Open Access

Monotonicity and non-monotonicity regions of topological entropy for Lorenz-like families with infinite derivatives

M.I. Malkin
M.I. Malkin
 and 
K.A. Safonov
K.A. Safonov
   | Nov 16, 2020
Applied Mathematics and Nonlinear Sciences's Cover Image
About this article
Previous Article
Next Article
Cite
Published Online: Nov 16, 2020
Page range: 293 - 306
Received: Jan 08, 2020
Accepted: Apr 11, 2020
DOI: https://doi.org/10.2478/amns.2020.2.00052
Keywords
© 2020 M.I. Malkin et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Afraimovich-Bykov-Shilnikov geometric model of Lorenz attractor
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.)
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).
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.
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.
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.
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.
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
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.
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
Sketch of creation of the trivial lacuna

Fig. 11

The graph of topological entropy with respect to parameter c
The graph of topological entropy with respect to parameter c

Fig. 12

The graph of topological entropy with respect to parameter ɛ
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.
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.
eISSN:
2444-8656
Language:
English
Publication timeframe:
Volume Open
Journal Subjects:
Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics
Journal RSS Feed