ELSFWIER Physica A 211 (1994) 218-233 2D universality of period-doubling bifurcations in 3D conservative reversible mappings Stavros Komineasa, Michael N. Vrahatisb, Tassos Bountisb zyxwvutsrqponmlk ‘Department of Physics, University of Crete, GR-714.09 Hera&on, Crete, Greece bDepartment of Mathematics, University of Patras, GR-261.10 Patras, Greece Received 20 April 1993; Revised 5 March 1994 zyxwvutsrqponmlkjihgfedcbaZYX Abstract Infinite sequences of period-doubling bifurcations are known to occur generically (i.e. with codimension 1) not only in dissipative 1D systems but also in 2D conservative systems, described by area-preserving mappings. In this paper, we study a 3D volume- preserving, reversible mapping and show that it does possess period 2”(m = 1,2, . . .) orbits, with stability intervals whose length decreases rapidly, with increasing m. Varying one parameter of the system we find that these orbits always bifurcate out of one another with the usual stability exchange and universal properties of period-doubling sequences of 2D-conservative maps. This raises the interesting question whether these 3D reversible maps possess an analytic integral which would render them essentially 2-dimensional. 1. Introduction It is well known that infinite sequences of period-doubling bifurcations occur generically and constitute a universal route to chaos in dissipative dynamical systems [ 11. Such sequences are observed as one parameter of the system is varied (i.e. they are of codimension 1) and their universality is related to the fact that locally their dynamics is one-dimensional. In the case of 2D area-preserving mappings, period-doubling bifurcations are also generic and occur in a very similar way, only with different universal constants [2,3]. The two eigenvalues of the Jacobian return matrix of a period 2” orbit collide at -1 and split off on the real axis, as a stable 2”+‘-period orbit is “born” with its eigenvalues entering the unit circle at +l. In higher-dimensional conservative systems, however, the situation is a lot less clear. One generally finds that the eigenvalues of a 2” periodic orbit split off the unit circle at some point other than -1. This is the so-called phenomenon of complex instability [4,5], which is not associated with the simultaneous appear- 037%4371/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved