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 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 F 0 = driving force strength ω = driving angular frequency t = dimensionless time Motion is periodic for some values of F 0, but chaotic for others Driven here with F 0 Damped here with r

Random Oscillating Magnetic Pendulum (R.O.M.P.) –Non-linear equation of motion Where b, C are amplitudes of damping and the driving force, respecitively 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

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.

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 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 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) Right Well Left Well

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)

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 Region Frequency ~0.65 Hz Chaotic Region Frequency ~0.80 Hz Above Chaotic Region Frequency ~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

Mapping You call x n+1 =f(x n ) mapping With f(α,x n ) 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, x n ) generates a set of x n, 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 BrunnerChaos and Stability = 4 to solve the iteration graphically easier and to get a better overview, we draw the 45 0 line in the plot x n+1 = x n α = 3.1 α = 4

Bifurcation Diagram Δα is the range in which the program varies α Initial x is equal to x 0,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 Da n = a n - a n-1 be the width between successive period doublings Da n+1 n anan Da dndn lim n®¥ d n » is called the Feigenbaum´s number d Alexander BrunnerChaos 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 BrunnerChaos and Stability

Concepts of Chaos Theory Lyapunov Exponents 1 2 Alexander BrunnerChaos and Stability

‹ Lyapunov Exponents

Application to the Logistic Map As we found out, a > 0 means chaos and a < 0 indicates nonchaotic behaviour Alexander BrunnerChaos 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 – – – – –Thornton, Steven T. and Jerry B. Marion. Classical Dynamics of Particles and Systems., Chapter 4: Nonlinear Oscillations and Chaos R.O.M.P. – – – Mapping and Lyapunov Exponents –Theoretische Physik I: Mechanik by Matthias Bartelmann, Kapitel 14: Strabilitaet und Chaos –