Sambhus, NiranjanSridhar, S.2015-03-012015-03-012015-03-01http://hdl.handle.net/11007/2931We investigate the orbital structure of a model triaxial star cluster, centered around a supermassive black hole (BH), appropriate to galactic nuclei. Sridhar and Touma {1999) proved that the presence of the BH enforces some regularity in the dynamics within the radius of influence of the BH. We employ their averaging method to reduce the degrees of freedom from three to two. Numerical orbit integrations, together with Poincare surfaces of section allow us to draw a global portrait of the orbital structure; in our calculations we employ a model cluster potential that is triaxial and harmonic. The averaged dynamics of the axisymmetric case is integrable, and we present a detailed comparison of orbits in oblate and prolate axisymmetric potentials. Both cases support resonant orbits with fixed values of eccentricity, inclination, and periapse, whose lines of nodes rotates steadily. These occur for all values of oblateness, but only for axis ratio greater than two, in the prolate case; we identify this phenomenon with the (in)stability of the long axis orbit. We then systematically explore significantly triaxial potentials, possessing small oblateness, or prolateness. Resonant orbits and their families are studied both numerically, and through secular perturbation theory. Chaos is highly suppressed for all the cases we studied, and we obtain effective third integrals. Some of the orbits appear to reinforce the shape of the potential; we provide phase space, as well as real space portraits of these orbits. A particularly promising resonant orbit exists in highly prolate, triaxial potentials.enStellar OrbitsArticle