The Schnakenberg system is a planar mass-action system for two chemical species (Schnakenberg 1979). A dimensionless form for
the concentrations 
and 
is
 |  |  |
(1)
|
 |  |  |
(2)
|
where 
and primes denote derivatives of 
and 
with respect to time 
. The opposite signs of 
represent conversion of one unit of 
into one of 
by autocatalysis. Adding the two equations gives 
. Writing 
, the unique positive equilibrium
point is 
. The Jacobian
at this point has determinant 
and trace 
, so its two eigenvalues
are purely imaginary when the trace vanishes. If 
and 
with 
, varying 
through 
makes the pair of eigenvalues
cross the imaginary axis. The resulting supercritical Hopf
bifurcation produces a small attracting limit cycle
on the side where 
. Adding diffusion
terms gives a spatially dependent system of partial
differential equations for the two concentrations.
