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We use both  N ‐body simulations and integration in fixed potentials to explore the stability and the long‐term secular evolution of self‐consistent, equilibrium, non‐rotating, triaxial spheroidal galactic models. More specifically, we consider Dehnen models built with the Schwarzschild method. We show that short‐term stability depends on the degree of velocity anisotropy (radially anisotropic models are subject to rapid development of radial‐orbit instability). Long‐term stability, on the other hand, depends mainly on the properties of the potential and, in particular, on whether it admits a substantial fraction of strongly chaotic orbits. We show that in the case of a weak density cusp (γ= 1 Dehnen model) the  N ‐body model is remarkably stable, while the strong‐cusp (γ= 2) model exhibits substantial evolution of shape away from triaxiality, which we attribute to the effect of chaotic diffusion of orbits. The different behaviour of these two cases originates from the different phase space structure of the potential; in the weak‐cusp case there exist numerous resonant orbit families that impede chaotic diffusion. We also find that it is hardly possible to affect the rate of this evolution by altering the fraction of chaotic orbits in the Schwarzschild model, which is explained by the fact that the chaotic properties of an orbit are not preserved by the  N ‐body evolution. There are, however, parameters in Schwarzschild modelling that do affect the stability of an  N ‐body model, so we discuss the recipes of building a ‘good’ Schwarzschild model.

Chaotic mixing and the secular evolution of triaxial cuspy galaxy models built with Schwarzschild’s method