  Attractors in asymmetric neural networks with deterministic parallel dynamics
were shown to present a "chaotic" regime at symmetry eta < 0.5, where the
average length of the cycles increases exponentially with system size, and an
oscillatory regime at high symmetry, where the typical length of the cycles is
2. We show, both with analytic arguments and numerically, that there is a sharp
transition, at a critical symmetry $\e_c=0.33$, between a phase where the
typical cycles have length 2 and basins of attraction of vanishing weight and a
phase where the typical cycles are exponentially long with system size, and the
weights of their attraction basins are distributed as in a Random Map with
reversal symmetry. The time-scale after which cycles are reached grows
exponentially with system size $N$, and the exponent vanishes in the symmetric
limit, where $T\propto N^{2/3}$. The transition can be related to the dynamics
of the infinite system (where cycles are never reached), using the closing
probabilities as a tool.
  We also study the relaxation of the function $E(t)=-1/N\sum_i |h_i(t)|$,
where $h_i$ is the local field experienced by the neuron $i$. In the symmetric
system, it plays the role of a Ljapunov function which drives the system
towards its minima through steepest descent. This interpretation survives, even
if only on the average, also for small asymmetry. This acts like an effective
temperature: the larger is the asymmetry, the faster is the relaxation of $E$,
and the higher is the asymptotic value reached. $E$ reachs very deep minima in
the fixed points of the dynamics, which are reached with vanishing probability,
and attains a larger value on the typical attractors, which are cycles of
length 2.
