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The thermal instability with a piecewise power law cooling function is investigated using one- and three-dimensional simulations with periodic and shearing-periodic boundary conditions in the presence of constant thermal diffusion and kinematic viscosity coefficients. Consistent with earlier findings, the flow behavior depends on the average density, <rho>. When $\bra{\rho}$ is in the range 1-5 x 10^{-24} g cm^{-3} the system is unstable and segregates into cool and warm phases with temperatures of roughly 100 and 10^4 K, respectively. However, in all cases the resulting average pressure <p> is independent of <rho> and just a little above the minimum value. For a constant heating rate of 0.015 erg g^{-1} s^{-1}, the mean pressure is around 24 x 10^{-14} dyn (corresponding to p/k_B ~ K cm^{-3}). Cool patches tend to coalesce into bigger ones. In all cases investigated there is no sustained turbulence, which is in agreement with earlier results. Simulations in which turbulence is driven by a body force show that when rms velocities of between 10 and 30 km/s are obtained, the resulting dissipation rates rates are comparable to the thermal energy input rate. The resulting mean pressures are then about 30 x 10^{-14} dyn, corresponding to p/k_B ~ 2170 K cm^{-3}. This is comparable to the value expected for the Galaxy. Differential rotation tends to make the flow two-dimensional, that is, uniform in the streamwise direction, but this does not lead to instability.

Given a polynomial or a rational map f we associate to it a space of maps. We introduce local coordinates in this space, which are essentially the set of critical values of the map. Then we consider an arbitrary periodic orbit of f with multiplier \rho\not=1 as a function of the local coordinates, and establish a simple connection between the dynamical plane of f and the function \rho in the space associated to f. The proof is based on the theory of quasiconformal deformations of rational maps. As a corollary, we show that multipliers of non-repelling periodic orbits are also local coordinates in the space.

Consider the systems \(\left( 1 \right)\left( {sI - B} \right)\frac{{dv}}{{ds}} = \left( {\rho I - A} \right)v\) and \(\left( 2 \right)\frac{{dx}}{{dt}} = \left( {B + t^{ - 1} A} \right)x\) , where s and t are variables, ρ is a parameter, and A and B = diag(λ1,...,λn ) are n by n matrices. (1) has only regular singular points, and (2) has an irregular singular point at t = ∞. Several kinds of special solutions having particular behavior near singular points were selected in previous papers. In the present paper, the author shows how (2) results from (1) in a process of confluence as ρ → ∞. It is analyzed how the special solutions of (1) converge to those of (2) in that process. As a consequence new proofs of earlier results about connection problems are obtained.