Self-similarity of period-doubling branching in 3-D reversible mappings

We consider 3-dimensional (3-D) mappings which are reversible, i.e. possess a time-reversal symmetry. In the mappings studied here, the reversibility is such that it guarantees the existence of one-parameter families (curves) of symmetric periodic orbits in the phase space. It is found that such curves can often intersect one another. In particular, a curve of n-cycles can intersect a curve of 2n-cycles, which in turn can intersect a curve of 4n-cycles etc. We show that the tree of branching curves of successively-doubled periods in the 3-D phase space possesses some geometric self-similarity. In particular, we identify the scaling factors a = -4.018 .. . . fl = 16.363... and 6 = 8.721 ... . Previously-studied 1-parameter 2-D (area-preserving) reversible mappings, in which these scalings also occur, are special cases of 3-D reversible mappings in that they have an integral of motion. The more general mappings studied here have no such integral. We discuss the reasons for our result in terms of normal forms.