#
D_{3} Dihedral Logistic Map of Fractional Order^{ †}

^{1}

^{2}

^{3}

^{4}

^{*}

^{†}

## Abstract

**:**

_{3}dihedral logistic map of fractional order is introduced. The map presents a dihedral symmetry D

_{3}. It is numerically shown that the construction and interpretation of the bifurcation diagram versus the fractional order requires special attention. The system stability is determined and the problem of hidden attractors is analyzed. Furthermore, analytical and numerical results show that the chaotic attractor of integer order, with D

_{3}symmetries, looses its symmetry in the fractional-order variant.

## 1. Introduction

## 2. D_{3} Dihedral Logistic Map of IO

## 3. Dihedral Logistic Map of FO

## 4. Stability of Fixed Points

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

## 5. Bifurcation Diagrams

- ${Q}_{1}$:
- Should the BD be considered as the “reunion” of all BSs?
- ${Q}_{2}$:
- Considering the intensive numerical experiments which indicate that different initial conditions generates different BSs, how many such BSs can be finally obtained and which one of these BSs should be considered the “right” BD?

**Remark**

**2.**

- (i)
- Beside the dependence on initial conditions, because fractional derivatives are nonlocal operators, they present the so called memory effect which means that the actual behavior is not only influenced by the actual state of the underlying system but also by the events happened in the past. Therefore, beside the dependence on the initial conditions, at every moment n, the solutions ${x}_{n}$ and ${y}_{n}$ depends not only on ${x}_{0}$ and ${y}_{0}$ but also on all previous values ${x}_{k}$ and ${y}_{k}$, for $k=1,2,\dots ,n-1$.
- (ii)
- Because the decay rate of the solutions in the asymptotically stable case is ${n}^{-q}$ [40] (see also [14]), smaller values of q implies bigger errors, while to bigger values of q, close to 1, errors are smaller. In Figure 7, the graph of ${n}^{-q}$ is represented as function of n for different values of q. For clarity, only first few dozens of values n have been considered. The circles at $n=30$ indicates the order of errors for each considered values of q. If one considers ${n}_{max}=$ 10,000, from the curve $q=0.01$ one obtains 10,000${}^{-0.01}=0.9120$, while the curve $q=0.9$, gives 10,000${}^{-0.9}=2.5119e-004$.

## 6. Hidden Attractors

## 7. Symmetry Broken by the Fractional Order

**Theorem**

**3.**

**Proof.**

## 8. 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.**A ${D}_{3}$-symmetric image of the DLM of IO (7). In red is indicated the counterclockwise rotation with ${120}^{\circ}$ applied to the point A to obtain the point B which is symmetric with respect the line (b).

**Figure 2.**Stability domain S in the case of the fixed point ${X}_{0}^{*}$. Tick line represents the asymptotic stability range of q for ${X}_{0}^{*}$.

**Figure 3.**DLM of FO. (

**a**) Bifurcation diagram vs. q. The bifurcation-like point P indicates the beginning of the stability of ${X}_{0}^{*}$ for $q>0.8512$; (

**b**) zoomed image; (

**c**) time series tending to the asymptotically stable fixed point ${X}_{0}^{*}$ for $q>0.8512$; (

**d**) phase portrait with the orbit points indicating the evolution of the orbit to ${X}_{0}^{*}$; (

**e**) zoomed area of the bifurcation diagram.

**Figure 4.**Another bifurcation diagram vs. $q\in (0,0.15]$ for initial conditions $[-0.5,0.1]$, $[0.2,0.1]$, $[-0.01,0.1]$, $[0.4,0.1]$, $[0.1,0.1]$.

**Figure 5.**Bifurcation diagrams of the DLM of FO and IO versus the initial conditions ${x}_{0}\in [-0.5,0.5]$ and ${y}_{0}=0.8$. (

**a**) The FO case $q=0.03$; (

**b**) the IO case.

**Figure 6.**Sketch of diagrams of bifurcations. (

**a**) The IO case; (

**b**) FO case. ${u}_{0i}$, $i=1,2,\dots ,5$ are initial conditions.

**Figure 8.**Two orbits of the DLM of FO for $q=0.1$. (

**a**) Time series of a periodic-like orbit from ${[{x}_{0},{y}_{0}]}_{1}=[-0.5,-0.1]$; (

**b**) phase portrait of the orbit and a zoom indicating the slow orbit convergence; (

**c**) time series of a two-band quasiperiodic-like orbit from ${[{x}_{0},{y}_{0}]}_{2}=[0.2,0.1]$. The zoom indicates the alternate pattern of the orbit between the two subsets ${\mathcal{A}}_{1}$ and ${\mathcal{A}}_{2}$ (red and brown) of the quasiperiodic-like attractor; (

**d**) phase portrait of the orbit ${[{x}_{0},{y}_{0}]}_{2}=[0.2,0.1]$.

**Figure 9.**Hidden chaotic attractors of the DLMFO with broken-symmetry. (

**a**) $q=0.03$ and ${[{x}_{0},{y}_{0}]}_{5}=[0.1,-0.7]$. (

**b**) $q=0.03$ and ${[{x}_{0},{y}_{0}]}_{3}=[0.01,0.01]$; (

**c**) $q=0.01$, and $[{x}_{0},{y}_{0}]=[-0.01,-0.01]$.

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Danca, M.-F.; Kuznetsov, N.
*D*_{3} Dihedral Logistic Map of Fractional Order. *Mathematics* **2022**, *10*, 213.
https://doi.org/10.3390/math10020213

**AMA Style**

Danca M-F, Kuznetsov N.
*D*_{3} Dihedral Logistic Map of Fractional Order. *Mathematics*. 2022; 10(2):213.
https://doi.org/10.3390/math10020213

**Chicago/Turabian Style**

Danca, Marius-F., and Nikolay Kuznetsov.
2022. "*D*_{3} Dihedral Logistic Map of Fractional Order" *Mathematics* 10, no. 2: 213.
https://doi.org/10.3390/math10020213