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Applications of the min-max symbols of multimodal maps

José M. Amigó

José M. Amigó

Centro de Investigación Operativa, Universidad Miguel Hernández, Avda. de la Universidad s/nElche (Alicante), Spain
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Amigó, José M.
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Ángel Giménez

Ángel Giménez

Centro de Investigación Operativa, Universidad Miguel Hernández, Avda. de la Universidad s/nElche (Alicante), Spain
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Giménez, Ángel
  
Jan 01, 2016
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Published Online: Jan 01, 2016
Page range: 87 - 98
Received: Jun 24, 2015
Accepted: Dec 17, 2015
DOI: https://doi.org/10.21042/AMNS.2016.1.00008
Keywords
© 2016 José M. Amigó, Ángel Giménez, published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Fig. 1

Four bad symbols with respect to the critical line y = ci. The rest are of the form MIj, Mck with j,k < i, and mIj, mck with j,k > i.
Four bad symbols with respect to the critical line y = ci. The rest are of the form MIj, Mck with j,k < i, and mIj, mck with j,k > i.

Fig. 2

Graph of the trimodal maps fν1,ν2
Graph of the trimodal maps fν1,ν2

Fig. 3

Plot h(f0,ν2) in bits vs ν2, 0 < ν2 ≤ 1 (ε = 10−4,Δν2 = 0.001).
Plot h(f0,ν2) in bits vs ν2, 0 < ν2 ≤ 1 (ε = 10−4,Δν2 = 0.001).

Fig. 4

Level sets of h(fν1,ν2) in bits vs ν1,ν2, 0 ≤ ν1,ν2 ≤ 1 and ν1 ≠ ν2 (ε = 10−4,Δν1 = Δν2 = 0.01).
Level sets of h(fν1,ν2) in bits vs ν1,ν2, 0 ≤ ν1,ν2 ≤ 1 and ν1 ≠ ν2 (ε = 10−4,Δν1 = Δν2 = 0.01).

Fig. 5

Graph of the trimodal maps gν1,ν2
Graph of the trimodal maps gν1,ν2

Fig. 6

Plot h(g0,ν2) in bits vs ν2, 0 < ν2 ≤ 1 (ε = 10−4,Δν2 = 0.001).
Plot h(g0,ν2) in bits vs ν2, 0 < ν2 ≤ 1 (ε = 10−4,Δν2 = 0.001).

Fig. 7

Level sets of h(gν1,ν2) in bits vs ν1,ν2, 0 ≤ ν1,ν2 ≤ 1 and ν1 ≠ ν2 (ε = 10−4,Δν1 = Δν2 = 0.01).
Level sets of h(gν1,ν2) in bits vs ν1,ν2, 0 ≤ ν1,ν2 ≤ 1 and ν1 ≠ ν2 (ε = 10−4,Δν1 = Δν2 = 0.01).

Performances when computing h(f0_1,0_9) in bits with the bimodal map_

precisionhn
ε = 10−40.655591287672179
ε = 10−50.643433302022565
ε = 10−60.6395788596031786
ε = 10−70.6383595747515645

Transition rules for l-modal maps with a positive shape_

ωni$\begin{array}{} \displaystyle \omega _n^i \end{array}$→ωn+1i$\begin{array}{} \displaystyle \omega_{n+1}^{i} \end{array}$
mIodd, MIeven→mγn+1i$\begin{array}{} \displaystyle {m^{\gamma _{n + 1}^i}} \end{array}$
mIeven, MIodd→Mγn+1i$\begin{array}{} \displaystyle {M^{\gamma _{n + 1}^i}} \end{array}$
mceven, Mceven→mγn+1i$\begin{array}{} \displaystyle {m^{\gamma _{n + 1}^i}} \end{array}$
mcodd, Mcodd→Mγn+1i$\begin{array}{} \displaystyle {M^{\gamma _{n + 1}^i}} \end{array}$

Transition rules for l-modal maps with a negative shape_

ωni$\begin{array}{} \displaystyle \omega _n^i \end{array}$→ωn+1i$\begin{array}{} \displaystyle \omega_{n+1}^{i} \end{array}$
mIodd, MIeven→Mγn+1i$\begin{array}{} \displaystyle {M^{\gamma _{n + 1}^i}} \end{array}$
mIeven, MIodd→mγn+1i$\begin{array}{} \displaystyle {m^{\gamma _{n + 1}^i}} \end{array}$
mceven, Mceven→Mγn+1i$\begin{array}{} \displaystyle {M^{\gamma _{n + 1}^i}} \end{array}$
mcodd, Mcodd→mγn+1i$\begin{array}{} \displaystyle {m^{\gamma _{n + 1}^i}} \end{array}$

Performances when computing h(g0_1,0_9) in bits with the trimodal map_

precisionhn
ε = 10−41.02013528493203
ε = 10−51.00638666069640
ε = 10−61.002020495722023
ε = 10−71.000639265386394
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