Brusselator with Two Coupled Cells
The equations reflect the Brusselator
kinetics in each cell.
There are two kinds
of reactions. For a homogeneous reaction, both cells exhibit
The reaction is said to be inhomogeneous if this identity does
The branching behavior is depicted in
Figure 1. The branch indicated by a full line and the branch that
bifurcates off at
(dashed lines) consist of stationary
the stability is exchanged. All the other
branches consist of periodic orbits. For the value
is a Hopf bifurcation into periodic homogeneous orbits. First, the
emerging branch is unstable; it gains stability at
via a subcritical pitchfork bifurcation.
That is, for
there are stable homogeneous
oscillations, which are ``encircled" by unstable inhomogeneous
orbits. The two branches of stationary inhomogeneous solutions,
which exist for
lose their stability at the two Hopf
bifurcation points with
A remarkable part of the solutions is the
Hopf bifurcation at
Because all of the solutions merging in
the bifurcation point
are homogeneous, the coupling terms vanish; the
equations of the two cells decouple. Consequently, the same
solutions can be calculated by solving only the subsystem of the
first two equations (n=2). In this way, both the homogeneous
equilibria and the homogeneous oscillations are obtained.
However, the stability behavior is different. The bifurcations
into inhomogeneous solutions do not exist for the reduced system, and there
is an exchange of stability at
that by the process of coupling two identical cells the stability
behavior can change drastically. In particular, the homogeneous
orbits in the range
lose stability, caused by the onset of additional bifurcations.
This serves as an example for the phenomenon that relaxing a
restriction can enrich the bifurcation behavior.
Concerning the underlying ``spaces" of admissible
functions, distinguish between the space of smooth vector
functions with independent components and the small subspace of
homogeneous functions satisfying y1=y3, y2=y4.
Two-cells Brusselator, branching diagram y2(0) versus .
Postscript-Files for better printing results: