Approximation and Computation of Lyapunov Exponents, Global Errors, and Functional Traveling Waves

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Van Vleck9973393The investigator and his colleagues study issues in computational differential equations: shadowing and related topics of global error analysis for initial value problems, computation of stability information and Lyapunov exponents, and analysis and computation of lattice differential equations, i.e., differential equations that are discrete in space and continuous in time. This project has as a central theme the combination of dynamical systems ideas with numerical analysis ideas. The focus is on the development and analysis of efficient, accurate numerical techniques and on analytical and computational techniques that help identify interesting dynamic properties of differential equations.The investigator and his colleagues study methods for understanding the propagation of errors that occur during the numerical computation of differential equations. This understanding is fundamental because models of physical and biological systems are often solved using computational techniques. The research on lattice differential equations has applications in the area of materials modeling. The models being considered allow for the inclusion of both microscopic and macroscopic effects, which will become more important as advances in computer technology allow for larger, more sophisticated models of materials.