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Chaos Presentation

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Chaos Presentation

  1. 1. Using a Symbolic Mechanics Program to Model Chaotic Dynamical Systems By Albert Yang Saltire Software
  2. 2. Project Introduction • Saltire Software’s Mechanical Expressions (a symbolic mechanics software) is used to model chaotic dynamical systems – Enables the analysis of complex dynamical systems • Three tests are identified to classify chaotic nature in dynamical systems – Sensitivity Test – Mixing Test – Fourier Transform Test
  3. 3. Definitions • Differential Equation: A mathematical equation which relates a function, and its derivatives • Dynamical System: Mathematically, it is a concept where a fixed equation relates the position of a point to the time. • Deterministic system: A system where the outcomes and results of a system are defined by the initial conditions. • Phase Space: A phase space is a diagram which outlines and shows all possible configurations in a dynamical system.
  4. 4. Chaos Theory • Chaos theory deals with the study of sensitive dynamical systems – A small change in initial conditions may result in a huge change in the end state • Examples of natural chaotic dynamical systems include: – Weather – Solar System/Galaxy – Double pendulum • Chaos theory is so aptly named because the behavior tends to be chaotic
  5. 5. Construction of the Flywheel
  6. 6. Construction of the Flywheel
  7. 7. Flywheel Animation
  8. 8. Flywheel Motion = 2 ∙ (−5 + 33 − 20 ∙ cos 𝜃 𝑡 − 20 ∙ cos 𝜑 𝑡 + 8 ∙ cos 𝜃 + 𝜑 𝑡 ) ∙ (−5 ∙ sin 𝜃 𝑡 + 2 ∙ sin 𝜃 𝑡 + 𝜑 𝑡 33 − 20 ∙ cos 𝜃 𝑡 − 20 ∙ cos 𝜑 𝑡 + 8 ∙ cos 𝜃 𝑡 + 𝜑 𝑡
  9. 9. Flywheel Motion
  10. 10. Chaos Tests • There are 3 main tests that can be used to test for chaotic behavior in a dynamical system • They are: • Sensitivity Test Mixing Test Transform Test
  11. 11. Sensitivity Test • The Sensitivity Test tests for the sensit- ivity to a change in initial conditions.
  12. 12. Sensitivity Test
  13. 13. Mixing Test • Topological mixing is when the phase space of the dynamical system is completely filled. • If at some time in the system, it reaches the same point with the same velocity, then the motion has to be the same. • An example of the phase space of periodic motion
  14. 14. Mixing Test • The phase space of the flywheel system:
  15. 15. Fourier Transform Tests • Fourier Transforms can be used to transform functions from the time domain to the frequency domain – Instead of time being the dependent variable, frequency is • A Discrete Fourier Transform (DFT) is used to transform functions whose actual equation is not known – A Fast Fourier Transform (FFT) is the efficient method of solving
  16. 16. Fourier Transform Tests Periodic Function Flywheel Function
  17. 17. Analysis of the Tests • Now the question is: do these tests always work? • Answer: No. Take the following system: 5 10 5 5 10 5 A 1 B C 2 3 y x
  18. 18. Analysis of the Tests • Sensitivity Test:
  19. 19. Analysis of the Tests • Mixing Test:
  20. 20. Analysis of the Tests • Fourier Transform Test
  21. 21. Analysis of the Tests • Sensitivity Test Passes • Mixing Test Doesn’t Pass • Fourier Transform Test Passes • The main conclusion that can be made here is that all of the tests are required to make sure that a system is indeed chaotic – One test may fail where the others may not. •
  22. 22. Conclusions • A question: why bother with the numerics of chaos if they aren’t guaranteed to be accurate? • By analyzing specific patterns that aren’t affected too much by the buildup of error in the system, systems can be categorized as chaotic or non-chaotic. – There is little dependence on the actual numbers being outputted • The tests have been shown to be effective ones, although with limitations • Conclusions can be made that there are three reliable tests in order to determine the presence of chaos in a dynamical system.
  23. 23. Citations
  24. 24. Acknowledgements • Everyone at Saltire Software who helped develop Mechanical Expressions. It’s an amazing program. • Everyone at Maplesoft who helped develop Maple. It’s another amazing program. • Mentor, Phil Todd. Hours of mentoring, the original flywheel design, and always more questions to ask and more things to investigate. • Mom and Dad • ASE Coordinators and Volunteers for making this entire thing possible

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