  People are often divided into conformists and contrarians, the former tending
to align to the majority opinion in their neighborhood and the latter tending
to disagree with that majority. In practice, however, the contrarian tendency
is rarely followed when there is an overwhelming majority with a given opinion,
which denotes a social norm. Such reasonable contrarian behavior is often
considered a mark of independent thought, and can be a useful strategy in
financial markets.
  We present the opinion dynamics of a society of reasonable contrarian agents.
The model is a cellular automaton of Ising type, with antiferromagnetic pair
interactions modeling contrarianism and plaquette terms modeling social norms.
We introduce the entropy of the collective variable as a way of comparing
deterministic (mean-field) and probabilistic (simulations) bifurcation
diagrams.
  In the mean field approximation the model exhibits bifurcations and a chaotic
phase, interpreted as coherent oscillations of the whole society. However, in a
one-dimensional spatial arrangement one observes incoherent oscillations and a
constant average.
  In simulations on Watts-Strogatz networks with a small-world effect the mean
field behavior is recovered, with a bifurcation diagram that resembles the
mean-field one, but using the rewiring probability as the control parameter.
Similar bifurcation diagrams are found for scale free networks, and we are able
to compute an effective connectivity for such networks.
