Normal Forms for Reversible (1:1) Resonance Vector Fields and Applications

Marco Antonio Teixeira and Jiazhong Yang, Normal forms for reversible (1:1) resonance vector fields and applicationsThis article is devoted to studying 4-dimensional reversible vector fields having (1:1) resonance at an equilibrium. By giving an efficient normal form we are able to describe all invariant sets and to establish a relationship between them and certain parameters reflected in the normal form. As a result, we can study the local dynamics around the equilibrium point for the most vector fields in our class. Important conclusions concerning the existence of invariant tori and periodic orbits are obtained. Moreover, we give an analogue of the Lyapunov-Devaney theorem in the (1:1) reversible case, which can be treated as a limit case where two pairs of non-resonant purely imaginary eigenvalues tend to accumulate.