# Finite-Time Stabilization of Unstable Orbits in the Fractional Difference Logistic Map

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. H-Ranks and Algebraic Complexity

#### 2.2. Classical Logistic Map and Types of Convergence

#### 2.3. Fractional Logistic Map

## 3. The Unstable Orbits of the Fractional Difference Logistic Map

#### 3.1. The Existence of the Unstable Period-1 Orbit at $a=3$

#### 3.2. The Existence of the Non-Asymptotic Convergence to the Unstable Period-1 Orbit

## 4. The Memory Effects and the Naive Control Scheme

#### 4.1. The Naive Control Scheme for the Classical Logistic Map

#### 4.2. The Naive Control Scheme for the Fractional Difference Logistic Map

## 5. The Proposed Scheme Based on a Single Control Impulse

## 6. The Proposed Scheme Based on Multiple Control Impulses

## 7. Concluding Remarks

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Galor, O. Discrete Dynamical Systems; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Smith, H.L.; Thieme, H.R. Dynamical Systems and Population Persistence; American Mathematical Soc.: Providence, RI, USA, 2011; Volume 118. [Google Scholar]
- Hasegawa, Y. Control Problems of Discrete-Time Dynamical Systems; Springer: Berlin/Heidelberg, Germany, 2013; Volume 447. [Google Scholar]
- Yang, X.; Deng, W.; Yao, J. Neural network based output feedback control for DC motors with asymptotic stability. Mech. Syst. Signal Process.
**2022**, 164, 108288. [Google Scholar] [CrossRef] - Vinoth, S.; Sivasamy, R.; Sathiyanathan, K.; Unyong, B.; Vadivel, R.; Gunasekaran, N. A novel discrete-time Leslie–Gower model with the impact of Allee effect in predator population. Complexity
**2022**, 2022, 6931354. [Google Scholar] [CrossRef] - Zheng, Y.; Yu, J. Stabilization of multi-rotation unstable periodic orbits through dynamic extended delayed feedback control. Chaos Solitons Fractals
**2022**, 161, 112362. [Google Scholar] [CrossRef] - Rodríguez-Núñez, J.M.; de León, A.; Molinar-Tabares, M.E.; Flores-Acosta, M.; Castillo, S. Computational chaos control based on small perturbations for complex spectra simulation. Simulation
**2022**, 98, 835–846. [Google Scholar] [CrossRef] - Hulka, T.; Matousek, R.; Lozi, R.P. Stabilization of Higher Periodic Orbits of Chaotic maps using Permutation-selective Objective Function. In Proceedings of the 2022 IEEE Workshop on Complexity in Engineering (COMPENG), IEEE, Florence, Italy, 18–20 July 2022; pp. 1–5. [Google Scholar]
- Bramburger, J.J.; Kutz, J.N.; Brunton, S.L. Data-driven stabilization of periodic orbits. IEEE Access
**2021**, 9, 43504–43521. [Google Scholar] [CrossRef] - Weng, Y.; Zhang, Q.; Cao, J.; Yan, H.; Qi, W.; Cheng, J. Finite-time model-free adaptive control for discrete-time nonlinear systems. IEEE Trans. Circuits Syst. II Express Briefs 2023. [CrossRef]
- Edelman, M. Maps with power-law memory: Direct introduction and Eulerian numbers, fractional maps, and fractional difference maps. Handb. Fract. Calc. Appl.
**2019**, 2, 47–63. [Google Scholar] - Chen, L.; Yin, H.; Yuan, L.; Machado, J.T.; Wu, R.; Alam, Z. Double color image encryption based on fractional order discrete improved Henon map and Rubik’s cube transform. Signal Process. Image Commun.
**2021**, 97, 116363. [Google Scholar] [CrossRef] - Zhu, L.; Jiang, D.; Ni, J.; Wang, X.; Rong, X.; Ahmad, M.; Chen, Y. A stable meaningful image encryption scheme using the newly-designed 2D discrete fractional-order chaotic map and Bayesian compressive sensing. Signal Process.
**2022**, 195, 108489. [Google Scholar] [CrossRef] - Liu, Z.; Xia, T.; Wang, T. Dynamic analysis of fractional-order six-order discrete chaotic mapping and its application in information security. Optik
**2023**, 272, 170356. [Google Scholar] [CrossRef] - Coll, C.; Herrero, A.; Ginestar, D.; S’anchez, E. The discrete fractional order difference applied to an epidemic model with indirect transmission. Appl. Math. Model.
**2022**, 103, 636–648. [Google Scholar] [CrossRef] - Abbes, A.; Ouannas, A.; Shawagfeh, N.; Grassi, G. The effect of the Caputo fractional difference operator on a new discrete COVID-19 model. Results Phys.
**2022**, 39, 105797. [Google Scholar] [CrossRef] - Chu, Y.M.; Bekiros, S.; Zambrano-Serrano, E.; Orozco-L’opez, O.; Lahmiri, S.; Jahanshahi, H.; Aly, A.A. Artificial macro-economics: A chaotic discrete-time fractional-order laboratory model. Chaos Solitons Fractals
**2021**, 145, 110776. [Google Scholar] [CrossRef] - Peng, Y.; Liu, J.; He, S.; Sun, K. Discrete fracmemristor-based chaotic map by Grunwald–Letnikov difference and its circuit implementation. Chaos Solitons Fractals
**2023**, 171, 113429. [Google Scholar] [CrossRef] - Edelman, M.; Jacobi, R. Power-Law Memory in Living Species and the Distribution of Lifespans. In Proceedings of the APS March Meeting Abstracts, Virtual, 15–19 March 2021; Volume 2021, p. L14-003. [Google Scholar]
- Zambrano-Serrano, E.; Bekiros, S.; Platas-Garza, M.A.; Posadas-Castillo, C.; Agarwal, P.; Jahanshahi, H.; Aly, A.A. On chaos and projective synchronization of a fractional difference map with no equilibria using a fuzzy-based state feedback control. Phys. Stat. Mech. Its Appl.
**2021**, 578, 126100. [Google Scholar] [CrossRef] - Lu, Q.; Zhu, Y.; Li, B. Necessary optimality conditions of fractional-order discrete uncertain optimal control problems. Eur. J. Control.
**2023**, 69, 100723. [Google Scholar] [CrossRef] - Yao, Y.; Wu, L.B. Backstepping control for fractional discrete-time systems. Appl. Math. Comput.
**2022**, 434, 127450. [Google Scholar] [CrossRef] - Shahamatkhah, E.; Tabatabaei, M. Containment control of linear discrete-time fractional-order multi-agent systems with time-delays. Neurocomputing
**2020**, 385, 42–47. [Google Scholar] [CrossRef] - Edelman, M. Universal fractional map and cascade of bifurcations type attractors. Chaos Interdiscip. J. Nonlinear Sci.
**2013**, 23, 033127. [Google Scholar] [CrossRef] [Green Version] - Edelman, M. Fractional maps and fractional attractors. Part II: Fractional difference caputo α-families of maps. Discontinuity Nonlinearity Complex.
**2015**, 4, 391–402. [Google Scholar] [CrossRef] [Green Version] - Kaslik, E.; Sivasundaram, S. Non-existence of periodic solutions in fractional-order dynamical systems and a remarkable difference between integer and fractional-order derivatives of periodic functions. Nonlinear Anal. Real World Appl.
**2012**, 13, 1489–1497. [Google Scholar] [CrossRef] [Green Version] - Diblík, J.; Fečkan, M.; Pospíšil, M. Nonexistence of periodic solutions and S-asymptotically periodic solutions in fractional difference equations. Appl. Math. Comput.
**2015**, 257, 230–240. [Google Scholar] [CrossRef] - Franklin, G.F.; Powell, J.D.; Emami-Naeini, A.; Powell, J.D. Feedback Control of Dynamic Systems; Prentice Hall: Upper Saddle River, NJ, USA, 2002; Volume 4. [Google Scholar]
- Piunovskiy, A.; Plakhov, A.; Torres, D.F.; Zhang, Y. Optimal impulse control of dynamical systems. Siam J. Control. Optim.
**2019**, 57, 2720–2752. [Google Scholar] [CrossRef] [Green Version] - Lu, G.; Landauskas, M.; Ragulskis, M. Control of divergence in an extended invertible logistic map. Int. J. Bifurc. Chaos
**2018**, 28, 1850129. [Google Scholar] [CrossRef] - Landauskas, M.; Ragulskis, M. A pseudo-stable structure in a completely invertible bouncer system. Nonlinear Dyn.
**2014**, 78, 1629–1643. [Google Scholar] [CrossRef] - Navickas, Z.; Ragulskis, M.; Karaliene, D.; Telksnys, T. Weak and strong orders of linear recurring sequences. Comput. Appl. Math.
**2018**, 37, 3539–3561. [Google Scholar] [CrossRef] - Petkevičiūtė-Gerlach, D.; Timofejeva, I.; Ragulskis, M. Clocking convergence of the fractional difference logistic map. Nonlinear Dyn.
**2020**, 100, 3925–3935. [Google Scholar] [CrossRef] - Kurakin, V.; Kuzmin, A.; Mikhalev, A.; Nechaev, A. Linear recurring sequences over rings and modules. J. Math. Sci.
**1995**, 76, 2793–2915. [Google Scholar] [CrossRef] - Bisgard, J. Analysis and Linear Algebra: The Singular Value Decomposition and Applications; American Mathematical Soc.: Providence, RI, USA, 2020; Volume 94. [Google Scholar]
- May, R.M. Simple mathematical models with very complicated dynamics. Nature
**1976**, 261, 459–467. [Google Scholar] [CrossRef] - Edelman, M. On stability of fixed points and chaos in fractional systems. Chaos Interdiscip. J. Nonlinear Sci.
**2018**, 28, 023112. [Google Scholar] [CrossRef] [Green Version] - Edelman, M. Evolution of systems with power-law memory: Do we have to die?(Dedicated to the Memory of Valentin Afraimovich). Demogr. Popul. Health Aging Health Expend.
**2020**, 50, 65–85. [Google Scholar] - Edelman, M. Stability of fixed points in generalized fractional maps of the orders 0< α< 1. Nonlinear Dyn.
**2023**, 111, 10247–10254. [Google Scholar]

**Figure 1.**The bifurcation diagram of (4) is depicted in part (

**a**); the plot of H-ranks is shown in part (

**b**). Note that parameter values $N=50$ and $\epsilon ={10}^{-10}$ were used for the computation of H-ranks.

**Figure 2.**Three types of convergence processes to an orbit of a classical logistic map at $a=3.3$. Parts (

**a**–

**c**) correspond to asymptotic convergence to the stable period-2 orbit, non-asymptotic convergence to the stable period-2 orbit, and non-asymptotic convergence to the unstable period-1 orbit respectively. Plots on the left side depict the values of H-ranks for ${x}_{0}\in [0,1]$ and $a=3.3$. Plots on the right side illustrate the trajectories of the classical logistic map started from different initial conditions marked by a black circle in the corresponding plots of H-ranks.

**Figure 3.**The bifurcation diagram of (1) for $\alpha =0.8$ is depicted in part (

**a**); the plot of H-ranks is shown in part (

**b**). Parameter values $N=50$ and $\epsilon ={10}^{-10}$ were used for the computation of H-ranks.

**Figure 4.**Types of convergence in the fractional logistic map for parameter values $\alpha =0.8,a=3$. Part (

**a**) depicts asymptotic convergence to the period-2 asymptotically stable attractor. Part (

**b**) temporary stabilization of the trajectory to the unstable period-1 attractor. Plots on the right represent trajectories of the fractional logistic map; plots on the left depict the H-ranks for initial conditions ${x}_{0}\in (0,1)$; the black circle indicates the initial condition from which the trajectories on the right begin. Note that in (

**b**), the black circle is on the red band of high H-ranks.

**Figure 5.**The naive stabilization scheme does not work for the fractional difference logistic map ($\alpha =0.8,a=3$, and the stabilization threshold $\delta =0.05$). Part (

**a**) depict H-ranks of fractional logistic map trajectories starting from the initial condition, where the black circle denotes the initial condition, chosen in such a way that the logistic map trajectory is temporarily stabilized. In part (

**b**), fractional logistic map trajectories are depicted. Red dotted lines represent ${x}^{*}\pm \delta $; the green dashed line represents the iteration at which the stabilization is performed; the black line represents the initial part of the trajectory for which $|{x}_{n}-{x}^{*}|<\delta $ holds true; the gray solid line is the continuation of the black trajectory if no stabilization is performed; the blue line is the trajectory after performing naive stabilization by setting ${x}_{86}={x}^{*}$. Part (

**c**) depicts the H-ranks computed time-forward starting from the moment of stabilization, where the black circle denotes the position of the trajectory after the stabilization impulse ${x}^{*}$. Part (

**d**) is a zoomed plot of the boxed area present in part (

**b**), which details the stabilization process: when the initial trajectory intersects the red dotted line, the control impulse (red arrow) shifts the trajectory to ${x}^{*}$.

**Figure 6.**The weights ${G}_{k}^{\alpha}$ for the fractional logistic map (1). While the most recent iterations have the biggest impact on the next elements of the map, the memory horizon does reach the initial condition.

**Figure 7.**A single control impulse can stabilize the unstable orbit of the fractional difference logistic map for a finite time. The parameters of the model are set to: $\alpha =0.8$; $a=3$; $\delta =0.05$. Red horizontal dotted lines denote the tolerance corridor around the unstable period-1 orbit: ${x}^{*}\pm \delta $. Black lines represent the trajectories of the fractional difference logistic map before stabilization impulse. Grey solid lines show how the trajectory would evolve if the control is not applied. Vertical green dashed lines denote the iteration number where the control impulse is applied. Blue lines represent trajectories after the stabilization impulse. Parts (

**b**,

**d**) depict the patterns of H-ranks computed time-forward starting from the stabilization moment. The black circles in (

**b**,

**d**) show the position of the map trajectory after the control impulse. Note that the unstable period-1 is stabilized in both panels (

**a**,

**c**). However, the transient processes are different right after the control impulse.

**Figure 8.**A proper selection of the control impulse is a key to the proposed scheme. Violation of the boundedness of the fractional logistic map with parameters $\alpha =0.8$, $a=3$, $\delta =0.05$ is observed in panels (

**a**,

**c**). In (

**a**,

**c**), red dotted lines represent ${x}^{*}\pm \delta $; black lines represent the trajectories of the fractional logistic map before the impulse; gray solid lines show how the black lines would continue if the control impulse is not applied; blue lines represent trajectories after the stabilization impulse; the green dashed line represents the iteration at which the impulse is applied. Parts (

**b**,

**d**) depict the H-ranks computed time-forward starting from the stabilization impulse in (

**a**,

**c**), respectively. The white circle in (

**b**) shows the trajectory after performing the stabilization impulse in (

**a**); the black circle in (

**d**) shows the trajectory after performing the stabilization impulse in (

**d**). Note that the blue line in (

**c**) dips below zero before asymptotically converging to the asymptotically stable period-2 orbit.

**Figure 9.**Realization of the proposed control scheme for the fractional difference logistic map at $\alpha =0.8$, $a=3$, $\delta =0.05$ in part (

**a**). Blue and purple lines denote stabilized trajectories; the pink line shows the trajectory asymptotically converging to the asymptotically stable period-2 orbit. Parts (

**b**–

**d**) depict the H-ranks computed time-forward starting from the first, second and third stabilization moments, respectively; black circles show the position of the map trajectories after respective control impulses.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Uzdila, E.; Telksniene, I.; Telksnys, T.; Ragulskis, M.
Finite-Time Stabilization of Unstable Orbits in the Fractional Difference Logistic Map. *Fractal Fract.* **2023**, *7*, 570.
https://doi.org/10.3390/fractalfract7080570

**AMA Style**

Uzdila E, Telksniene I, Telksnys T, Ragulskis M.
Finite-Time Stabilization of Unstable Orbits in the Fractional Difference Logistic Map. *Fractal and Fractional*. 2023; 7(8):570.
https://doi.org/10.3390/fractalfract7080570

**Chicago/Turabian Style**

Uzdila, Ernestas, Inga Telksniene, Tadas Telksnys, and Minvydas Ragulskis.
2023. "Finite-Time Stabilization of Unstable Orbits in the Fractional Difference Logistic Map" *Fractal and Fractional* 7, no. 8: 570.
https://doi.org/10.3390/fractalfract7080570