# On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions

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## Abstract

**:**

## 1. Introduction

## 2. Periodic Structure of PDDS on Generalized Independent Local Functions

**Theorem**

**1.**

**Proof.**

- If $i\in {A}_{m}$ is such that ${x}_{i}^{m}=1$ and ${F}_{i}={x}_{i}\vee {M}_{i}$, then ${x}_{i}^{m+1}=1$ and so $i\in {A}_{m+1}$.
- If $i\in {A}_{m}$ is such that ${x}_{i}^{m}=0$ and ${F}_{i}={x}_{i}\wedge {M}_{i}$, then ${x}_{i}^{m+1}=0$ and so $i\in {A}_{m+1}$.

**Example**

**1.**

**Theorem**

**2.**

**Proof.**

**Case 1:**${F}_{i}={x}_{i}^{\prime}\vee {M}_{i}$ and ${M}_{i}\left(x\right)={M}_{i}({x}_{p+1}^{0},\dots ,{x}_{n}^{0})=1$. Then ${x}_{i}^{m}=1$ for each $m\ge 1$.

**Case 2:**${F}_{i}={x}_{i}^{\prime}\vee {M}_{i}$ and ${M}_{i}\left(x\right)={M}_{i}({x}_{p+1}^{0},\dots ,{x}_{n}^{0})=0$. Then ${x}_{i}^{m}={\left({x}_{i}^{m-1}\right)}^{\prime}$ for each $m\ge 1$. In particular, for each $m\ge 1$ we have ${x}_{i}^{m}=\left(\right)open="\{"\; close>\begin{array}{cc}{\left({x}_{i}^{0}\right)}^{\prime},\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}m\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{odd}\hfill \\ {x}_{i}^{0},\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}m\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{even}\hfill \end{array}$.

**Case 3:**${F}_{i}={x}_{i}^{\prime}\wedge {M}_{j}$ and ${M}_{i}\left(x\right)={M}_{i}({x}_{p+1}^{0},\dots ,{x}_{n}^{0})=0$. Then ${x}_{i}^{m}=0$ for each $m\ge 1$.

**Case 4:**${F}_{i}={x}_{i}^{\prime}\wedge {M}_{i}$ and ${M}_{i}\left(x\right)={M}_{i}({x}_{p+1}^{0},\dots ,{x}_{n}^{0})=1$. Then ${x}_{i}^{m}={\left({x}_{i}^{m-1}\right)}^{\prime}$ for each $m\ge 1$. In particular, for each $m\ge 1$ we have ${x}_{i}^{m}=\left(\right)open="\{"\; close>\begin{array}{cc}{\left({x}_{i}^{0}\right)}^{\prime},\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}m\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{odd}\hfill \\ {x}_{i}^{0},\hfill & \mathrm{if}\phantom{\rule{4.pt}{0ex}}m\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}\mathrm{even}\hfill \end{array}$.

**Remark**

**1.**

**Example**

**2.**

**Example**

**3.**

## 3. Periodic Structure of the Composition of Conjugate PDDS

**Theorem**

**3.**

- (i)
- x is a periodic point of $F\circ G$ if, and only if, x is a periodic point of $F\circ H$.
- (ii)
- ${\mathrm{Per}}_{t}(F\circ H)={\mathrm{Per}}_{t}(F\circ G)$ provided that t is odd.
- (iii)
- ${\mathrm{Per}}_{t}(F\circ H)={\mathrm{Per}}_{\frac{t}{2}}(F\circ G)$ provided that t is even.

**Proof.**

- (i)
- Let x be a periodic point of $F\circ G$. Then, for some $t\ge 1$, ${(F\circ G)}^{t}\left(x\right)=x$. Therefore ${(F\circ H)}^{2t}\left(x\right)=x$, which means that x is a periodic point of $F\circ H$.Conversely, if x is a periodic point of $F\circ H$ then ${(F\circ H)}^{s}\left(x\right)=x$ for some $s\ge 1$. Therefore ${(F\circ G)}^{s}\left(x\right)={(F\circ H)}^{2s}\left(x\right)=x$, and so x is a periodic point of $F\circ G$.

- (ii)
- Assume that t is an odd number. Then, from $t|2{t}^{\prime}$ we get that $t|{t}^{\prime}$. Moreover, since ${(F\circ G)}^{t}\left(x\right)={(F\circ H)}^{2t}\left(x\right)=x$, we have that ${t}^{\prime}|t$. Hence, $t={t}^{\prime}$ and the conclusion follows.
- (iii)
- Assume now that t is an even number. Then ${(F\circ G)}^{\frac{t}{2}}\left(x\right)={(F\circ H)}^{t}\left(x\right)=x$ and so ${t}^{\prime}|\frac{t}{2}$, which jointly $t|2{t}^{\prime}$ yields that ${t}^{\prime}=\frac{t}{2}$ and the proof finishes. □

**Example**

**4.**

**Corollary**

**1.**

**Proof.**

**Example**

**5.**

**Proposition**

**1.**

**Corollary**

**2.**

- $x=({x}_{1},\dots ,{x}_{n})$ is a fixed point for $F\circ G$.
- x is a 2-periodic point of the NAND-PDS over D.
- $\{j\phantom{\rule{4pt}{0ex}}:\phantom{\rule{4pt}{0ex}}{x}_{j}=1\}\in \mathsf{\Theta}=\{\overline{{A}_{D}\left(Q\right)}\phantom{\rule{4pt}{0ex}}:\phantom{\rule{4pt}{0ex}}Q\in P\left(V\right)\}$.

**Proof.**

**Corollary**

**3.**

- $x=({x}_{1},\dots ,{x}_{n})$ is a fixed point for $G\circ F$.
- x is a 2-periodic point of the NOR-PDS over D.
- $\{j\phantom{\rule{4pt}{0ex}}:\phantom{\rule{4pt}{0ex}}{x}_{j}=0\}\in \mathsf{\Theta}=\{\overline{{A}_{D}\left(Q\right)}\phantom{\rule{4pt}{0ex}}:\phantom{\rule{4pt}{0ex}}Q\in P\left(V\right)\}$.

**Corollary**

**4.**

**Proof.**

**Corollary**

**5.**

**Proof.**

**Corollary**

**6.**

**Proof.**

- It is easy to see that $x=(1,\dots ,1)$ is a 2-periodic point of $F\circ \copyright $. Hence, by Theorem 3, x is a fixed point of $F\circ G$.
- If $n=3$, $x=(1,1,0)$ is a point of period 3 of $F\circ \copyright $. Thus, by Theorem 3, x is a point of period 3 of $F\circ G$.
- Finally, if $n\ge 4$, $x=(1,1,0,\dots ,0)$ is a point of period $2n$ for $F\circ \copyright $ (see [27] (Corollary 3)). Therefore, by Theorem 3, x is a point of period n for $F\circ G$. □

**Corollary**

**7.**

**Proof.**

**Corollary**

**8.**

**Proof.**

**Corollary**

**9.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Graph associated with $F({x}_{1},{x}_{2},{x}_{3})=({x}_{1}\wedge ({x}_{2}\vee {x}_{3}),{x}_{2}\wedge ({x}_{1}^{\prime}\vee {x}_{3}^{\prime}),{x}_{3}\vee ({x}_{2}\wedge {x}_{1}))$.

**Figure 2.**Phase portrait of the system $F({x}_{1},{x}_{2},{x}_{3})=({x}_{1}\wedge ({x}_{2}\vee {x}_{3}),{x}_{2}\wedge ({x}_{1}^{\prime}\vee {x}_{3}^{\prime}),{x}_{3}\vee ({x}_{2}\wedge {x}_{1}))$.

**Figure 3.**Graph associated with the system $F({x}_{1},{x}_{2},{x}_{3})=({x}_{1}\wedge {x}_{2},{x}_{2}\wedge ({x}_{1}^{\prime}\vee {x}_{3}^{\prime}),{x}_{3}\vee {x}_{1}).$

**Figure 4.**Phase portrait of the system $F({x}_{1},{x}_{2},{x}_{3})=({x}_{1}\wedge {x}_{2},{x}_{2}\wedge ({x}_{1}^{\prime}\vee {x}_{3}^{\prime}),{x}_{3}\vee {x}_{1}).$

**Figure 5.**Graph associated with $F({x}_{1},{x}_{2},{x}_{3})=({x}_{1}^{\prime}\vee {x}_{3},{x}_{2}^{\prime}\vee {x}_{3},{x}_{3})$.

**Figure 6.**Phase portrait of the system $F({x}_{1},{x}_{2},{x}_{3})=({x}_{1}^{\prime}\vee {x}_{3},{x}_{2}^{\prime}\vee {x}_{3},{x}_{3})$.

**Figure 7.**Graph associated with $F({x}_{1},{x}_{2},{x}_{3})=({x}_{1}^{\prime}\wedge {x}_{3},{x}_{2}^{\prime}\vee {x}_{3}^{\prime},{x}_{1}\vee {x}_{3})$.

**Figure 8.**Phase portrait of the system $F({x}_{1},{x}_{2},{x}_{3})=({x}_{1}^{\prime}\wedge {x}_{3},{x}_{2}^{\prime}\vee {x}_{3}^{\prime},{x}_{1}\vee {x}_{3})$.

**Figure 9.**Graph associated with $F({x}_{1},{x}_{2},{x}_{3})=({x}_{1}^{\prime}\vee {x}_{3}^{\prime},{x}_{2}^{\prime}\wedge {x}_{3}^{\prime},{x}_{3}^{\prime}\wedge {x}_{1}\wedge {x}_{2})$.

**Figure 10.**Phase portrait of the system $F({x}_{1},{x}_{2},{x}_{3})=({x}_{1}^{\prime}\vee {x}_{3}^{\prime},{x}_{2}^{\prime}\wedge {x}_{3}^{\prime},{x}_{3}^{\prime}\wedge {x}_{1}\wedge {x}_{2}).$

**Figure 11.**Graph associated with $F({x}_{1},{x}_{2},{x}_{3})=({x}_{1}^{\prime}\vee {x}_{2}\vee {x}_{3},{x}_{2}^{\prime}\wedge {x}_{1}^{\prime},{x}_{3}\wedge {x}_{1}^{\prime})$.

**Figure 12.**Phase portrait of the system $F({x}_{1},{x}_{2},{x}_{3})=({x}_{1}^{\prime}\vee {x}_{2}\vee {x}_{3},{x}_{2}^{\prime}\wedge {x}_{1}^{\prime},{x}_{3}\wedge {x}_{1}^{\prime})$.

**Figure 13.**Graph associated with $F\circ \copyright ({x}_{1},{x}_{2},{x}_{3},{x}_{4})=({x}_{1}\vee {x}_{2}^{\prime},{x}_{1}\vee {x}_{2}^{\prime}\vee {x}_{3}\vee {x}_{4}^{\prime},{x}_{2}^{\prime}\vee {x}_{3},{x}_{2}^{\prime}\vee {x}_{4}^{\prime})$.

**Figure 14.**Phase portrait of $F\circ \copyright ({x}_{1},{x}_{2},{x}_{3},{x}_{4})=({x}_{1}\vee {x}_{2}^{\prime},{x}_{1}\vee {x}_{2}^{\prime}\vee {x}_{3}\vee {x}_{4}^{\prime},{x}_{2}^{\prime}\vee {x}_{3},{x}_{2}^{\prime}\vee {x}_{4}^{\prime})$.

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**MDPI and ACS Style**

Aledo, J.A.; Barzanouni, A.; Malekbala, G.; Sharifan, L.; Valverde, J.C.
On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions. *Mathematics* **2020**, *8*, 1088.
https://doi.org/10.3390/math8071088

**AMA Style**

Aledo JA, Barzanouni A, Malekbala G, Sharifan L, Valverde JC.
On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions. *Mathematics*. 2020; 8(7):1088.
https://doi.org/10.3390/math8071088

**Chicago/Turabian Style**

Aledo, Juan A., Ali Barzanouni, Ghazaleh Malekbala, Leila Sharifan, and Jose C. Valverde.
2020. "On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions" *Mathematics* 8, no. 7: 1088.
https://doi.org/10.3390/math8071088