Using a Symbolic Mechanics Program to Model Chaotic Dynamical Systems

Using a Symbolic Mechanics Program to Model Chaotic Dynamical SystemsBy Albert YangSaltire Software

Project IntroductionSaltire Softwares Mechanical Expressions (a symbolic mechanics software) is used to model chaotic dynamical systemsEnables the analysis of complex dynamical systemsThree tests are identified to classify chaotic nature in dynamical systemsSensitivity TestMixing TestFourier Transform Test

DefinitionsDifferential Equation: A mathematical equation which relates a function, and its derivativesDynamical 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.

Chaos TheoryChaos theory deals with the study of sensitive dynamical systemsA small change in initial conditions may result in a huge change in the end stateExamples of natural chaotic dynamical systems include:WeatherSolar System/GalaxyDouble pendulumChaos theory is so aptly named because the behavior tends to be chaotic

Construction of the Flywheel

Construction of the Flywheel

Flywheel Animation

Flywheel Motion

Flywheel Motion

Chaos TestsThere are 3 main tests that can be used to test for chaotic behavior in a dynamical systemThey are:Sensitivity TestMixing TestTransform Test

Sensitivity TestThe Sensitivity Test tests for the sensit- ivity to a change in initial conditions.

Sensitivity Test

Mixing TestTopological 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

Mixing TestThe phase space of the flywheel system:

Fourier Transform TestsFourier Transforms can be used to transform functions from the time domain to the frequency domainInstead of time being the dependent variable, frequency isA Discrete Fourier Transform (DFT) is used to transform functions whose actual equation is not knownA Fast Fourier Transform (FFT) is the efficient method of solving

Fourier Transform Tests Periodic Function Flywheel Function

Analysis of the TestsNow the question is: do these tests always work?Answer: No. Take the following system:

Analysis of the TestsSensitivity Test:

Analysis of the TestsMixing Test:

Analysis of the TestsFourier Transform Test

Analysis of the TestsSensitivity Test PassesMixing Test Doesnt PassFourier 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 chaoticOne test may fail where the others may not.

ConclusionsA question: why bother with the numerics of chaos if they arent guaranteed to be accurate?By analyzing specific patterns that arent 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 outputtedThe tests have been shown to be effective ones, although with limitationsConclusions can be made that there are three reliable tests in order to determine the presence of chaos in a dynamical system.

Citations

AcknowledgementsEveryone at Saltire Software who helped develop Mechanical Expressions. Its an amazing program.Everyone at Maplesoft who helped develop Maple. Its 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 DadASE Coordinators and Volunteers for making this entire thing possible