# Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations

- Beatrice Paternoster
^{1}and - Leonid Shaikhet
^{2}Email author

**2008**:718408

https://doi.org/10.1155/2008/718408

© B. Paternoster and L. Shaikhet. 2008

**Received: **6 December 2007

**Accepted: **9 May 2008

**Published: **5 June 2008

## Abstract

It is supposed that the fractional difference equation , has an equilibrium point and is exposed to additive stochastic perturbations type of that are directly proportional to the deviation of the system state from the equilibrium point . It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V. Kolmanovskii and L. Shaikhet general method of Lyapunov functionals construction can be successfully used for getting of sufficient conditions for stability in probability of equilibrium points of the considered stochastic fractional difference equation. Numerous graphical illustrations of stability regions and trajectories of solutions are plotted.

## 1. Introduction—Equilibrium Points

Recently, there is a very large interest in studying the behavior of solutions of nonlinear difference equations, in particular, fractional difference equations [1–38]. This interest really is so large that a necessity appears to get some generalized results.

is investigated. Here , , , , are known constants. Equation (1.1) generalizes a lot of different particular cases that are considered in [1–8, 16, 18, 19, 20, 22, 23, 24, 32, 35, 37].

then (1.1) has not equilibrium points.

Remark 1.1.

If and , then (1.1) has only one point of equilibrium: . If , then (1.1) has only one point of equilibrium: .

Remark 1.2.

Consider the case , . If , then (1.1) has only one point of equilibrium: . If , then each solution is an equilibrium point of (1.1).

## 2. Stochastic Perturbations, Centering, and Linearization—Definitions and Auxiliary Statements

Let be a probability space and let be a nondecreasing family of sub- -algebras of , that is, for , let be the expectation, let , , be a sequence of -adapted mutually independent random variables such that , .

Note that the equilibrium point of (1.1) is also the equilibrium point of (2.1).

It is clear that the stability of the trivial solution of (2.2) is equivalent to the stability of the equilibrium point of (2.1).

Two following definitions for stability are used below.

Definition 2.1.

The trivial solution of (2.2) is called stable in probability if for any and there exists such that the solution satisfies the condition for any initial function such that .

Definition 2.2.

The trivial solution of (2.3) is called mean square stable if for any there exists such that the solution satisfies the condition for any initial function such that . If, besides, , for any initial function , then the trivial solution of (2.3) is called asymptotically mean square stable.

The following method for stability investigation is used below. Conditions for asymptotic mean square stability of the trivial solution of constructed linear equation (2.3) were obtained via V. Kolmanovskii and L. Shaikhet general method of Lyapunov functionals construction [44–46]. Since the order of nonlinearity of (2.2) is more than 1, then obtained stability conditions at the same time are [47–49] conditions for stability in the probability of the trivial solution of nonlinear equation (2.2) and therefore for stability in probability of the equilibrium point of (2.1).

Lemma 2.3.

then the trivial solution of (2.3) is asymptotically mean square stable.

Lemma 2.4.

then the trivial solution of (2.3) is asymptotically mean square stable.

Consider also the necessary and sufficient condition for asymptotic mean square stability of the trivial solution of (2.3).

Lemma 2.5 ([46]).

Corollary 2.6.

If, in particular, , then condition (2.10) is the necessary and sufficient condition for asymptotic mean square stability of the trivial solution of (2.3) for .

Remark 2.7.

Remark 2.8.

As it follows from results of [47–49] the conditions of Lemmas 2.3, 2.4, 2.5 at the same time are conditions for stability in probability of the equilibrium point of (2.1).

## 3. Stability of Equilibrium Points

From conditions (2.4), (2.6) it follows that . Let us check if this condition can be true for each equilibrium point.

So, . It means that the condition holds only for one from the equilibrium points and . Namely, if , then ; if , then ; if , then . In particular, if , then via Remark 1.1 and (2.3) we have , . Therefore, if , if , if .

So, via Remark 2.7, we obtain that equilibrium points and can be stable concurrently only if corresponding and are negative concurrently.

As it follows from Remark 2.7 this point of equilibrium cannot be asymptotically stable.

Corollary 3.1.

Then the equilibrium point is stable in probability.

The proof follows from (2.3), Lemma 2.3, and Remark 2.8.

Corollary 3.2.

Then the equilibrium point is stable in probability.

Proof.

It means that the condition of Lemma 2.4 holds. Via Remark 2.8 the proof is completed.

Corollary 3.3.

The proof follows from (2.3), (2.10), (2.11).

## 4. Examples

Example 4.1.

In Figure 1, the region where the points of equilibrium are absent (white region), the region where both points of equilibrium and are there but unstable (yellow region), the region where the point of equilibrium is stable only (red region), the region where the point of equilibrium is stable only (green region), and the region where both points of equilibrium and are stable (cyan region) are shown in the space of ( ). All regions are obtained via condition (3.11) for . In Figures 2, 3 one can see similar regions for and , accordingly, that were obtained via conditions (3.11), (3.12). In Figure 4 it is shown that sufficient conditions (3.3) and (3.4), (3.5) are enough close to necessary and sufficient conditions (3.11), (3.12): inside of the region where the point of equilibrium is stable (red region) one can see the regions of stability of the point of equilibrium that were obtained by condition (3.3) (grey and green regions) and by conditions (3.4), (3.5) (cyan and green regions). Stability regions obtained via both sufficient conditions of stability (3.3) and (3.4), (3.5) give together almost whole stability region obtained via necessary and sufficient stability conditions (3.11), (3.12).

Example 4.2.

Different particular cases of this equation were considered in [2–5, 16, 22, 23, 37].

Since then condition (4.6), (4.7) is better than (4.6), (4.8).

In particular, from (4.10) it follows that for , (this case was considered in [3, 23]) the equilibrium point is stable if and only if . Note that in [3] for this case the condition only is obtained.

In Figure 9 four trajectories of solutions of (4.2) in the case , , , are shown: (1) , , , (red line, stable solution); (2) , , , (brown line, unstable solution); (3) , , , (blue line, unstable solution); (4) , , , (green line, stable solution).

For example, from (4.12) it follows that for , (this case was considered in [22, 37]), the equilibrium point is stable if and only if . In Figure 10 four trajectories of solutions of (4.2) in the case , , , are shown: (1) , , , (red line, stable solution); (2) , , , (brown line, unstable solution); (3) , , , (blue line, unstable solution); (4) , , , (green line, stable solution).

Via simulation of a sequence of mutually independent random variables consider the behavior of the equilibrium point by stochastic perturbations. In Figure 11 one thousand trajectories are shown for , , , , . In this case, stability condition (4.12) holds ( ) and therefore the equilibrium point is stable: all trajectories go to . Putting , we obtain that stability condition (4.12) does not hold ( ). Therefore, the equilibrium point is unstable: in Figure 12 one can see that 1000 trajectories fill the whole space.

Example 4.3.

Note that for particular case , , , in [35] it is shown that the equilibrium point is locally asymptotically stable if ; and for particular case , , , in [18] it is shown that the equilibrium point is locally asymptotically stable if . It is easy to see that both these conditions follow from (4.21).

Similar results can be obtained for the equation that was considered in [1].

Example 4.4.

Note that in [24] equation (4.18) was considered with and positive , . There it was shown that equilibrium point is locally asymptotically stable if and only if that is a part of conditions (4.25).

In Figure 15 the region where the points of equilibrium are absent (white region), the region where the both points of equilibrium and are there but unstable (yellow region), the region where the point of equilibrium is stable only (red region), the region where the point of equilibrium is stable only (green region) and the region where the both points of equilibrium and are stable (cyan region) are shown in the space of ( , ). All regions are obtained via conditions (4.25), (4.26) for . In Figures 16 similar regions are shown for .

Consider the point (Figure 15) with , . In this point both equilibrium points and are unstable. In Figure 17 two trajectories of solutions of (4.22) are shown with the initial conditions , and , . In Figure 18 two trajectories of solutions of (4.22) with the initial conditions , and , are shown in the point (Figure 15) with . One can see that the equilibrium point is stable and the equilibrium point is unstable. In the point (Figure 15) with , the equilibrium point is unstable and the equilibrium point is stable. Two corresponding trajectories of solutions are shown in Figure 19 with the initial conditions and , . In the point (Figure 15) with , both equilibrium points and are stable. Two corresponding trajectories of solutions are shown in Figure 20 with the initial conditions , and , .

## Authors’ Affiliations

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