A TASTE OF CHAOS By Adam T., Andy S., and Shannon R.
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Transcript of A TASTE OF CHAOS By Adam T., Andy S., and Shannon R.
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 xtDrxkyrykI
r
The Chaotic Oscillator:Equations of Motion
So far, contributions from spring force terms appear linear, but…
...
)(21221
2
frictionmagneticgravity
relaxedeqrelaxedeq tDkyykxxkI
rkk
I
r
?21 CC
Driving function
…Yeah.
The Chaotic Oscillator:Equations of Motion
...
)(21221
2
frictionmagneticgravity
relaxedeqrelaxedeq tDkyykxxkI
rkk
I
r
)()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( eqI
rmgFr
),(
3
1~R
5
1~R
5
1~R
…velocity-dependent damping?!
or
(experiment?)
The Chaotic Oscillator:Equations of Motion
5
1~R
nm
meqrelaxedeqrelaxedeq R
yk
I
rmgyykxxk
I
rkk
I
r
)(
)cos()sin(1221
2
)cos(222meqmm rrrrR
))cos(
(cos2
1
R
rry meqm
nR
fCCCC
)(
))(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 Results
Initial 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