# Stability Results for Two-Dimensional Systems of Fractional-Order Difference Equations

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Remark**

**1.**

**Proposition**

**1.**

**Definition**

**5.**

- a.
- The trivial solution of (1) is called stable if for any $\epsilon >0$ there exists $\delta =\delta \left(\epsilon \right)>0$ such that for every ${x}_{0}\in {\mathbb{R}}^{n}$ satisfying $\parallel {x}_{0}\parallel <\delta $ we have $\parallel \phi (nh,{x}_{0})\parallel <\epsilon $ for any $n\ge 0$.
- b.
- The trivial solution of (1) is called asymptotically stable if it is stable and there exists $\rho >0$, such that $\underset{n\to \infty}{lim}\phi (nh,{x}_{0})=0$ whenever $\parallel {x}_{0}\parallel <\rho $.
- c.
- The trivial solution of (1) is called$\mathcal{O}\left({n}^{-q}\right)$-asymptotically stableif it is stable and there exists $\rho >0$, such that for any $\parallel {x}_{0}\parallel <\rho $ one has $\parallel \phi (n,{x}_{0})\parallel =\mathcal{O}\left({n}^{-q}\right)$ as $n\to \infty .$

## 3. Stability Results for Systems of Two Fractional-Order Difference Equations

**Theorem**

**1.**

- 1.
- System (2) is $\mathcal{O}\left({n}^{-q}\right)$-globally asymptotically stable if and only if all the roots of ${\Delta}_{A}\left(z\right)$ are inside the unit circle ($\left|z\right|<1$), where $q=min\{{q}_{1},{q}_{2}\}$.
- 2.
- If $det\left(A\right)\ne 0$ and ${\Delta}_{A}\left(z\right)$ has at least one root outside the closed unit circle ($\left|z\right|\ge 1$), system (2) is unstable.

## 4. Fractional-Order Independent Results

**Theorem**

**2**

**.**System (2) is unstable for any choice of the fractional orders ${q}_{1}$ and ${q}_{2}$ if one of the following conditions hold:

- 1.
- $det\left(A\right)<0$;
- 2.
- $det\left(A\right)>0$, $0<h<1$ and $(1-h)({a}_{11}+{a}_{22})\ge {(1-h)}^{2}det\left(A\right)+1$;
- 3.
- ${a}_{11}>0$ and ${a}_{11}{a}_{22}\ge det\left(A\right)>0$.

**Proof.**

- 1.
- Since ${\Delta}_{A}\left(1\right)={h}^{{q}_{1}+{q}_{2}}\delta <0$ and ${\Delta}_{A}(\infty )=\infty $, because the function ${\Delta}_{A}$ is continuous, it follows that it has at least one real root in the interval $(1,\infty )$. From Theorem 1 it follows that system (2) is unstable.
- 2.
- We have:$$\begin{array}{cc}\hfill {\Delta}_{A}\left(\frac{1}{1-h}\right)& ={h}^{{q}_{1}+{q}_{2}}\left({\displaystyle \frac{1}{{(1-h)}^{2}}}-{a}_{11}{\displaystyle \frac{1}{1-h}}-{a}_{22}{\displaystyle \frac{1}{1-h}}+\delta \right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\displaystyle \frac{{h}^{{q}_{1}+{q}_{2}}}{{(1-h)}^{2}}}\left(1-({a}_{11}+{a}_{22})(1-h)+\delta {(1-h)}^{2}\right)\le 0.\hfill \end{array}$$Because ${\Delta}_{A}(\infty )=\infty $, $\frac{1}{1-h}>1$ and ${\Delta}_{A}$ is a continuous function on $(1,\infty )$, it follows that ${\Delta}_{A}$ has at least one root in the interval $\left(\frac{1}{1-h},\infty \right)\subset \left(1,\infty \right)$ and, consequently, from Theorem 1 system (2) is unstable.
- 3.
- If ${a}_{11}{a}_{22}\ge \delta $, with ${a}_{11}>0$ and ${a}_{22}>0$ then$${\Delta}_{A}\left(z\right)=[z{(1-{z}^{-1})}^{{q}_{1}}-{a}_{11}{h}^{{q}_{1}}]\xb7[z{(1-{z}^{-1})}^{{q}_{2}}-{a}_{22}{h}^{{q}_{2}}]+{h}^{{q}_{1}+{q}_{2}}(\delta -{a}_{11}{a}_{22}).$$Denoting ${\Delta}_{1}\left(z\right)=z{(1-{z}^{-1})}^{{q}_{1}}-{a}_{11}{h}^{{q}_{1}}$, we have that ${\Delta}_{1}\left(1\right)=-{a}_{11}{h}^{{q}_{1}}<0$ and ${\Delta}_{1}(\infty )=\infty $, therefore there exists ${z}_{0}\in (1,\infty )$ such that ${\Delta}_{1}\left({z}_{0}\right)=0$. Hence, ${\Delta}_{A}\left({z}_{0}\right)={h}^{{q}_{1}+{q}_{2}}(\delta -{a}_{11}{a}_{22})\le 0$ and as ${\Delta}_{A}(\infty )=\infty $, it follows that ${\Delta}_{A}$ has a real root in the interval $[{z}_{0},\infty )\subset (1,\infty )$. Thus, according to Theorem 1 system (2) is unstable.

## 5. Fractional-Order Dependent Results

- the line$$l(\delta ,{q}_{1},{q}_{2},h)\phantom{\rule{3.33333pt}{0ex}}:\phantom{\rule{1.em}{0ex}}{a}_{11}{\left(\frac{h}{2}\right)}^{{q}_{1}}+{a}_{22}{\left(\frac{h}{2}\right)}^{{q}_{2}}+\delta {\left(\frac{h}{2}\right)}^{{q}_{1}+{q}_{2}}+1=0$$
- for ${q}_{1}\ne {q}_{2}$, the smooth parametric curve$$\mathrm{\Gamma}(\delta ,{q}_{1},{q}_{2},h)\phantom{\rule{3.33333pt}{0ex}}:\phantom{\rule{1.em}{0ex}}\left\{\begin{array}{c}{a}_{11}={\rho}_{2}({q}_{1},{q}_{2},\theta )u{(\theta ,h)}^{{q}_{1}}-\delta {\rho}_{1}({q}_{1},{q}_{2},\theta )u{(\theta ,h)}^{-{q}_{2}}\hfill \\ {a}_{22}=\delta {\rho}_{2}({q}_{1},{q}_{2},\theta )u{(\theta ,h)}^{-{q}_{1}}-{\rho}_{1}({q}_{1},{q}_{2},\theta )u{(\theta ,h)}^{{q}_{2}}\hfill \end{array}\right.,$$
- for ${q}_{1}={q}_{2}=:q$, the line$$\mathrm{\Lambda}(\delta ,q,h):\phantom{\rule{1.em}{0ex}}{a}_{11}+{a}_{22}=-2\sqrt{\delta}cos\left[(2-q)arccos\left(\frac{h}{2}{\delta}^{1/2q}\right)\right].$$

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

- 1.
- Considering ${a}_{ii}^{0}=\underset{\theta \to 0}{lim}{a}_{ii}\left(\theta \right)$, it follows that $({a}_{11}^{0},{a}_{22}^{0})$ belongs to the line $l(\delta ,{q}_{1},{q}_{2},h)$;
- 2.
- $\underset{\theta \to \pi /2}{lim}\left({a}_{11}\left(\theta \right),{a}_{22}\left(\theta \right)\right)=(-\infty ,\infty )$;
- 3.
- ${a}_{11}\left(\theta \right){a}_{22}\left(\theta \right)<\delta $, for all $\theta \in \left(0,{\displaystyle \frac{\pi}{2}}\right)$.

**Proof.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**3**

- 1.
- If ${q}_{1}\ne {q}_{2}$, system (2) is $\mathcal{O}\left({n}^{-q}\right)$-asymptotically stable (where $q=min\{{q}_{1},{q}_{2}\}$) if and only if $({a}_{11},{a}_{22})$ is in the domain $S(\delta ,{q}_{1},{q}_{2},h)$ situated above the line $l(\delta ,{q}_{1},{q}_{2},h)$ and below the curve $\mathrm{\Gamma}(\delta ,{q}_{1},{q}_{2},h)$.
- 2.
- If ${q}_{1}={q}_{2}:=q$, system (2) is $\mathcal{O}\left({n}^{-q}\right)$-asymptotically stable if and only if$$-\delta {\left(\frac{h}{2}\right)}^{q}-{\left(\frac{h}{2}\right)}^{-q}<{a}_{11}+{a}_{22}<-2\sqrt{\delta}cos\left[(2-q)arccos\left(\frac{h}{2}{\delta}^{1/2q}\right)\right].$$

**Proof.**

## 6. A Discrete FitzHugh-Nagumo Neuronal Model

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Cottone, G.; Paola, M.D.; Santoro, R. A novel exact representation of stationary colored Gaussian processes (fractional differential approach). J. Phys. A Math. Theor.
**2010**, 43, 085002. [Google Scholar] [CrossRef] - Engheia, N. On the role of fractional calculus in electromagnetic theory. IEEE Antennas Propag. Mag.
**1997**, 39, 35–46. [Google Scholar] [CrossRef] - Henry, B.I.; Wearne, S.L. Existence of Turing instabilities in a two-species fractional reaction-diffusion system. SIAM J. Appl. Math.
**2002**, 62, 870–887. [Google Scholar] [CrossRef][Green Version] - Heymans, N.; Bauwens, J.C. Fractal rheological models and fractional differential equations for viscoelastic behavior. Rheol. Acta
**1994**, 33, 210–219. [Google Scholar] [CrossRef] - Mainardi, F. Fractional Relaxation-Oscillation and Fractional Phenomena. Chaos Solitons Fractals
**1996**, 7, 1461–1477. [Google Scholar] [CrossRef] - Ionescu, C.; Lopes, A.; Copot, D.; Machado, J.T.; Bates, J. The role of fractional calculus in modeling biological phenomena: A review. Commun. Nonlinear Sci. Numer. Simul.
**2017**, 51, 141–159. [Google Scholar] [CrossRef] - Brandibur, O.; Kaslik, E. Stability properties of a two-dimensional system involving one Caputo derivative and applications to the investigation of a fractional-order Morris-Lecar neuronal model. Nonlinear Dyn.
**2017**, 90, 2371–2386. [Google Scholar] [CrossRef][Green Version] - Brandibur, O.; Kaslik, E. Stability of two-component incommensurate fractional-order systems and applications to the investigation of a FitzHugh-Nagumo neuronal model. Math. Methods Appl. Sci.
**2018**, 41, 7182–7194. [Google Scholar] [CrossRef][Green Version] - Mozyrska, D.; Wyrwas, M. Explicit criteria for stability of fractional h-difference two-dimensional systems. Int. J. Dyn. Control
**2017**, 5, 4–9. [Google Scholar] [CrossRef][Green Version] - Mozyrska, D.; Wyrwas, M. Stability by linear approximation and the relation between the stability of difference and differential fractional systems. Math. Methods Appl. Sci.
**2017**, 40, 4080–4091. [Google Scholar] [CrossRef] - Mozyrska, D.; Wyrwas, M. Fractional Linear Equations with Discrete Operators of Positive Order. Adv. Model. Control Non-Integer-Order Syst. Lect. Notes Electr. Eng.
**2015**, 320, 47–58. [Google Scholar] - Mozyrska, D.; Wyrwas, M. Stability of Linear Systems with Caputo Fractional-, Variable-Order Difference Operator of Convolution Type. In Proceedings of the 2018 41st International Conference on Telecommunications and Signal Processing (TSP), Athens, Greece, 4–6 July 2018; pp. 1–4. [Google Scholar]
- Sabatier, J.; Farges, C. On stability of commensurate fractional order systems. Int. J. Bifurc. Chaos
**2012**, 22, 1250084. [Google Scholar] [CrossRef] - Li, C.; Ma, Y. Fractional dynamical system and its linearization theorem. Nonlinear Dyn.
**2013**, 71, 621–633. [Google Scholar] [CrossRef] - Čermák, J.; Kisela, T. Stability properties of two-term fractional differential equations. Nonlinear Dyn.
**2015**, 80, 1673–1684. [Google Scholar] [CrossRef] - Wang, Z.; Yang, D.; Zhang, H. Stability analysis on a class of nonlinear fractional-order systems. Nonlinear Dyn.
**2016**, 86, 1023–1033. [Google Scholar] [CrossRef] - Kukushkin, M.V. Asymptotics of eigenvalues for differential operators of fractional order. Fract. Calc. Appl. Anal.
**2019**, 22, 658–680. [Google Scholar] [CrossRef][Green Version] - Kukushkin, M.V. On One Method of Studying Spectral Properties of Non-selfadjoint Operators. In Abstract and Applied Analysis; Hindawi: London, UK, 2020; Volume 2020. [Google Scholar]
- FitzHugh, R. Impulses and physiological states in theoretical models of nerve membrane. Biophys. J.
**1961**, 1, 445–466. [Google Scholar] [CrossRef][Green Version] - Anastasio, T.J. The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Biol. Cybern.
**1994**, 72, 69–79. [Google Scholar] [CrossRef] - Lundstrom, B.N.; Higgs, M.H.; Spain, W.J.; Fairhall, A.L. Fractional differentiation by neocortical pyramidal neurons. Nat. Neurosci.
**2008**, 11, 1335–1342. [Google Scholar] [CrossRef] - Weinberg, S.H. Membrane capacitive memory alters spiking in neurons described by the fractional-order Hodgkin-Huxley model. PLoS ONE
**2015**, 10, e0126629. [Google Scholar] [CrossRef] [PubMed] - Du, M.; Wang, Z.; Hu, H. Measuring memory with the order of fractional derivative. Sci. Rep.
**2013**, 3, 3431. [Google Scholar] [CrossRef] [PubMed] - Mozyrska, D.; Wyrwas, M. Stability of discrete fractional linear systems with positive orders. Sci. Direct
**2017**, 50, 8115–8120. [Google Scholar] - Elaydi, S. An Introduction to Difference Equations; Springer Science & Business Media: New York, NY, USA, 2005. [Google Scholar]
- Mozyrska, D.; Wyrwas, M. The Z-transform method and delta type fractional difference operators. Discret. Dyn. Nat. Soc.
**2015**, 852734. [Google Scholar] [CrossRef][Green Version] - Brandibur, O.; Kaslik, E. Exact stability and instability regions for two-dimensional linear autonomous systems of fractional-order differential equations. arXiv
**2019**, arXiv:1910.07237. [Google Scholar] - Gorenflo, R.; Mainardi, F. Fractional calculus, integral and differential equations of fractional order. In Fractals and Fractional Calculus in Continuum Mechanics; Carpinteri, A., Mainardi, F., Eds.; CISM Courses and Lecture Notes; Springer: Wien, Austria, 1997; Volume 378, pp. 223–276. [Google Scholar]
- Li, Y.; Chen, Y.; Podlubny, I. Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica
**2009**, 45, 1965–1969. [Google Scholar] [CrossRef] - Wyrwas, M.; Mozyrska, D. On Mittag–Leffler Stability of Fractional Order Difference Systems. Adv. Model. Control Non-Integer-Order Systems. Lect. Notes Electr. Eng.
**2015**, 320, 209–220. [Google Scholar] - Kuznetsov, Y.A. Elements of Applied Bifurcation Theory; Springer Science & Business Media: New York, NY, USA, 2004; Volume 112. [Google Scholar]
- Mercik, S.; Weron, K. Stochastic origins of the long-range correlations of ionic current fluctuations in membrane channels. Phys. Rev. E
**2001**, 63, 051910. [Google Scholar] [CrossRef] [PubMed] - Magin, R.L. Fractional calculus models of complex dynamics in biological tissues. Comput. Math. Appl.
**2010**, 59, 1586–1593. [Google Scholar] [CrossRef][Green Version] - Diethelm, K.; Ford, N.; Freed, A. A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlinear Dyn.
**2002**, 29, 3–22. [Google Scholar] [CrossRef] - Teka, W.; Marinov, T.M.; Santamaria, F. Neuronal spike timing adaptation described with a fractional leaky integrate-and-fire model. PLoS Comput. Biol.
**2014**, 10, e1003526. [Google Scholar] [CrossRef] - Teka, W.; Stockton, D.; Santamaria, F. Power-Law Dynamics of Membrane Conductances Increase Spiking Diversity in a Hodgkin-Huxley Model. PLoS Comput. Biol.
**2016**, 12, e1004776. [Google Scholar] [CrossRef] [PubMed] - Teka, W.W.; Upadhyay, R.K.; Mondal, A. Spiking and bursting patterns of fractional-order Izhikevich model. Commun. Nonlinear Sci. Numer. Simul.
**2018**, 56, 161–176. [Google Scholar] [CrossRef] - Mondal, A.; Sharma, S.K.; Upadhyay, R.K.; Mondal, A. Firing activities of a fractional-order FitzHugh-Rinzel bursting neuron model and its coupled dynamics. Sci. Rep.
**2019**, 9, 15721. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**Stability regions $S(\delta ,{q}_{1},{q}_{2},h)$ in the $({a}_{11},{a}_{22})$-plane, for several values of the discretization step h. The boundary of the stability region corresponding to the continuous-time counterpart [8] is represented by green curve.

**Figure 2.**Superposition of stability regions displayed in the $({q}_{1},{q}_{2})$-plane for equilibrium states $({v}^{\ast},{w}^{\ast})$ of system (12) (with parameter values: $r=0.08$, $d=0.8$), considering several values of the equilibrium membrane potential ${v}^{\ast}$ between 0 and $0.98$, taking the discretization steps $0.125$, $0.25$, $0.5$ and 1, respectively (the smallest region plotted in blue corresponds to the largest step size $h=1$).

**Figure 3.**Evolution of membrane potential of system (11), with parameter values $r=0.08$, $c=0.7$, $d=0.8$, and $I=1.25$ for different values of the fractional orders.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Brandibur, O.; Kaslik, E.; Mozyrska, D.; Wyrwas, M. Stability Results for Two-Dimensional Systems of Fractional-Order Difference Equations. *Mathematics* **2020**, *8*, 1751.
https://doi.org/10.3390/math8101751

**AMA Style**

Brandibur O, Kaslik E, Mozyrska D, Wyrwas M. Stability Results for Two-Dimensional Systems of Fractional-Order Difference Equations. *Mathematics*. 2020; 8(10):1751.
https://doi.org/10.3390/math8101751

**Chicago/Turabian Style**

Brandibur, Oana, Eva Kaslik, Dorota Mozyrska, and Małgorzata Wyrwas. 2020. "Stability Results for Two-Dimensional Systems of Fractional-Order Difference Equations" *Mathematics* 8, no. 10: 1751.
https://doi.org/10.3390/math8101751