Bifurcations of families of invariant curves in four-dimensional reversible mappings

Abstract

The bifurcation of families of closed invariant curves near a fixed point of a four-dimensional reversible mapping as two pairs of its multipliers collide and move off the unit circle, is studied with reference to an analogous bifurcation in reversible vector fields. A description of the principal features of the bifurcation is presented by way of a conjecture, and for a particular class of maps an analytic expression is derived for distinguishing normal from inverted bifurcations. Evidence in support of the conjecture is indicated in terms of an order-by-order perturbation scheme as well as of numerical computations.