1. Using a Symbolic Mechanics
Program to Model Chaotic
Dynamical Systems
By Albert Yang
Saltire Software
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. 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. 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
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. Sensitivity Test
• The Sensitivity Test
tests for the sensit-
ivity to a change in
initial conditions.
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
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
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. 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.
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
