Everything you ever wanted to know about Dehnen . . . . but were afraid to ask

This paper focuses on the statistical properties of chaotic orbit ensembles evolved in triaxial generalisations of the Dehnen potential which have been proposed recently to model realistic ellipticals that have a strong density cusp and manifest significant deviations from axisymmetry. Allowance is made for a possible supermassive black hole, as well as low amplitude friction, noise, and periodic driving which can mimic irregularities associated with discreteness effects and/or an external environment. The chaos exhibited by these potentials is quantified by determining (1) how the relative number of chaotic orbits depends on the steepness of the cusp, as probed by γ, the power law exponent with which density diverges, and MBH , the black hole mass; (2) how the size of the largest Lyapunov exponent varies with γ and MBH ; and (3) the extent to which Arnold webs significantly impede phase space transport, both with and without perturbations. The most important conclusions dynamically are (1) that, in the absence of irregularities, chaotic orbits tend to be extremely ‘sticky,’ so that different pieces of the same chaotic orbit can behave very differently for times ∼ 10000tD or more, but (2) that even very low amplitude perturbations can prove efficient in erasing many – albeit not all – of these differences. The implications of these facts are discussed both for the structure and evolution of real galaxies and for the possibility of constructing approximate near-equilibrium models using Schwarzschild’s method. For example, when trying to use Schwarzschild’s method to construct model galaxies containing significant numbers of chaotic orbits, it seems advantageous to build libraries with chaotic orbits evolved in the presence of low amplitude friction and noise, since such noisy orbits are more likely to represent reasonable approximations to time-independent building blocks. Much of the observed qualitative behaviour can be reproduced with a toy potential given as the sum of an anisotropic harmonic oscillator and a spherical Plummer potential, which suggests that the results may be generic.