Stability of invariant curves in four-dimensional reversible mappings near 1:1 resonance

Recent work on reversible Neimark-Sacker bifurcation in 4D maps is summarized. Linear stability analysis of families of invariant curves appearing in this bifurcation is presented by (a) referring to the analogous stability problem in reversible Hopt bifurcation in vector fields and (b) perturbatively calculating a set of quantities, termed quasi-multipliers, for the invariant curves. In particular, the critical rotation numbers corresponding to transition from elliptic to hyperbolic invariant curves on the subthreshold side in the so-called inverted bifurcation are calculated. Results of numerical iterations corroborating the above analysis are presented. The question of exploring the structure of the phase space close to the invariant curves is briefly addressed in the conclusion.