Using Numerical Integration to Further Understand Dynamical Systems
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Some equations are impossible to solve analytically, so a numerical integrator must be used. Mainly studied the Lorenz System and the double pendulum, both of which are chaotic systems. Studied return map and symbolic entropy of Lorenz system. Studied fluctuation of total mechanical energy found in double pendulum, a conservative system. Further understanding chaos theory gives a plethora of applications, such as identifying large scale features in real world systems like ocean circulation. There seems to be an underlying order in the Lorenz system according to consistent measurements of its entropy (measurements were made using different partitions). The total energy in the double pendulum fluctuates, as is expected from a numerical integrator, but it does so in a sporadic and nonmonotonic way.