Figure 7. Dynamics on a 3-dimensional system (A = 3×3 matrix).

Presented is the three dimensional space defined by a₁₁, a₁₂, and a₁₃, the first row elements of the matrix A. The goal is defined by two pairs of input-output vectors. Empty circle: optimal non-modular solutions. Full circle: modular solutions. A typical trajectory is shown for a number of different cases. Lines represent all configurations that achieve the goal (satisfy Av ₁ = u ₁ and Av ₂ = u ₂). (A) A Constant goal G₁ = { [v ₁₁ = (1,−1,−1.4), u ₁₁ = (1,−2.4,0.4)]; [v ₁₂ = (0.5,1.2,−1.9), u ₁₂ = (0.5,−0.7,3.1) ] }. (B) Modularly varying goals. G ₁ as above, and G₂ = { [ v ₁₁ = (1,1.7,−0.7), u ₁₁ = (1,1,2.4) ]; [ v ₁₂ = (−0.7,−2.3,−1.1), u ₁₂ = (−0.7,−3.4,−1.2) ] }. Switching rate is E = 100/r time steps. (C) Modularly varying goals with nearly identical modules: G₁ = { [ (1,1.7,−0.7), (1,1,2.4) ]; [ (−0.7,−2.3,−1.1), (−0.7,−3.4,−1.2) ] } and G₂ = { [ (1,−1,−1.4), (1+η,−2.4,0.4) ]; [ (0.5,1.2,−1.9), (0.5+0.5η,−0.7,3.1) ] }. The distance between the two modular solutions for each of the goals is η = 0.1. Zoom in: adaptation dynamics between the modular solutions. (D) Random non-modular varying goals: G₁ = { [ (−2.5,1,1), (0,1,1) ]; [ (5.4,−1,1), (3,−1,1) ] }, G₂ = { [ (1.1,1,1), (1.1,1,1) ]; [ (0.6,−1,1), (0.6,−1,1) ] }. E = 100/r time steps. There is no solution that solves both goals well, and therefore the dynamics lead to ‘confusion’, a situation where none of the goals are achieved.