Emergence of chaotic behaviour in linearly stable systems

Strong nonlinear effects combined with diffusive coupling may give rise to unpredictable evolution in spatially extended deterministic dynamical systems even in the presence of a fully negative spectrum of Lyapunov exponents. This regime, denoted as ``stable chaos'', has been so far mainly characterized by numerical studies. In this manuscript we investigate the mechanisms that are at the basis of this form of unpredictable evolution generated by a nonlinear information flow through the boundaries. In order to clarify how linear stability can coexist with nonlinear instability, we construct a suitable stochastic model. In the absence of spatial coupling, the model does not reveal the existence of any self-sustained chaotic phase. Nevertheless, already this simple regime reveals peculiar differences between the behaviour of finite-size and that of infinitesimal perturbations. A mean-field analysis of the truly spatially extended case clarifies that the onset of chaotic behaviour can be traced back to the diffusion process that tends to shift the growth rate of finite perturbations from the quenched to the annealed average. The possible characterization of the transition as the onset of directed percolation is also briefly discussed as well as the connections with a synchronization transition.Comment: 30 pages, 8 figures, Submitted to Journal of Physics

Hi Spin Temperatures and Heating Requirements in Outer Regions of Disk Galaxies

ABSRACT:We show how to use 21-cm emission and absorption studies to estimate the heat inputs to the neutral gas in low pressure environments, such as in outer disks of spiral galaxies, in galactic halos or in intergalactic space. For a range of model parameters we calculate the gas kinetic and spin temperatures ($T_K$ and $T_S$) and the relation between $T_S$ and the heat input to the gas. We outline the conditions for a ``two phase medium'' to exist. We find that although $T_S$ can be much smaller than $T_K$, $T_S$ is always $\gg 3$ K for column densities greater that $5 \times 10^{18}$ cm$^{-2}$. This excludes the possibility that relevant HI masses at the periphery of galaxies are invisible at 21-cm in emission. The outermost interstellar gas in a disk galaxy is more directly affected by external processes and in this paper we estimate the intensity of the extragalactic background at energies close to 0.1 keV by comparing our theoretical results with HI emission/absorption studies. We take into account the possibility that some energy produced in the inner regions affects the energy balance in outer regions. We find that in the absence of any other local heat source QSO dominated background models are still compatible with the spin temperature limits derived for the two best documented HI emission/absorption studies in outer regions.Comment: 24 pages, 8 figures ARCETRI-PR-93-2