Chaotic, staggered, and polarized dynamics in opinion forming: The contrarian effect.

We reconsider the no-tie-breaking two-state Galam contrarian model of opinion dynamics for update groups of size 3. While the initial model assumes a constant density of contrarians a for both opinions, the density now depends for each opinion on its global support. Proportionate contrarians are thus found to indeed preserve the main results of the former case. However, restricting the contrarian behavior to only the current collective majority makes the dynamics more complex with additional features. For a density a < a(c) = 1/9 of one-sided contrarians, a chaotic basin is found in the 50-50 region separated from two majority-minority point attractors, one on each side. For 1/9 < a less similar to 0.301 only the chaotic basin survives. In the range a > 0.301 the chaotic basin disappears and the majority starts to alternate between the two opinions with a staggered flow toward two point attractors. We then study the effect of both decoupling the local update time sequence from the contrarian behavior activation and a smoothing of the majority rule. A status quo-driven bias for contrarian activation is also considered. Introduction of unsettled agents driven in the debate on a contrarian basis is shown only to shrink the chaotic basin. The model may shed light on recent apparent contradictory elections with on the one hand very close results as in the United States in 2000 and in Germany in 2002 and 2005, and on the other hand, a huge majority as in France in 2002.