Bifurcations in Hamiltonian systems with a reflecting symmetry

A reflecting symmetry q 7→ −q of a Hamiltonian system does not leave the symplectic structure dq∧dp invariant and is therefore usually asso- ciated with a reversible Hamiltonian system. However, if q 7→ −q leads to H 7→ −H, then the equations of motion are invariant under the re- flection. This imposes strong restrictions on equilibria with q = 0. I will discuss the resulting bifurcations in one degree of freedom.