# Coupled Discrete Fractional-Order Logistic Maps

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## Abstract

**:**

## 1. Introduction

## 2. A Model of Coupled FO Logistic Maps

## 3. Bounds and Global Dynamics

**Theorem**

**1.**

**Theorem**

**2.**

**Proof.**

- (1)
- If $S=\{x\ge 1+\delta ,\phantom{\rule{0.166667em}{0ex}}y\ge -\frac{1}{3}+\delta \}$ and (12) holds, then (7) implies that$$\begin{array}{ccc}\hfill x\left(n\right)& =& x\left(0\right)+\frac{p}{\Gamma \left(q\right)}\sum _{i=1}^{{n}_{0}}\frac{\Gamma (n-i+q)}{\Gamma (n-i+1)}(3y(i-1)+1)x(i-1)(1-x(i-1))\hfill \\ & & +\frac{p}{\Gamma \left(q\right)}\sum _{i={n}_{0}+1}^{n}\frac{\Gamma (n-i+q)}{\Gamma (n-i+1)}(3y(i-1)+1)x(i-1)(1-x(i-1))\hfill \\ & \le & x\left(0\right)+\frac{p}{\Gamma \left(q\right)}\sum _{i=1}^{{n}_{0}}(3y(i-1)+1)x(i-1)(1-x(i-1))\hfill \\ & & -3{\delta}^{2}(1+\delta )\frac{p}{\Gamma \left(q\right)}\sum _{i={n}_{0}+1}^{n}\frac{\Gamma (n-i+q)}{\Gamma (n-i+1)},\hfill \end{array}$$
- (2)
- If $S=\{x\le -\delta ,\phantom{\rule{0.166667em}{0ex}}y\le -\frac{1}{3}-\delta \}$ and (12) holds, then (7) implies that$$\begin{array}{ccc}\hfill x\left(n\right)& \ge & x\left(0\right)+\frac{p}{\Gamma \left(q\right)}\sum _{i=1}^{{n}_{0}}(3y(i-1)+1)x(i-1)(1-x(i-1))\hfill \\ & & +3{\delta}^{2}(1+\delta )\frac{p}{\Gamma \left(q\right)}\sum _{i={n}_{0}+1}^{n}\frac{\Gamma (n-i+q)}{\Gamma (n-i+1)},\hfill \end{array}$$
- (3)
- If $S=\{x\le \frac{4}{3}-\delta ,\phantom{\rule{0.166667em}{0ex}}y\ge 1+\delta \}$ and (12) holds, then (7) implies that$$\begin{array}{ccc}\hfill y\left(n\right)& \le & y\left(0\right)+\frac{p}{\Gamma \left(q\right)}\sum _{i=1}^{{n}_{0}}(-3x(i-1)+4)y(i-1)(1-y(i-1))\hfill \\ & & -3{\delta}^{2}(1+\delta )\frac{p}{\Gamma \left(q\right)}\sum _{i={n}_{0}+1}^{n}\frac{\Gamma (n-i+q)}{\Gamma (n-i+1)},\hfill \end{array}$$
- (4)
- If $S=\{x\ge \frac{4}{3}+\delta ,\phantom{\rule{0.166667em}{0ex}}y\le -\delta \}$ and (12) holds, then (7) implies that$$\begin{array}{ccc}\hfill y\left(n\right)& \ge & y\left(0\right)+\frac{p}{\Gamma \left(q\right)}\sum _{i=1}^{{n}_{0}}(-3x(i-1)+4)y(i-1)(1-y(i-1))\hfill \\ & & +3{\delta}^{2}(1+\delta )\frac{p}{\Gamma \left(q\right)}\sum _{i={n}_{0}+1}^{n}\frac{\Gamma (n-i+q)}{\Gamma (n-i+1)},\hfill \end{array}$$

## 4. Non-Existence of Hidden Attractors

**Proposition**

**1.**

**Proof.**

**Proof.**

**Proposition**

**2.**

**Remark**

**2.**

- (i)
- The shapes of BSs approximately preserve the shapes for different initial conditions, but move along the p-axis.
- (ii)
- This delay-like phenomenon with respect to the initial conditions (the BSs seem to move “forward” or “backward” on the BDs (see the dotted lines I, $II$, and $III$ in Figure 5, as the initial conditions are changing) was already found in a continuous-time FO system [47], where the “delay” was observed with respect to the integration step-size of the numerical method used.
- (iii)
- It is interesting to compare the above results with the case of the Feigenbaum attractor of the IO logistic map $x(n+1)=px\left(n\right)(1-x(n\left)\right)$, for the limiting value ${p}_{\infty}=3.569946\dots $ [48], which, however, is not an attracting set and for which there is no sensitive dependence on initial conditions.

## 5. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Stability regions of equilibria. (

**a**) Stability region of ${E}_{1}$ (grey plot); (

**b**) Stability region of ${E}_{5}$ (grey plot); (

**c**) Instability region of all equilibria (light red plot); (

**d**) Phase plot of a representative orbit from the point within the stability region of ${E}_{1}$, with $p=0.5$ and $q=0.5$ (Figure 1a) from initial condition close to ${E}_{1}$; (

**e**) phase plot of a representative orbit from a point within the stability region of ${E}_{1}$, with $p=0.1$ and $q=0.5$ (Figure 1b) from initial condition close to ${E}_{5}$.

**Figure 2.**(

**a**) The BD for $p=0.55$ vs. q obtained for $({x}_{0},{y}_{0})=(0.1,0.3)$; (

**b**) the BD and for $q=0.4$ vs. $p\in (0.3,0.6)$ for $({x}_{0},{y}_{0})=(0.1,0.3)$.

**Figure 3.**(

**a**) The BD for $p=0.55$ vs. q with two BSs: one obtained for $({x}_{0},{y}_{0})=(0.1,0.3)$, in Figure 2a (red plot), and one for a second initial condition $({x}_{0},{y}_{0})=(0.5,0.5)$ (blue plot); (

**b**) the BD for $q=0.4$ vs. $p\in (0.3,0.6)$ with two BSs: one for $({x}_{0},{y}_{0})=(0.1,0.3)$, in Figure 2b (red plot), and a new one for $({x}_{0},{y}_{0})=(0.9,0.6)$ (blue plot). Dotted line in the BD in Figure 2b indicates the existence of two different attractors: a four-periodic-like orbit (red bullets) and a two-period chaotic band orbit (dark blue segments).

**Figure 4.**Two attractors for $p=0.55$ and $q=0.28$: (

**a**) The four-periodic-like attractor for $({x}_{0},{y}_{0})=(0.1,0.3)$; (

**b**) the two-period chaotic-like attractor for $({x}_{0},{y}_{0})=(0.5,0.5)$; (

**c**) zoomed region around the point 1 of the periodic-like orbit underlines the slow convergence of the orbit; (

**d**) time series of the periodic-like orbit.

**Figure 5.**The BDs of system (8) for $p=0.55$. (

**a**) Initial conditions $(0.1,0.3)$ (red BS) and $(0.5,0.5)$ (blue BS); (

**b**) Initial conditions $(0.1,0.3)$ (red BS), $(0.5,0.5)$ (blue BS) and $(0.01,0.7)$ (green plot).

**Figure 6.**The BDs of system (8) for $p=0.55$, obtained with the same three different initial conditions but with different maximum iteration numbers: (

**a**) 3500 iterations; (

**b**) 5000 iterations.

**Table 1.**Spectrum $\sigma $ of eigenvalues of J evaluated at equilibria ${E}_{i}$, $i=1,2,\dots ,5$.

E | $\mathit{\sigma}\left(\mathit{J}\right)$ |
---|---|

${E}_{1}$ | $(-\frac{4p}{3}l,\frac{4p}{3}l)$ |

${E}_{2}$ | $(p,4p)$ |

${E}_{3}$ | $(-p,p)$ |

${E}_{4}$ | $(-4p,4p)$ |

${E}_{5}$ | $(-4p,-p)$ |

${\mathit{E}}_{1}$ | ${\mathit{E}}_{2,3,4}$ | ${\mathit{E}}_{5}$ |
---|---|---|

stable for $p<3\times {2}^{q-2}{cos}^{q}\frac{\pi}{2(2-q)}$ | Unstable for all p and $q\in (0,1)$ | stable for $p<{2}^{q-2}$ |

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**MDPI and ACS Style**

Danca, M.-F.; Fečkan, M.; Kuznetsov, N.; Chen, G. Coupled Discrete Fractional-Order Logistic Maps. *Mathematics* **2021**, *9*, 2204.
https://doi.org/10.3390/math9182204

**AMA Style**

Danca M-F, Fečkan M, Kuznetsov N, Chen G. Coupled Discrete Fractional-Order Logistic Maps. *Mathematics*. 2021; 9(18):2204.
https://doi.org/10.3390/math9182204

**Chicago/Turabian Style**

Danca, Marius-F., Michal Fečkan, Nikolay Kuznetsov, and Guanrong Chen. 2021. "Coupled Discrete Fractional-Order Logistic Maps" *Mathematics* 9, no. 18: 2204.
https://doi.org/10.3390/math9182204