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## A TASTE OF CHAOS

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**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**Driven harmonic oscillator accessory**Mechanical Oscillator Photo gate Rotary motion sensor Springs Magnet DC Power supply Point mass (of unknown origins) Materials Of Chaos!**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 The Apparatus**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!**The Chaotic Oscillator:Equations of Motion**(Newton’s Law – angular version of F=ma)**The Chaotic Oscillator:Equations of Motion**So far, contributions from spring force terms appear linear, but…**The Chaotic Oscillator:Equations of Motion**Driving function …Yeah.**The Chaotic Oscillator:Equations of Motion**Time-dependent driving function D(t) Gravity? Magnetic force?! ? … ?!! “dipole-induced dipole interaction”? Friction, etc.?!?! ??!!? …velocity-dependent damping?!**The Chaotic Oscillator:Equations of Motion**or (experiment?)**Solving non-linear equations**• 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**“Inevitably, underlying instabilities begin to**appear…” “God help us, we’re in the hands of engineers” -Dr. Ian Malcolm, fictional chaotician**Question:**What do you get when you cross a shark with a telescope?**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:**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.**Ladies and Gentlemen, I give you…**Somethin’ Trickier • We can write a 2nd-order ODE as two coupled 1st-order ODEs. • Then we have Runge-Kutta recurrence relations**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 Results**Initial conditions: Start from right eq. position. ωi = 0**The Results**Initial conditions: Start from left eq. position. ωi = 0**Motion of the Grimace**Grimace time**“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**Thanks everyone…**Keep it chaotic