  We study a cellular automaton opinion formation model of Ising type, with
antiferromagnetic pair interactions modeling anticonformism, and ferromagnetic
plaquette terms modeling the social norm constraints. For a sufficiently large
connectivity, the mean-field equation for the average magnetization (opinion
density) is chaotic. This "chaoticity" would imply irregular coherent
oscillations of the whole society, that may eventually lead to a sudden jump
into an absorbing state, if present.
  However, simulations on regular one-dimensional lattices show a different
scenario: local patches may oscillate following the mean-field description, but
these oscillations are not correlated spatially, so the average magnetization
fluctuates around zero (average opinion near one half). The system is chaotic,
but in a microscopic sense where local fluctuations tend to compensate each
other.
  By varying the long-range rewiring of links, we trigger a small-world effect.
We observe a bifurcation diagram for the magnetization, with period doubling
cascades ending in a chaotic phase. As far as we know, this is the first
observation of a small-world induced bifurcation diagram.
  The social implications of this transition are also interesting. In the
presence of strong "anticonformistic" (or "antinorm") behavior, efforts for
promoting social homogenization may trigger violent oscillations.
