Evolution of a multimodal map induced by an equivariant vector field

It has been shown that the topological characterization of an equivariant system should preferably be achieved by working in a fundamental domain generated by the symmetry properties appearing in the phase space. In this paper, we discuss the case when the equivariance of the studied system is taken into account to study the evolution of the population of periodic orbits when a control parameter is varied. The Burke-Shaw system is considered here as an example. It is shown that the equivariance of this system may be used to reduce the multimodal first-return map in a Poincare section to a unimodal map. A relationship between four-symbol sequences and two-symbol sequences is given. The non-trivial evolution of the orbit spectrum of a multimodal map is then predicted from the much simpler unimodal map to which the multimodal map reduces.