Fig. 1. Graph of the reaction curves R₁ and R₂ of equation q₁=r₁(q₂) and q₂=r₂(q₁), respectively.

Fig. 2. Numerical representation of the attractors and the basins of attraction for the duopoly game with myopic adjustment. (a) For the benchmark (no spillover) case with parameters a=10, b=0.5, v₁=v₂=0.23, c₁=c₂=2, γ₁₂=γ₂₁=0, the Nash equilibrium E_* is stable (a stable node). (b) With the same parameters a, b, v_i, c_i as in (a) and asymmetric spillover parameters γ₁₂=0 and γ₂₁=1 the Nash equilibrium is unstable (a saddle point) and the generic trajectory starting from the white region converges to a stable cycle of period 2. The two figures are obtained by taking a grid of initial conditions and generating, for each of them, a numerically computed trajectory of the duopoly map. If the trajectory is diverging then a grey dot is painted in the point corresponding to the initial condition, otherwise a white dot is painted.

Fig. 3. (a) For a=10, b=0.5, v₁=v₂=0.32, c₁=c₂=2, γ₁₂=0, γ₂₁=0.25, the Nash equilibrium E_* is unstable and an attracting chaotic area exists around it. (b) With the same parameters a, b, v_i, c_i as in (a) and γ₁₂=0, γ₂₁=1 a larger chaotic area exists around the unstable Nash equilibrium.

Fig. 4. (a) With the same parameters a, b, v_i, c_i as in Fig. 3 and γ₁₂=0.2, γ₂₁=0.25 the portion of the chaotic area close to the diagonal is more frequently visited by the iterated point. (b) With the same set of parameters as in (a) the versus time representation of the difference q₁(t)−q₂(t) is represented along a generic chaotic trajectory.

Fig. 5. (a) The reaction curves R_i, i=1,2, and the lines ω_i⁻¹, i=1,2, of (22) and (23), respectively, are represented for the benchmark (no spillover) case with parameters a=10, b=0.5, v₁=v₂=0.25, c₁=3, c₂=4, γ₁₂=γ₂₁=0. (b) The reaction curves R_i, i=1,2, and the lines ω_i⁻¹, i=1,2, are represented for the same parameters a, b, v_i, c_i as in (a) and asymmetric spillover parameters γ₁₂=0 and γ₂₁=3.

Fig. 6. (a) For a=10, b=0.5, v₁=0.25, v₂=0.3, c₁=4, c₂=3, γ₁₂=2, γ₂₁=4 the critical curves LC₋₁ and LC are represented, together with the boundaries which separate from . (b) With the same parameters a, b, v_i, c_i as in (a) and γ₁₂=3, γ₂₁=7 a portion of belongs to the region Z₂, and consequently “lakes” of are nested inside .