Journal on Policy and Complex Systems
plitude that does not decay to zero in the long run ( Figure 2 ).
In the ( s1 , s2 , s3 ) phase space , there is an attractor or limit cycle ( Figure 3 ): trajectories starting from different initials conditions approach the attractor over the long run .
Because of the existence of the attractor , differences in the initial conditions values do not decay to zero ( as for stable fixed points ), nor do they diverge ( chaos , strange attractor ) after a long time . This is shown in Figure 4 .
When we further increase the intra-group couplings , the oscillations period increases and the sinusoidal dependence changes drastically ( Figure 5 ).
The shape of the corresponding attractor approaches a rectangle ( Figure 6 ).
For J1 = J2 = J3 > 0.617 , the attractor is replaced by a stable fixed point , i . e . s1 , s2 , and s3 approach fixed values , independent of initial conditions .
Figure 1 . J1 = J2 = J3 = 0.15 , K12 = - 0.2 , K21 = 0.2 , K13 = 0 , K31 = 0.2 , K23 = 0 , K32 = 0.2 .
Figure 2 . J1 = J2 = J3 = 0.25 , K12 = - 0.2 , K21 = 0.2 , K13 = 0 , K31 = 0.2 , K23 = 0 , K32 = 0.2 .
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