Jeroen S.W. Lamb:

Local bifurcations in k-symmetric dynamical systems

Abstract:

A map $U:\Real^d\mapsto\Real^d$ is called a (reversing)
$k$-symmetry of the dynamical system represented by the map
$L:\Real^d\mapsto\Real^d$ if $k$ is the smallest positive integer 
for which $U$ is a (reversing) symmetry of the $k$th iterate of $L$.  We
study generic local bifurcations of fixed points that are invariant under
the action of a compact Lie group $\Si$ that is a reversing $k$-symmetry
group of the map $L$, on the basis of a normal form approach. 

We derive normal forms relating the local bifurcations of $k$-symmetric maps
to local steady-state bifurcations of symmetric flows of vector fields.    
Alternatively, we also discuss the derivation of normal forms entirely
within the framework of Taylor expansions of maps.

We illustrate our results with some examples.