The integration of differential equations is normally
carried out using Runge-Kutta type methods that attempt to match terms
in a Taylor series expansion. In this talk, I will describe numerical
methods based on symbolic dynamics, a fundamental concept in nonlinear
dynamics. Unlike Runge-Kutta type methods, methods based on symbolic
dynamics can integrate chaotic systems with no loss of accuracy.
Because of this accuracy, it is possible to obtain plots of the
fractal structure of the Lorenz attractor. The fractal structure of
the Lorenz attractor was inferred by Lorenz in 1963 but was not
explicitly exhibited until now. The talk will outline further
applications to the Lorenz equations and to the three-body problem.
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