  Systems such as fluid flows in channels and pipes or the complex
Ginzburg-Landau system, defined over periodic domains, exhibit both continuous
symmetries, translational and rotational, as well as discrete symmetries under
spatial reflections or complex conjugation. The simplest, and very common
symmetry of this type is the equivariance of the defining equations under the
orthogonal group O(2). We formulate a novel symmetry reduction scheme for such
systems by combining the method of slices with invariant polynomial methods,
and show how it works by applying it to the Kuramoto-Sivashinsky system in one
spatial dimension. As an example, we track a relative periodic orbit through a
sequence of bifurcations to the onset of chaos. Within the symmetry-reduced
state space we are able to compute and visualize the unstable manifolds of
relative periodic orbits, their torus bifurcations, a transition to chaos via
torus breakdown, and heteroclinic connections between various relative periodic
orbits. It would be very hard to carry through such analysis in the full state
space, without a symmetry reduction such as the one we present here.
