A TASTE OF CHAOS

By Adam T., Andy S., and Shannon R.

“You've never heard of Chaos theory? Non-linear equations?”

-Dr. Ian Malcolm, fictional chaotician

A TASTE OF CHAOS

• Aperiodic (not a repeated pattern of motion)

• Unpredictable due to sensitive dependence on initial conditions

• Not random… completely deterministic• Governed by non-linear equations of

motion (not just terms like x or x’, but also xn, (x’)n … although not all non-linear eq.’s are chaotic)

• Examples: weather (“butterfly effect”), circuits, fluid dynamics, etc.

Experiment

Chaotic motion

Materials Of Chaos!• Driven harmonic oscillator accessory• Mechanical Oscillator• Photo gate• Rotary motion sensor• Springs• Magnet• DC Power supply• Point mass (of unknown origins)

The Apparatus

• Mechanical oscillator drive

• Springs• Magnet

• Point Mass

• Sinusoidal driving force on spring 1

• Linear Restoring force• Sinusoidal varying

damping force• Sinusoidal varying

force of gravity

Initial conditions and settings

• Potential wells can create harmonic oscillations depending on initial conditions and settings

• Example changing driving amplitude created enough tension in spring one giving the point mass enough energy

for……Chaos!

A Well with potential

Potential Wells

The Chaotic Oscillator:Equations of Motion

(Newton’s Law – angular version of F=ma)

...121 frictionmagneticgravityspringspringI

I

...

))(()( 21

frictionmagneticgravity

relaxedeqrelaxedeq xtDrxkyrykIr

The Chaotic Oscillator:Equations of Motion

So far, contributions from spring force terms appear linear, but…

...

)(21221

2

frictionmagneticgravity

relaxedeqrelaxedeq tDkyykxxkIrkk

Ir

?21 CC

Driving function

…Yeah.

The Chaotic Oscillator:Equations of Motion

...

)(21221

2

frictionmagneticgravity

relaxedeqrelaxedeq tDkyykxxkIrkk

Ir

)()cos(2)( 22ddd rdrtdrdtD

Time-dependent driving function D(t)

Gravity? Magnetic force?! ? … ?!!“dipole-induced dipole interaction”?

Friction, etc.?!?! ??!!?

The Chaotic Oscillator:Equations of Motion

)sin( eqIrmgFr

),(

3

1~R

5

1~R

5

1~R

…velocity-dependent damping?!

or

(experiment?)

The Chaotic Oscillator:Equations of Motion

5

1~R

n

mmeqrelaxedeqrelaxedeq R

ykIrmgyykxxk

Irkk

Ir

)()cos()sin(1221

2

)cos(222meqmm rrrrR

))cos(

(cos2

1

Rrr

y meqm

nRfCCCC)())(cos()sin( 40321

)(21 CC

Torque vs. angle

• Analytical techniques of little use in non-linear situations

• We rely on numerical methods of solving the eqn’s of motion

• Due to extreme sensitivity, small computational errors can have drastic effects…

• Thus, advances in technology have been historically necessary for sophisticated studies of chaos

Solving non-linear equations

“Inevitably, underlying instabilities

begin to appear…”

“God help us, we’re in the

hands of engineers”

-Dr. Ian Malcolm, fictional chaotician

What do you get when you cross

a shark with a telescope?

Question:

Answer:

=X ||

The Runge-Kutta Method

The Solution to All Our Problems

(Or at least the first-order differential equation ones)

Numerical Solutions to ODEs

• Most differential equations have no analytical solution.

• We must approximate them numerically.– Euler

– Improved Euler– Runge-Kutta

• Trade-off: Computational ease vs. Accuracy

Classical Runge-Kutta

• Approximate solution of first-order ODEs.• Know initial conditions.• Choose step size.

• Recurrence relation:

Classical Runge-Kutta

2nd Order ODEs

• Classical Runge-Kutta is excellent… unless you’re us.

• Our equation of motion

is second order.• Thus, we need a slightly more tricky

method of approximation.

Somethin’ Trickier

• We can write a 2nd-order ODE as two coupled 1st-order ODEs.

• Then we have Runge-Kutta recurrence relations

Ladies and Gentlemen, I give you…

Somethin’ Trickier

• Notice that K1 and I1 are determined by initial conditions.

• Notice, also, that all other Ki and Ii are dependent on the preceding Kis and Iis.

Our Equation of Motion

• We can apply this technique to our equation of motion.

• Set

• Thus,

• And we have two coupled 1st-order equations.• Excellent…

Our Equation of Motion

• But wait! That mysterious magnetic/gravitational/frictional acceleration term is not known….

•But we can find the angular acceleration due to these forces at a given time or a given position…

Our Equation of Motion

• After we know these points, we can interpolate with a spline.

• But first, we must collect the data.

Data for Spline

• Creating a representation of force for gravity, magnetism, and lets say umm friction.

• Removal of springs and driving force• Rotating point mass and disk combination• Plot acceleration vs. position (hopeful

representation)

The Spline Interpolation

This is a clever subtitle.

Spline Interpolation

• The problem:– We have a set of discrete points.– We need a continuous function.

• The solution:– Spline interpolation!

Types of Splines

• Linear spline– Simply connect the dots

• Quadratic spline– Takes into account four points

• Cubic spline– Si(xi)=Si+1(xi)– Twice continuous differentiable

Types of Splines

• Linear spline– Simply connect the dots

• Quadratic spline– Takes into account four points

• Cubic spline– Si(xi)=Si+1(xi)– Twice continuous differentiable

Quadratic Spline

• The interpolation gives a different function between every two points.

• The coefficients of the spline are given by the recurrence relation

Our Spline (Take 1)

• Find {ti,θi} and {tj,ωj}.

• Use a spline interpolation to form functions t(θ) and α(t).

• Obtain α(θ) by way of α(t(θ)).

Our Spline (Take 1)

• Spline of {θi,ti} to get t(θ).– Uses the equation on the last slide.

• α(t) found by differentiating the spline of {tk,ωk}. (dω(t)/dt = α(t).)

• Same recurrence relation for zis as before.

Our Spline (Take 1)

• This method of determining α(θ) was abandoned.

– We realized that DataStudio will record {θi,αi}.

Our Spline (Take 2)

• A quadratic spline was calculated a data set {θi,αi}.

• Here’s a sample portion of the spline.

Return to Runge-Kutta

Endgame

We are now able to approximate the solution θ(t).

The ResultsInitial conditions:

Start from right eq. position.

ωi = 0

The ResultsInitial conditions:

Start from left eq. position.

ωi = 0

Motion of Chaos

Motion of the GrimaceG

rimac

e

“That is one big pile of $@!*”

-Dr. Ian Malcolm,Fictional chaotician

“That is one big pile of $@!*”

-Dr. Ian Malcolm,Fictional chaotician

Poincare Plot

• Periodic data points instead of a constant stream

• Less cluttered evaluation of data• Puts harmonic motion in the spotlight

Poincare plot

Thanks everyone…

Keep it chaotic