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.

In a previous paper (Voglis et al., Paper I), we demonstrated that, in a rotating galaxy with a strong bar, the unstable asymptotic manifolds of the short-period family of unstable periodic orbits around the Lagrangian points L 1 or L 2 create correlations among the apocentric positions of many chaotic orbits, thus supporting a spiral structure beyond the bar. In this paper, we present evidence that the unstable manifolds of all the families of unstable periodic orbits near and beyond corotation contribute to the same phenomenon. Our results refer to a Nbody simulation, a number of drawbacks of which, as well as the reasons why these do not significantly affect the main results, are discussed. We explain the dynamical importance of the invariant manifolds as due to the fact that they produce a phenomenon of 'stickiness' slowing down the rate of chaotic escape in an otherwise non-compact region of the phase space. We find a stickiness time of the order of 100 dynamical periods, which is sufficient to support a long-living spiral structure. Manifolds of different families become important at different ranges of values of the Jacobi constant. The projections of the manifolds of all the different families in the configuration space produce a pattern due to the 'coalescence' of the invariant manifolds. This follows closely the maxima of the observed m = 2 component near and beyond corotation. Thus, the manifolds support both the outer edge of the bar and the spiral arms.

We estimate the two-point correlation function in redshift space of the recently compiled H I Parkes All-Sky Survey neutral hydrogen (H I) sources catalogue, which if modelled as a power law, ξ (r) = (r 0 /r) γ , the best-fitting parameters for the H I selected galaxies are found to be r 0 = 3.3 ± 0.3 h −1 Mpc with γ = 1.38 ± 0.24. Fixing the slope to its universal value γ = 1.8, we obtain r 0 = 3.2 ± 0.2 h −1 Mpc.Comparing the measured two-point correlation function with the predictions of the concordance cosmological model ( = 0.74), we find that at the present epoch the H I selected galaxies are antibiased with respect to the underlying matter fluctuation field with their bias value being b 0 0.68. Furthermore, dividing the H I galaxies into two richness subsamples we find that the low-mass H I galaxies have a very low present bias factor (b 0 0.48), while the high-mass H I galaxies trace the underlying matter distribution as the optical galaxies (b 0 1). Using our derived present-day H I galaxy bias we estimate their redshift-space distortion parameter, and correct accordingly the correlation function for peculiar motions. The resulting real-space correlation length is r re 0 = 1.8 ± 0.2 h −1 Mpc and r re 0 = 3.9 ± 0.6 h −1 Mpc for the low-and high-mass H I galaxies, respectively. The low-mass H I galaxies appear to have the lowest correlation length among all extragalactic populations studied to date. In order to corroborate these results we have correlated the IRAS-Point Source Catalogue for Redshift (PSCz) reconstructed density field, smoothed over scales of 5 h −1 Mpc, with the positions of the H I galaxies, to find that indeed the H I galaxies are typically found in negative overdensity regions (δρ/ρ PSCz 0), even more so the low-mass H I galaxies.Finally, we also study the redshift evolution of the H I galaxy linear bias factor and find that the H I-galaxy population is antibiased up to z ∼ 1.3. While at large redshifts z ∼ 3, we predict that the H I galaxies are strongly biased. Our bias evolution predictions are consistent with the observational bias results of Lyα galaxies.

In the presence of a strong m = 2 component in a rotating galaxy, the phase-space structure near corotation is shaped to a large extent by the invariant manifolds of the short-period family of unstable periodic orbits terminating at L 1 or L 2 . The main effect of these manifolds is to create robust phase correlations among a number of chaotic orbits large enough to support a spiral density wave outside corotation. The phenomenon is described theoretically by soliton-like solutions of a Sine-Gordon equation. Numerical examples are given in an N-body simulation of a barred spiral galaxy. In these examples, we demonstrate how the projection of unstable manifolds in configuration space reproduces essentially the entire observed bar-spiral pattern.