Chaos You Can Play In May Tan Lim, Erin Miller, Nicky Grigg, Aaron - - PowerPoint PPT Presentation

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Chaos You Can Play In May Tan Lim, Erin Miller, Nicky Grigg, Aaron - - PowerPoint PPT Presentation

Apr 28, 2023 220 likes •413 views

Chaos You Can Play In May Tan Lim, Erin Miller, Nicky Grigg, Aaron Clauset SFI Complex System Summer School 3 July 2003 Outline Experimental setup Equations of Motion The Lorentz Equations Mathematical Simulation Data

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SLIDE 1

May Tan Lim, Erin Miller, Nicky Grigg, Aaron Clauset SFI Complex System Summer School 3 July 2003

Chaos You Can Play In

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SLIDE 2
  • Experimental setup
  • Equations of Motion
  • The Lorentz Equations
  • Mathematical Simulation
  • Data Analysis
  • Getting Lucky

Outline

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SLIDE 3

Experimental Setup

Diagram from Strogatz (1994)

Tracking the fluorescent ball color CCD camera (fish eye lens) shutter speed = 1/2000 s NI frame grabber + LabView 6.0

Wheel Diameter 25cm Cup Diameter 6.6cm Cup Volume 400mL Inclination Angle 15 deg

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SLIDE 4

Waterwheel in Action

Watch for the change in behavior

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SLIDE 5

Equations of Motion

2) Mass change in each cup 1) Angle change for each cup 3) Torque balance of entire wheel

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SLIDE 6

Equations of Motion

2) Mass change in each cup 1) Angle change for each cup 3) Torque balance of entire wheel Note: Q = 0 for m > mmax

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SLIDE 7

Equations of Motion

2) Mass change in each cup 1) Angle change for each cup 3) Torque balance of entire wheel Note: Q = 0 for m > mmax

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SLIDE 8

Leak Rate

Mass Time (by 100’s of ms)

Our assumption – Potential energy per unit volume at top of liquid is equal to kinetic energy per unit volume of leaking water.

So…

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SLIDE 9
  • Lorenz system
  • Discrete vs.

continuous distribution of mass

  • Take lowest order

term in Fourier expansion, then change variables

  • Completeness of

model relative to experiment

Limitations of Strogatz Model

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SLIDE 10

Simulated Mass Regimes

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SLIDE 11

Omega Regimes

Lorenz Equations Waterwheel Equations

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SLIDE 12

Model Agreement

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SLIDE 13
  • reconstruction preserves

topological features

  • delay coordinate (tau) embedding
  • average mutual entropy
  • global false nearest neighbors, d
  • d not associated with

dimensionality of original system

Phase Space Reconstruction

NN NN NN NN

y(k) = [s(k), s(k+T), ... , s(k+(d-1)T ] y (k) = [s (k), s (k+T), ... , s (k+(d-1)T]

E E

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SLIDE 14

Lorenz and Model Attractors

Lorenz Model

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SLIDE 15

Model/Reality Agreement

Model Data

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SLIDE 16
  • simulating x and x+delta
  • local Lyapunov exponent - nearby points

separate exponentially in time

Sensitivity to Initial Conditions

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SLIDE 17

Sensitivity to Initial Conditions

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SLIDE 18

Special thanks to

  • Andrew Belmonte
  • Ray Goldstein
  • CSSS Experimental Lab Sponsors