Fig. 1. For μ₁=μ₂=3.4 the duopoly game with quadratic reaction functions and naive expectations has two stable steady-states (Nash equilibria) E₁ and E₂ and a stable cycle C₂ of period two. These attractors are represented in the strategy space . The light and dark grey regions represent the basins of E₁ and E₂, respectively, the white region represents the basin of C₂.

Fig. 2. Space of the parameters Ω={(μ,α) μ>0,0≤α≤1} for the map T under the assumption of homogeneous players. Ω^s(O) represent the set of parameters such that the fixed point O is asymptotically stable, Ω^s(E_S) represent the set of parameters such that the fixed point E_S is asymptotically stable, Ω^s(E_i) represents the common stability region of E₁ and E₂, Ω^s(E_i,C₂) represents the subset of Ω^s(E_i) where the stable cycle C₂ coexists with the two stable Nash equilibria E₁ and E₂. The portion of the curve of equation α=1/(μ+1) included inside Ω^s(E_i) represents the set of parameters at which the transition between simply connected and nonconnected basins of the stable Nash equilibria E₁ and E₂ occurs.

Fig. 3. Representation of the basins of the equilibria E₁ and E₂ in the case of homogeneous behavior. The meaning of the colors is the same as in Fig. 1. (a) With parameters μ₁=μ₂=μ=3.4 and α₁=α₂=α=0.2<1/(μ+1), the fixed point O=(0,0) belongs to the region Z₂ between the two branches LC^(a) and LC^(b) of the critical set LC, and the basins of E₁ and E₂ are simply connected sets. (b) With parameters μ=3.5 and α=0.5>1/(μ+1), the fixed point O=(0,0) belongs to the region Z₄ bounded by LC^(b), and the basins of E₁ and E₂ are nonconnected sets.

Fig. 4. Basins of E₁ and E₂ when players are heterogeneous with respect to their adaptive expectation rules. The colors have the same meaning as in the previous figures. (a) With μ₁=μ₂=3.6 and α₁=0.55, α₂=0.7 the stable set of the saddle point E_S, which constitute the boundary between the two basins and , is entirely included in the regions Z₂ and Z₀. The basins are connected sets. (b) For μ₁=μ₂=3.6 and α₁=0.59, α₂=0.7 a portion of the stable set of the saddle point E_S belongs to the region Z₄, hence the preimages of the portion H₀ of inside Z₄ constitute nonconnected portion of nested inside .

Fig. 5. With μ₁=μ₂=μ=3.8 and α₁=0.85, α₂=0.8, the equilibria E₁ and E₂ are unstable. Two chaotic attractors A₁ and A₂ coexist, with basins and , represented by light and intermediate grey, respectively.

Fig. 6. (a) Critical curves of rank-0, obtained as the locus of points such that det(DT(x,y))=0. (b) Critical curves of rank-1, obtained as LC=T(LC₋₁). These curves separate the plane into three regions, denoted by Z₄, Z₂ and Z₀ whose points have four, two or no rank-1 preimages, respectively. (c) Riemann foliation of the plane. With each point of the region Z₄ four distinct inverses are associated, each defined on a different sheet of the foliation, whereas points of Z₂ are associated with two sheets. The projection on the phase plane of the folds connecting different sheets are the critical curves LC.