This paper investigates the relation between a branching process and a non-linear dynamical system in $\mathbb{C}^2$. This idea has previously been fruitful in many investigations, including that of the FKPP equation by McKean, Neveu, Bramson, and others. Our concerns here are somewhat different from those in other work: we wish to elucidate those features of the dynamical system which correspond to the long-term behaviour of the random process. In particular, we are interested in how the dimension of the global attractor corresponds to that of the tail $\sigma$-algebra of the process. The Poincar\'e--Dulac operator which (locally) intertwines the non-linear system with its linearization may sometimes be exhibited as a Fourier--Laplace transform of tail-measurable random variables; but things change markedly when parameters cross values giving the `primary resonance' in the Poincar\'e--Dulac sense. Probability proves effective in establishing {\it global} properties amongst which is a clear description of the global convergence to the attractor. Several of our probabilistic results are analogues of ones obtained by Kesten and Stigum, and by Athreya and Ney, for discrete branching processes. Our simpler context allows the use of It\^o calculus. Because the paper bridges two subjects, dynamical-system theory and probability theory, we take considerable care with the exposition of both aspects. For probabilist readers, we provide a brief guide to Poincar\'e--Dulac theory; and we take the view that in a paper which we hope will be read by analysts, it would be wrong to fudge any details of rigour in our probabilistic arguments. 1991 Mathematics Subject Classification: 60H30, 60J85, 34A20.