Two-dimensional invariant manifolds and global bifurcations:
some approximation and visualization studies
M.E. Johnson, M.S. Jolly, and I.G. Kevrekidis
Abstract
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.
This article appears as University
of Minnesota Supercomputer Institute Research Report Numbered 96/108,
May 1996 and is also to appear in
Numerical
Algorithms, Vol. 14, (1997), No. I-III.
Here you may find a Figure-less PostScript version,
a Figure-less .dvi, and the
figures
inline in html format.
maejohns@alumni.Princeton.EDU