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Fig. 1
Afraimovich-Bykov-Shilnikov geometric model of Lorenz attractor
Fig. 2
Bifurcation diagram for the Shimizu-Morioka system near the point Q(α = 0.608, λ = 1.044) which corresponds to the homoclinic figure-8 bifurcation to a saddle with zero saddle value. (The green region corresponds to parameter values where the Lorenz attractor exists.)
Fig. 3
The graph of topological entropy for the logistic map fa(x) = ax(1 − x).
Fig. 4
The bifurcation diagram for map Tc,ɛ. Green region correspond to the existence of Lorenz attractor.
Fig. 5
The kneading chart for map Tc,ɛ shows that the topological entropy as the function of c has a single minimum for ɛ in the interval [0,0.6] (more precise calculation gives ɛ ∈ [0,0.76]). Above the red line one has that Tc,ɛ is expanding (DTc,ɛ > 1), and there the topological entropy is monotone increasing in c.
Fig. 6
The graph of Tc,ɛ and the trajectory of the discontinuity point for c > 2.
Fig. 7
The second iteration of the map Tc,ɛ for ɛ = 0.7 with c = 1 (fig. a) and c = 1.1 (fig.b). In fig. b, a magnified fragment near the discontinuity point is shown.
Fig. 8
Saddle-node bifurcation for ɛ = 0.7
Fig. 9
Trajectory of the discontinuity point at the boundary of the LA2 region for ɛ = 0.4.
Fig. 10
Sketch of creation of the trivial lacuna
Fig. 11
The graph of topological entropy with respect to parameter c
Fig. 12
The graph of topological entropy with respect to parameter ɛ
Fig. 13
Illustration to theorem 4: above the dotted line one has the expanding condition, and in the whole green region there are no stable periodic orbits. PF curve indicates the pitchfork bifurcation.