Publications from the SCIGMA project and related work in computational dynamics, focusing on the visualization and computation of invariant manifolds in dynamical systems.
Authors: M.E. Johnson, M.S. Jolly, and I.G. Kevrekidis
Published: Numerical Algorithms, Vol. 14 (1997), No. I-III
Also appears as: University of Minnesota Supercomputer Institute Research Report 96/108, May 1996
We illustrate and discuss the computer-assisted study (approximation and visualization) of two-dimensional (un)stable manifolds of steady states and saddle-type limit cycles for flows in ℝⁿ. Our investigation highlights a number of computational issues arising in this task, along with our solutions and "quick-fixes" for some of these problems. Two examples illustrative of both successes and shortcomings of our current approach are presented. Representative "snapshots" demonstrate the dependence of two-dimensional invariant manifolds on a bifurcation parameter as well as their interactions. Such approximation and visualization studies are a necessary component of the computer-assisted study and understanding of global bifurcations.
Authors: I.G. Kevrekidis, M.S. Jolly, Y. Gao, and M.E. Johnson
Published: International Journal of Bifurcation and Chaos, 2001
We demonstrate, through a combination of numerical computation and visualization, the existence of a previously undetected global bifurcation in the Kuramoto-Sivashinsky equation. This bifurcation, which we term the "Oseberg transition," involves the creation of a heteroclinic connection between two saddle periodic orbits. The visualization techniques developed for this study reveal the intricate structure of the phase space and the complex interactions between invariant manifolds.
Extensive work on the Kuramoto-Sivashinsky equation (KSE), a paradigmatic model for spatiotemporal chaos, formed the core of the PhD thesis work at Princeton.
This work contributed to understanding how coherent structures emerge and interact in spatially extended systems, with applications to fluid dynamics and pattern formation.
Work on proving the existence of inertial manifolds for the shifted KSE (SKSE) under specific symmetry conditions, extending the theory of spectral barriers.
This theoretical work demonstrated how spatial inhomogeneities affect the long-time dynamics of PDEs, with implications for understanding pattern formation in non-uniform media.
SCIGMA (Stability Computations and Interactive Graphics for invariant Manifold Analysis) was documented in various forms:
Development of numerical methods for:
The SCIGMA project and associated research contributed to:
"The construction of software to be used by scientists and mathematicians alike in the study of phase-, parameter-, and physical-space of dynamical systems with the hopes of creating a means by which one can perform robust numerical experiments."
This work was conducted during the mid-1990s when scientific visualization was emerging as a field, SGI workstations were the pinnacle of graphics computing, and the web was just beginning to transform scientific communication. The vision of interactive, web-based scientific computation would take another decade to fully realize, but these early efforts helped pave the way.