ChaosDan Brunski

Dustin CombsSung 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

MotivationChaos 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

• 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

Driven here with F0

Damped here with r

tFxxrx cossin

Random Oscillating Magnetic Pendulum (R.O.M.P.)

– Non-linear equation of motion

http://www.thinkgeek.com/geektoys/cubegoodies/6758/

Where b, C are amplitudes of damping and the driving force, respecitively

http://www.physics.upenn.edu/courses/gladney/mathphys/subsection3_2_5.html

Demonstration of Chaos

Random Oscillating Magnetic Pendulum (R.O.M.P.)

Video displaying chaotic motion of R.O.M.P. with nine repelling magnets.

Right: Potential energy diagram of nine repelling magents

Potential energy diagram showing magnetic repelling peaks in a gravitational bowl

http://www.4physics.com:8080/phy_demo/ROMP/ROMP.html

• Sensitivity to initial conditions

Random Oscillating Magnetic Pendulum (R.O.M.P.)

A plot shows three close initial values yield three wildly varying results

Colors signify the final state of the pendulum given an initial value.

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:

Lorenzian waterwheel Lorenz attractor and the 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 measurementThe measurement of the amplitude

Mapping the Potential

Potential vs Position

-120

-100

-80

-60

-40

-20

0

-450 -400 -350 -300 -250 -200 -150 -100 -50 0 50

(theta)

-(o

meg

a)^

2

Potential vs Position

2

2

1 IcU

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

Two “wells” represent equilibrium points. In the lexicon of chaos theory, these are “strange attractors”.

Mapping the PotentialWe 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)

-180

-160

-140

-120

-100

-80

-60

-40

-20

0

-500 -400 -300 -200 -100 0

Driving Arm Down

Driving Arm Up

Right Well Left Well

-350 -300 -250 -200 -150 -100 -50

-7.5

-5

-2.5

2.5

5

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)

-350 -300 -250 -200 -150 -100 -50

-7.5

-5

-2.5

2.5

5

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

Below Chaotic RegionFrequency ~0.65 Hz

Chaotic RegionFrequency ~0.80 Hz

Above Chaotic RegionFrequency ~1.00 Hz

Chaos Data

•4.7 Volts

Chaotic Regions

* Damping distance of 0.3 cm yielded no chaotic points

• 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

Probing Lower Boundary

Frequency plotted:1000*f-400

Frequency plotted:1000*f-900

Frequency ~ 0.67 Hz

Frequency ~ 0.80 Hz

Left Well

Right Well

Left Well

The Chaotic Circuit

R = 47 kΩ

C = 0.1 μF

How it works

• Using Kirchoff’s Law

00

13

300

22

12

1

)(

)(

VR

RxDxx

R

Rx

xDVV

VVR

RV

R

R

dt

dVRC

xdt

dVRCV

xdt

dxRCV

v

v

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

Concepts of Chaos Theory Logistic Map

Chaos and Stability

We end up at the same point, no matter where we start

Feigenbaum‘s number

Pitchfork Bifurcation and Chaos

Alexander Brunner Chaos and Stability

= 4

to solve the iteration graphically easier and to get a better overview, we draw the 450 line in the plot

xn+1

= xn

α = 3.1

α = 4

Bifurcation Diagram

Δα is the range inwhich the programvaries 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

Concepts of Chaos Theory Feigenbaum‘s number

let Dan = an - an-1 be the width between successive period doublings

Dan+1

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 Feigenbaum´s number d

Alexander Brunner Chaos and Stability

Concepts of Chaos Theory Feigenbaum‘s Number

Facts

• The limit δ is a universal property when the function f (α,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)

Alexander Brunner Chaos and Stability

Concepts of Chaos Theory Lyapunov Exponents

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

Alexander Brunner Chaos and Stability

a

a < 0

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