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  1. A TASTE OF CHAOS By Adam T., Andy S., and Shannon R.

  2. “You've never heard of Chaos theory? Non-linear equations?” -Dr. Ian Malcolm, fictional chaotician

  3. 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.

  4. Experiment Chaotic motion

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

  6. 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

  7. 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!

  8. A Well with potential

  9. Potential Wells

  10. The Chaotic Oscillator:Equations of Motion (Newton’s Law – angular version of F=ma)

  11. The Chaotic Oscillator:Equations of Motion So far, contributions from spring force terms appear linear, but…

  12. The Chaotic Oscillator:Equations of Motion Driving function …Yeah.

  13. The Chaotic Oscillator:Equations of Motion Time-dependent driving function D(t) Gravity? Magnetic force?! ? … ?!! “dipole-induced dipole interaction”? Friction, etc.?!?! ??!!? …velocity-dependent damping?!

  14. The Chaotic Oscillator:Equations of Motion or  (experiment?)

  15. Torque vs. angle

  16. 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

  17. “Inevitably, underlying instabilities begin to appear…” “God help us, we’re in the hands of engineers” -Dr. Ian Malcolm, fictional chaotician

  18. Question: What do you get when you cross a shark with a telescope?

  19. Answer: | | = X

  20. The Runge-Kutta Method The Solution to All Our Problems (Or at least the first-order differential equation ones)

  21. 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

  22. Classical Runge-Kutta • Approximate solution of first-order ODEs. • Know initial conditions. • Choose step size. • Recurrence relation:

  23. Classical Runge-Kutta

  24. 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.

  25. 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

  26. 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.

  27. 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…

  28. 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…

  29. Our Equation of Motion • After we know these points, we can interpolate with a spline. • But first, we must collect the data.

  30. 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)

  31. The Spline Interpolation This is a clever subtitle.

  32. Spline Interpolation • The problem: • We have a set of discrete points. • We need a continuous function. • The solution: • Spline interpolation!

  33. 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

  34. 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

  35. Quadratic Spline • The interpolation gives a different function between every two points. • The coefficients of the spline are given by the recurrence relation

  36. 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)).

  37. 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.

  38. Our Spline (Take 1) • This method of determining α(θ) was abandoned. • We realized that DataStudio will record {θi,αi}.

  39. Our Spline (Take 2) • A quadratic spline was calculated a data set {θi,αi}. • Here’s a sample portion of the spline.

  40. Return to Runge-Kutta Endgame We are now able to approximate the solution θ(t).

  41. The Results Initial conditions: Start from right eq. position. ωi = 0

  42. The Results Initial conditions: Start from left eq. position. ωi = 0

  43. Motion of Chaos

  44. Motion of the Grimace Grimace time

  45. “That is one big pile of $@!*” -Dr. Ian Malcolm, Fictional chaotician

  46. “That is one big pile of $@!*” -Dr. Ian Malcolm, Fictional chaotician

  47. Poincare Plot • Periodic data points instead of a constant stream • Less cluttered evaluation of data • Puts harmonic motion in the spotlight

  48. Poincare plot

  49. Thanks everyone… Keep it chaotic