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.
To appear in
Numerical
Algorithms, Volume 14, (1997) No. I-III.
Figure 1. Triangulation of a part of an approximation of a stable
manifold of a steady state. For this computation, 10 points were
distributed on an initial ring linearly approximating the manifold.
Following the first six iterations of integration and redistribution,
29 points were needed to retain the average circumferential point density.
Figure 2. Schematic drawing of the initialization of computing
two-dimensional (un)stable manifolds of saddle-type limit cycles; see
text.
Figure 3. Triangulated representation of one
side of the stable manifold of a saddle-type limit cycle for the
Lorenz system at r=20.
Figure 4. Partial bifurcation diagram for the Arneodo {\it et al}. system
with respect to the parameter c for b=2.0. Solid lines: stable steady states; broken
lines: unstable steady states; full circles: stable limit cycles (a norm);
open circles: unstable limit cycles (a norm); notice the turning points at
the base of Silnikov period orbit ``corkscrews''. PD: Period doubling
bifurcation; HB: Hopf bifurcation; S marks the approximate location of
Silnikov heteroclinic loops.
Figure 5. Two views of the triangulation of the two-dimensional
$\WW^u_C$ for the Arneodo {\it et al.} example. The edge of
this ``cup'' is an attracting periodic solution, the multipliers of
which are negative and real and account for the wrapping of the cup
around its edge.
Figure 6. Phase space geometry for the Arneodo {\it et al.} example.
The top two panels show differently clipped views of the phase space at
c=3.0; the two-dimensional stable manifold of the origin (light
blue) ``folds" around one side of its unstable manifold (yellow)
which asymptotically
approaches a stable limit cycle (red).
The middle three and the lower left panels show elements of phase
space for c=3.9. (c) shows part of the two-dimensional unstable manifold
of C (the ``cup") after it has started intersecting with the stable
manifold of the origin; (c,d,e) show different clipped views of this intersection.
The lower right panel shows a view of the stable
manifold of the origin at c=7; notice the relative location of
the upper side of the unstable manifold of the origin as well
as its ``wrapping around" the stable manifold of the ``other"
fixed point C.
See text for further discussion.
Figure 7. Partial bifurcation diagram for the Lorenz equations for
b=8/3, \sigma=10. Critical values are r_H=24.74
and r_A\approx 24.06. We follow Sparrow's notation:
r_A corresponds to a heteroclinic connection between the origin
and the stable manifold of the
saddle limit cycles. The computations in this section were
performed for two values of r: r=20 and r=24.55.
Figure 8. The Lorenz system at r=24.55. The top two panels ((a)
and (b)) show the two-dimensional stable manifold of the origin in red
and (part of) the one-dimensional unstable manifold of the origin.
The lower left frame (c) shows the stable manifolds of each of a pair
of coexisting saddle-type limit cycles in blue (and red) and part of one side
of their unstable manifold in yellow (and olive green).
Figure 9. The Lorenz system at r=20.0. Frame (a) shows one side
of the stable manifold of each of two saddle-type limit cycles
(in red) along
with the one-dimensional unstable manifold of the origin
(in green) . Frame (b)
adds the unstable manifold of the limit cycle (in yellow)
to frame (a). Frame (c)
shows the triangulation of the unstable manifold of the limit cycle.
Frame (d) renders frame (c) and includes one side of the unstable
manifold of the origin.
maejohns@kerouac.princeton.edu
