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.