Example 4.1.

Consider (3.10) with , , . From (1.3) and (1.7)–(1.9) it follows that , and for any fixed and such that equation (3.10) has two points of equilibrium

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).

Consider now the behavior of solutions of (3.10) with in the points , , , of the space of () (Figure 1). In the point with , both equilibrium points and are unstable. In Figure 5 two trajectories of solutions of (3.10) are shown with the initial conditions , , and , . In Figure 6 two trajectories of solutions of (3.10) with the initial conditions , , and , are shown in the point with , . One can see that the equilibrium point is stable and the equilibrium point is unstable. In the point with , the equilibrium point is unstable and the equilibrium point is stable. Two corresponding trajectories of solutions are shown in Figure 7 with the initial conditions , , and , . In the point with , both equilibrium points and are stable. Two corresponding trajectories of solutions are shown in Figure 8 with the initial conditions , , and , . As it was noted above in this case, corresponding and are negative: and .

Example 4.2.

Consider the difference equation

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

Equation (4.2) is a particular case of (2.1) with

Suppose firstly that and consider two cases: (1) , (2) . In the first case,

In the second case,

In both cases, Corollary 3.1 gives stability condition in the form or

with

Corollary 3.2 in both cases gives stability condition in the form or (4.6) with

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

In the case , Corollary 3.3 gives stability condition in the form

or

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).

In the case , , Corollary 3.3 gives stability condition in the form

or

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.

Note also that if goes to zero all obtained stability conditions are violated. Therefore, by conditions the equilibrium point is unstable.

Example 4.3.

Consider the equation

(its particular cases were considered in [18, 19, 35]). Equation (4.13) is a particular case of (2.1) with , , , . From (1.7)–(1.9) it follows that by condition it has two equilibrium points

For equilibrium point sufficient conditions (3.3) and (3.4), (3.5) give

From (3.11), (3.12) it follows that an equilibrium point of (4.13) is stable in probability if and only if

For example, for from (4.15) we obtain

From (4.16) it follows

Similar for from (4.15) we obtain

From (4.16) it follows

Put, for example, . Then (4.13) has two equilibrium points: , . From (4.15)-(4.16) it follows that the equilibrium point is unstable and the equilibrium point is stable in probability if and only if

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].

In Figure 13 one thousand trajectories of (4.13) are shown for , , , , , . In this case stability condition (4.21) holds () and therefore the equilibrium point is stable: all trajectories go to zero. Putting , we obtain that stability condition (4.21) does not hold (). Therefore, the equilibrium point is unstable: in Figure 14 one can see that 1000 trajectories by the initial condition , fill the whole space.

Example 4.4.

Consider the equation

that is a particular case of (3.10) with , , , , , . As it follows from (1.4), (1.7)–(1.9) by conditions , , (4.22) has two equilibrium points

From (3.11), (3.12) it follows that an equilibrium point of (4.22) is stable in probability if and only if

Substituting (4.23) into (4.24), we obtain stability conditions immediately in the terms of the parameters of considered equation (4.22): the equilibrium point is stable in probability if and only if

the equilibrium point is stable in probability if and only if

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 , .

Consider the behavior of the equilibrium points of (4.22) by stochastic perturbations with . In Figure 21 trajectories of solutions are shown for , (the point in Figure 16) with the initial conditions , and . One can see that the equilibrium point (red trajectories) is stable and the equilibrium point (green trajectories) is unstable. In Figure 22 trajectories of solutions are shown for , (the point in Figure 16) with the initial conditions , and , . In this case both equilibrium points (red trajectories) and (green trajectories) are stable.