Dissipationless N‐body models of rotating galaxies, iso‐energetic to a non‐rotating model, are examined as regards the mass in regular and in chaotic motion. Iso‐energetic means that they have the same mass and the same binding energy and they are near the same scalar virial equilibrium, but their total amount of angular momentum is different. The values of their spin parameters λ are near the value λ= 0.22 of our Galaxy. We distinguish particles moving in regular and in chaotic orbits and we show that the spatial distribution of these two sets of particles is much different. The rotating models are characterized by larger fractions of mass in chaotic motion (up to the level of ≈65 per cent) compared with the fraction of mass in chaotic motion in the non‐rotating iso‐energetic model (which is on the level of ≈32 per cent). Furthermore, the Lyapunov numbers of the chaotic orbits in the rotating models become by about one order of magnitude larger than in the non‐rotating model. This impressive enhancement of chaos is produced, partly by the more complicated distribution of mass, induced by the rotation, but mainly by the resonant effects near corotation. Chaotic orbits are concentrated preferably in values of the Jacobi integral around the value of the effective potential at the corotation radius. We find that density waves form a central rotating bar embedded in a thin and a thick disc with exponential mean radial profile of the surface density. A surprising new result is that long living spiral arms are excited on the disc, composed almost completely by chaotic orbits. The bar excites an m= 2 mode of spiral waves on the surface density distribution of the disc, emanating from the corotation radius. The bar goes temporarily out of phase with respect to an excited spiral wave, but it comes in phase again in less that a period of rotation. As a consequence, spiral arms show an intermittent behaviour. They are deformed, fade or disappear temporarily, but they grow again re‐forming a well‐developed spiral pattern. Spiral arms are discernible up to 20 or 30 rotations of the bar (lasting for about a Hubble time). The relative power of the spiral m= 2 mode with respect to all other fluctuations on the surface density is initially about 50 per cent, but it is reduced by a factor of about 2 or 3 at the end of the Hubble time.
Two self-consistent (N-body) non-rotating equilibrium models of elliptical galaxies with smooth central density profiles (called 'Q' and 'C' models) are constructed, starting from quiet and clumpy cosmological initial conditions, respectively. Both models are triaxial. The Q model has an E7 maximum ellipticity in the inner parts and tends to E6 or E5 maximum ellipticity in the outer parts. The C model has a maximum ellipticity E4 in the inner parts and tends to an E2 or E1 in the outer parts.For each model, we identify the particles moving in chaotic orbits with the Lyapunov number exceeding a particular threshold (namely, 10 −2.8 , in units of the inverse radial periods of the particular orbits). At energy levels in the deepest 30 per cent of the potential well, no chaotic orbits were detected in the above limit of chaoticity. In the Q model, the detected chaotic part is 32 per cent of the total mass. This part has a nearly spherical distribution. It imposes limitations on the maximum ellipticity of the system, in spite of the fact that only a part of less than about 8 per cent of the total mass moves in chaotic orbits and is able to develop chaotic diffusion within a Hubble time. In the C model, the detected chaotic part is about 26 per cent of the total mass, but only less than 2 per cent can develop chaotic diffusion within a Hubble time.These chaotic components produce surface density profiles flatter than the profiles of the rest of the mass, particularly in the Q model. The two profiles intersect at a given distance, where the overall profile forms an observable hump, especially if the surface density profiles are taken along the shortest axis of the projection.
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