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Fig. 2. Numerical phase diagram for the density c (left) and hysteresis region for several
values of the noise and relaxation time T (right). Same color code as in Figure 1.
This map has three fixed points, the state x = 1 ( c = 0), the state x = 1
( c = 1) and a microscopically disordered state (0 ≤ c ≤ 1). The model is
obviously symmetric under the changes p 1 1 −p 2 , p 2 1 −p 1 and x → 1 −x ,
implying a fundamental equivalence of political opinions in this model. The
stability of fixed points marks the different phases, as shown in Fig. 1.
The stability of the state c =1( c = 0) corresponds to large social pressure
towards opinion 1 ( 1). The value of J determines if a change in social pressure
corresponds to a smooth or abrupt transition.
3.1 Phase Transitions and Damage Spreading in the Lattice Case
The numerical phase diagram of the model starting from a random initial state
with c 0 =0 . 5 is shown in Fig. 2. The scenario is qualitatively the same as
predicted by the mean-field analysis. In the upper-left part of the diagram both
states c = 0 and c = 1 are stable. In this region the final fate of the system
depends on the initial configuration.
Due to the symmetry of the model, the two second-order phase transition
curves meet at a bicritical point ( p t , 1 −p t ) where the first-order phase transition
line ends. Crossing the second-order phase boundaries on a line parallel to the
diagonal p 1 = p 2 , the density c exhibits two critical transitions, as shown in the
inset of the right panel of Fig. 2. Approaching the bicritical point the critical
region becomes smaller, and corrections to scaling increase. Finally, at the bicrit-
ical point, the two transitions coalesce into a single discontinuous (first-order)
one.
First-order phase transitions are usually associated to a hysteresis cycle due
to the coexistence of two stable states. To visualize the hysteresis loop (inset of
the right panel of Fig. 2) we modify the model slightly by letting p 0 =1 −p 3 =
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