Computational Mechanics of Coherent Structures in Spatiotemporal Systems

ORAL

Abstract

The use of computer simulation and numerical solutions have become common for handling increasingly complex mathematical models of physical phenomena. This has been most successful in nonlinear systems where analytic solutions are scarce, as exemplified by the discovery of deterministic chaos. As attention moves to higher dimensional systems, gaining insight from numerical solutions is no longer trivial. In particular, systems in which simple interactions propagate in a complicated manner to produce complex emergent behavior present serious difficulties for traditional mathematical analysis. Such difficulties are similar to those faced in the theory of computation. Thus a new approach to complex systems, computational mechanics, has been developed that employs the mathematical structures of computation theory to build intrinsic representations of temporal behavior, rather than relying solely on the equations of motion. A rigorous theory of coherent structures in fully discrete classical field theories using computational mechanics is given. The method is demonstrated on the simplest such systems that support emergent structures, namely elementary cellular automata. Results are compared with a similar, but distinct, approach using invariant sets of dynamical systems theory.

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Authors

  • Adam Rupe

    UC Davis

  • Ethan Anderes

    UC Davis, Lawrence Berkeley National Lab - ALS, National Institute of Standards and Technology - NCNR, Univ of California - Davis, University of California, Davis, California State University, Long Beach, University of California, Irvine, University of California, Merced, UC Merced, U Central Florida, Paul Scherrer Institute, Paul Scherrer Institute in Switzerland, Hartnell Comm Coll, University of Michigan, University of Nevada, Reno, National Security Technologies LLC, Livermore, California, Humboldt State University, Stanford University, San Diego State Univ, Institute of Mathematical Problems of Biology, Eindhoven University of Technology, University Of Nevada Reno, Univ of Nevada - Reno, University of Chicago, Physics Department of the University of Nevada, Reno, NV, USA, Institute for Academic Initiatives, PPC and Graduate School of Engineering, Osaka University, Joint Institute for High Temperatures, PPC and Graduate School of Engineering, Osaka University, Institute for Academic Initiatives, Joint Institute for High Temperatures, National Institute of Standards and Technology, Gaithersburg, MD, Department of Physics, UC Davis, Department of Physics & Astronomy, Univeristy of California Irvine, 92697, Department of Chemistry and of Physics, Univeristy of California Irvine, 92697, Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot, Israel 76100, Einstein Centre for Local-Realistic Physics, Cal State Long Beach, University of nevada, Reno, Nicolaus Copernicus University, Torun, Poland, University of Maryland, CERN, Univ of California, Davis