic Map

• As we found out, a gt 0 means chaos and a lt 0
  indicates nonchaotic behaviour

a
a lt 0
Alexander Brunner
Chaos and Stability
36
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

37
Useful Viewgraphs

• From Thornton
• Poincure through with side-by-side of 3-Space.
  (p. 168)
• Two point Poincure (p. 167)

38
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.com8080/phy_demo/ROMP/ROMP.ht
  ml
• 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