Bifurcation of limit cycles and symmetry
E.V. Nikolaev and E.E. Shnol
Institute for Mathematical Problems in Biology,
Pushchino, Moscow Region, 142 292, Russia
The aim of this communication is to discuss general features of bifurcations of limit cycles in the presence of symmetry.
Let 
, be a differential equation,
possessing a symmetry group 
. It is known
that every limit cycle C of the equation can be associated with an
appropriate pair of subgroups H and K, 
, such that 
, 
Fix K and
. The pair 
is called
the proper symmetry S of a cycle C. We call a limit cycle
C an F-cycle if H = K, an S-cycle if 
, and an FS-cycle when 
.
Let 
be a section to an arbitrary limit cycle C
and let P be its Poincar 
map. If C is an F-cycle,
then 
. So, we come to a problem on
bifurcations of fixed points of iterated maps with symmetry (see
Chossat and Golubitsky, 1988). If C is an S-cycle, then P can
be represented in the form 
(Fiedler, 1988). This allows
one to reduce an original problem to the analysis of bifurcations of
fixed points for maps Q without symmetry. The case of FS-cycles
is less trivial. If the symmetry 
of an FS-cycle is
commutative, i.e. H is commutative, then it can be reduced
to the study of an F-cycle with the smaller symmetry 
. Thus the case of non-commutative is
of greater interest.
Example. Given an FS-cycle C with 
, the square root Q of P
satisfies the equality 
. Here 
is the symmetry group of a regular n-gon and a is a
generator of 
. The symmetry can force a double
unit multiplier without the Jordan block to occur as a codimension
one case. It leads to the model system which is a particular case of
the normal form arising in a well-known problem of resonance 1:n
(Arnold, 1977). The principal deference between these two cases is
that here all the coefficients and parameter are real. Bifurcations
occurring in such n-resonance real model system can be
investigated completely for any 
, including the case n=4.