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.