2011ApJ...728..128D

We survey the properties of all orbit families in the rotating frame of a family of realistic triaxial potentials with central supermassive black holes (SMBHs). In such galaxies, most regular box orbits (vital for maintaining triaxiality) are associated with resonances which occupy two-dimensional surfaces in configuration space. For slow figure rotation all orbit families are largely stable. At intermediate pattern speeds a significant fraction of the resonant box orbits as well as inner long-axis tubes are destabilized by the "envelope doubling" that arises from the Coriolis forces and are driven into the destabilizing center. Thus, for pattern rotation periods 2x10⁸ yr ≲ T_p≲ 5x10⁹ yr, the two orbit families that are most important for maintaining triaxiality are highly chaotic. As pattern speed increases there is also a sharp decrease in the overall fraction of prograde short-axis tubes and a corresponding increase in the retrograde variety. At the highest pattern speeds (close to that of triaxial bars), box-like orbits undergo a sudden transition to a new family of stable retrograde loop-like orbits, which resemble orbits in three-dimensional bars, and circulate about the short axis. Our analysis implies that triaxial systems (with central cusps and SMBHs) could either be very rapidly rotating like fast bars or slowly rotating like triaxial elliptical galaxies or dark matter halos found in N-body simulations. Collisionless triaxial systems that remain in stable dynamical equilibrium are unlikely to have intermediate pattern speeds which produce a high level of stochasticity in both the box and inner long-axis tube orbit families.