ORDER IN THE DISTRIBUTION OF 3-D PERIODIC-ORBITS

We find the distribution of periodic orbits in a simple 3-D dynamical system, symmetric with respect to the ((x) over bar,(y) over bar) plane. The periodic orbits that intersect this plane perpendicularly are arranged along particular lines in the plane ((x) over bar (y) over bar). Along these lines there are certain ''basic orbits'', and between two consecutive basic orbits there are ''basic sequences'' of infinite sets of periodic orbits, whose multiplicities form arithmetic progressions. Between the orbits of these basic sequences there are higher order periodic orbits, forming Farey trees. Most of the lines are connected to each other at particular orbits on both their ends. However some lines are joined to others only at one end. Some lines have certain distortions or gaps. By decreasing one perturbation parameter we found that most orbits are produced by bifurcation from other orbits of low multiplicity like the simple periodic orbit 1a, or orbits that have bifurcated from 1a. However in some irregular cases the orbits disappear in pairs, as the perturbation decreases below a critical value.