Об устойчивости решений систем разностных уравнений в одном критическом случае
Рассматривается система нелинейных разностных уравнений, допускающая нулевое решение. С помощью метода функций Ляпунова изучается его устойчивость. Наряду с полной системой уравнений рассматривается линеаризованная система разностных уравнений. Известно, что если все корни характеристического уравне...
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Інститут прикладної математики і механіки НАН України
2008
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Об устойчивости решений систем разностных уравнений в одном критическом случае / А.О. Игнатьев // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 488-506. — Бібліогр.: 33 назв. — рос. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859621187727917056 |
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| author | Игнатьев, А.О. |
| author_facet | Игнатьев, А.О. |
| citation_txt | Об устойчивости решений систем разностных уравнений в одном критическом случае / А.О. Игнатьев // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 488-506. — Бібліогр.: 33 назв. — рос. |
| collection | DSpace DC |
| description | Рассматривается система нелинейных разностных уравнений, допускающая нулевое решение. С помощью метода функций Ляпунова изучается его устойчивость. Наряду с полной системой уравнений рассматривается линеаризованная система разностных уравнений. Известно, что если все корни характеристического уравнения линеаризованной системы по модулю меньше единицы, то нулевое решение полной системы асимптотически устойчиво. Если хотя бы один из корней характеристического уравнения по модулю больше единицы, то нулевое решение полной системы неустойчиво. В случае, когда часть корней характеристического уравнения по модулю меньше единицы, а часть равна единице, задача устойчивости не решается рассмотрением лишь линейных членов, и для ее решения нужно привлечь нелинейные слагаемые. Такой случай называется критическим. В настоящей работе рассмотрен критический случай одного корня, равного единице, когда задача устойчивости решается членами до третьего порядка малости в разложении правых частей исходных уравнений в ряды Маклорена.
A system of nonlinear difference equations which admits the zero solution is considered. Its stability is studied by means of Lyapunov’s direct method. Side by side with this system, a linearized system of difference equations is also considered. It is well known that if all roots of the characteristic equation of a linearized system lie within the unit circle on the complex plane, then the zero solution of the original full system is asymptotically stable. If at least one eigenvalue lies outside the unit disk, then the zero solution of the original system is unstable. In the case where the moduli of some eigenvalues are equal to unity, and the moduli of others are less than unity, the stability problem cannot be solved by considering only the linear terms. To solve this problem, it is necessary to use the terms of higher orders in expansions of the righthand sides of the original system of difference equations in Maclaurin series. Such case is called a critical one. In this paper, we consider the critical case where one eigenvalue is equal to unity, and the stability problem can be solved by involving terms up to the third order in expansions of the right-hand sides of the initial equations in Maclaurin series.
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| first_indexed | 2025-11-29T03:24:09Z |
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| fulltext |
Ukrainian Mathematical Bulletin
Volume 5 (2008), № 4, 479 – 497 UMB
On the stability of solutions of systems of
difference equations in a critical case
Aleksandr O. Ignatyev
(Presented by A. E. Shishkov)
Abstract. A system of nonlinear difference equations which admits
the zero solution is considered. Its stability is studied by means of Lya-
punov’s direct method. Side by side with this system, a linearized system
of difference equations is also considered. It is well known that if all roots
of the characteristic equation of a linearized system lie within the unit
circle on the complex plane, then the zero solution of the original full
system is asymptotically stable. If at least one eigenvalue lies outside
the unit disk, then the zero solution of the original system is unstable.
In the case where the moduli of some eigenvalues are equal to unity, and
the moduli of others are less than unity, the stability problem cannot be
solved by considering only the linear terms. To solve this problem, it is
necessary to use the terms of higher orders in expansions of the right-
hand sides of the original system of difference equations in Maclaurin
series. Such case is called a critical one. In this paper, we consider
the critical case where one eigenvalue is equal to unity, and the stabil-
ity problem can be solved by involving terms up to the third order in
expansions of the right-hand sides of the initial equations in Maclaurin
series.
2000 MSC. 39A11, 34K20.
Key words and phrases. Difference equations, Lyapunov function,
stability.
1. Introduction and basic definitions
The theory of discrete dynamical systems has grown tremendously in
the last decade. Difference equations can arise in a number of ways. They
may be the natural model of a discrete process (in combinatorics, for ex-
ample), or they can be a discrete approximation of a continuous process.
The development of the theory of difference systems has been strongly
promoted by advanced technologies in the scientific computation and by a
Received 25.07.2008
ISSN 1812 – 3309. c© Institute of Mathematics of NAS of Ukraine
480 On the stability of solutions...
large number of applications to models in biology, engineering, and other
sciences. For example, in papers [5,6,8,9,14], systems of difference equa-
tions were used as natural models of the dynamics of populations; in [10],
the difference equations were applied to a simulation in genetics; in [25],
the dynamics of an ecological system was also described by a system of
difference equations. The method of construction of difference schemes
for systems of differential equations is proposed in [2]. This method pro-
vides the consistency between differential and difference equations in the
sense of the stability of the zero solution (we note that, as in the case of
ordinary differential equations, the stability problem for any solution of
a difference equation is reduced to that of the zero solution).
Many evolution processes are characterized by the fact that, at certain
time moments, they experience abruptly a change of the state. These
processes are subjected to short-term perturbations, whose duration is
negligible in comparison with that of the process. Consequently, it is
natural to assume that these perturbations act instantaneously, that is,
in the form of impulses. Papers [31, 32] were the first articles in this
direction. The early works on differential equations with impulse effect
were summarized in monograph [33] in which the foundations of this
theory were described. In recent years, the study of impulsive systems has
attracted the increasing interest [16–18, 20–22, 28]. An impulsive system
consists of a continuous system which is governed by ordinary differential
equations and a discrete system which is governed by difference equations.
So the dynamics of impulsive systems essentially depends on properties
of the corresponding difference systems, and this confirms the importance
of studying the qualitative properties of difference systems.
The stability of a discrete process is the ability of the process to
resist the a priori unknown small influences. A process is said to be
stable if such disturbances do not change it. This property turns out to
be of utmost importance since an individual predictable process can be
physically realized, in general, only if it is stable in the corresponding
natural sense. One of the most powerful tools used in stability theory
is Lyapunov’s direct method. This method consists in the use of an
auxiliary function (the Lyapunov function).
Consider the system of difference equations
x(n + 1) = f(n, x(n)), f(n, 0) = 0, (1.1)
where n = 0, 1, 2, . . . is the discrete time, x(n) = (x1(n), . . . , xk(n))T ∈
A. O. Ignatyev 481
R
k, and f = (f1, . . . , fk)
T ∈ R
k. System (1.1) admits the trivial solution
x(n) = 0. (1.2)
By x(n, n0, x
0), we denote the solution of system (1.1) coinciding with
x0 = (x0
1, x
0
2, . . . , x0
k)
T for n = n0. Let Z+ be the set of nonnegative real
integers, and let Nn0
be the set of nonnegative real integers satisfying the
inequality n ≥ n0, Br = {x ∈ R
k : ‖x‖ ≤ r}.
By analogy with ordinary differential equations, we introduce the fol-
lowing definitions.
Definition 1.1. The trivial solution of system (1.1) is said to be stable
if, for any ε > 0, n0 ∈ Z+, there exists δ = δ(ε, n0) > 0 such that
‖x0‖ < δ implies ‖x(n, n0, x
0)‖ < ε for n > n0. Otherwise, the trivial
solution of system (1.1) is called unstable. If δ in this definition can be
chosen to be independent of n0 (i.e. δ = δ(ε)), then the zero solution of
system (1.1) is said to be uniformly stable.
Definition 1.2. Solution (1.2) of system (1.1) is said to be attract-
ing if, for any n0 ∈ Z+, there exists an η = η(n0) > 0 such that, for
any ε > 0 and x0 ∈ Bη, there exists a σ = σ(ε, n0, x
0) > 0 such that
‖x(n, n0, x
0)‖ < ε for all n ≥ n0 + σ.
In other words, solution (1.2) of system (1.1) is called attracting if
lim
n→∞
‖x(n, n0, x
0)‖ = 0. (1.3)
Definition 1.3. The trivial solution of system (1.1) is said to be uni-
formly attracting if, for some η > 0 and for each ε > 0, there exists a
σ = σ(ε) ∈ N such that ‖x(n, n0, x
0)‖ < ε for all n0 ∈ Z+, x0 ∈ Bη, and
n ∈ Nn0+σ.
In other words, solution (1.2) of system (1.1) is called uniformly at-
tracting if (1.3) holds uniformly in n0 ∈ Z+, x0 ∈ Bη.
Definition 1.4. The zero solution of system (1.1) is called
• asymptotically stable if it is both stable and attracting;
• uniformly asymptotically stable if it is both uniformly stable and
uniformly attracting.
482 On the stability of solutions...
Definition 1.5. The trivial solution of system (1.1) is said to be
exponentially stable if there exist M > 0 and η ∈ (0, 1) such that
‖x(n, n0, x
0)‖ < M‖x0‖ηn−n0 for n ∈ Nn0
.
A great number of papers is devoted to the investigation of the sta-
bility of solution (1.2) of system (1.1). The general theory of differ-
ence systems and the foundations of the stability theory are presented in
monographs [1, 7, 11, 13, 15, 19, 29, 30]. In paper [23], it is shown that if
system (1.1) is autonomous (i.e. f does not depend on n explicitly) or
periodic (i.e. there exists ω ∈ N such that f(n, x) ≡ f(n + ω, x)), then
the stability of solution (1.2) implies its uniform stability, and the asymp-
totic stability implies its uniform asymptotic stability. The asymptotic
stability of perturbed linear difference systems with periodic coefficients
was studied in [3]. Papers [4, 12, 24, 26, 27] considered the stability of
solutions of periodic and almost periodic systems.
Let us formulate the main theorems of Lyapunov’s direct method
about the stability of the zero solution of the system of autonomous
difference equations
x(n + 1) = f(x(n)). (1.4)
These statements have been mentioned in [13, Theorems 4.20 and 4.27].
They are related to the existence of an auxiliary function V (x); and the
analog of its derivative is the variation of V relative to (1.4) which is
defined as ∆V (x) = V (f(x)) − V (x).
Theorem A. If there exists a positive definite continuous function V (x)
such that ∆V (x) relative to (1.4) is a negative semidefinite function or
identically equals zero, then the trivial solution of system (1.4) is stable.
Theorem B. If there exists a positive definite continuous function V (x)
such that ∆V (x) relative to (1.4) is negative definite, then the trivial
solution of system (1.4) is asymptotically stable.
Theorem C. If there exists a continuous function V (x) such that ∆V (x)
relative to (1.4) is negative definite, and the function V is not positive
semidefinite, then the trivial solution of system (1.4) is unstable.
Consider the autonomous system
x(n + 1) = Ax(n) + X(x(n)), (1.5)
where A is a k × k nonsingular matrix, and X is a function such that
A. O. Ignatyev 483
lim
‖x‖→0
‖X(x)‖
‖x‖
= 0. (1.6)
According to [13, p. 175], we denote ρ(A) = max1≤i≤k |λi|, where λi (i =
1, . . . , k) are the roots of the characteristic equation
det(A − λIk) = 0. (1.7)
Here, Ik is the unit k × k matrix. In [1, Corollary 5.6.3 and Theorem
5.6.4], the following theorem was proved.
Theorem 1.1. If ρ(A) < 1, then the zero solution of system (1.5) is
asymptotically stable (moreover, the exponential stability holds in this
case). If ρ(A) > 1, then the zero solution of system (1.5) is unstable. If
ρ(A) ≤ 1 and the moduli of some eigenvalues of A are equal to unity,
then a function X(x) in system (1.5) can be chosen to be such that the
zero solution of system (1.5) is either stable or unstable.
Thus, it follows from Theorem 1.1 that the problem of stability of the
zero solution of system (1.5) can be solved by means of the system of the
linear approximation
x(n + 1) = Ax(n) (1.8)
(when ρ(A) < 1 or ρ(A) > 1). In the case ρ(A) = 1, we have a critical
case where the solution of the stability problem requires to use the terms
of higher degrees.
For studying the stability of the zero solution of system (1.5), Elaydi
[13] proposed to employ Lyapunov functions as a quadratic form
V (x) =
∑
i1+i2+···+ik=2,
ij≥0 (j=1,...,k)
bi1,i2,...,ikxi1
1 xi2
2 . . . x
ik
k , (1.9)
where bi1,i2,...,ik are constants. He formulated the following statement [13,
Corollary 4.31] without any proof.
Proposition 1.1. If ρ(A) > 1, then there exist a quadratic form V (x)
which is not positive semidefinite and a negative definite quadratic form
W (x) such that
W (x) = V (Ax) − V (x).
We will show here that Proposition 1.1 is not true and study the
stability problem in the critical case where one eigenvalue of the matrix
A is equal to unity. First, we show that Proposition 1.1 is false. To do
484 On the stability of solutions...
this, let us consider the system
x(n + 1) = Ax(n),
where x = ( x1
x2
) ∈ R
2, A = ( 1 0
0 2 ). Numbers 1 and 2 are the roots of its
characteristic equation, ρ(A) = 2 > 1. But, for any quadratic form
V (x) = b2,0x
2
1 + b1,1x1x2 + b0,2x
2
2,
we have
W (x) = V (Ax) − V (x) = b1,1x1x2 + 3b0,2x
2
2. (1.10)
The quadratic form (1.10) can be neither positive definite nor negative
definite. This example shows that Proposition 1.1 is not true.
2. Critical case where one eigenvalue is equal to unity
In this section, we consider the critical case where one root of the char-
acteristic equation (1.7) is equal to unity, i.e. we assume that Eq. (1.7)
has one root λ1 = 1, and other roots satisfy the conditions |λi| < 1 (i =
2, 3, . . . , k). The function X = (X1, . . . , Xk)
T is supposed to be holomor-
phic, and its expansion into a Maclaurin series begins with terms of the
second order of smallness. So, system (1.5) has the form
xj(n + 1) = aj1x1(n) + aj2x2(n) + · · · + ajkxk(n)
+ Xj(x1(n), . . . , xk(n)) (j = 1, . . . , k). (2.1)
Henceforth, we consider the critical case where the characteristic equation
of the system in the first approximation
xj(n + 1) = aj1x1(n) + aj2x2(n) + · · · + ajkxk(n) (j = 1, . . . , k) (2.2)
has one root equal to unity, and other k − 1 roots have moduli which are
less than unity.
In system (2.2), we introduce the variable y instead of one of the
variables xj by means of the substitution
y = β1x1 + β2x2 + · · · + βkxk, (2.3)
where βj (j = 1, . . . , k) are some constants which we choose to be such
that
y(n + 1) = y(n). (2.4)
Relations (2.3) and (2.4) yield
A. O. Ignatyev 485
y(n + 1) = β1x1(n + 1) + β2x2(n + 1) + · · · + βkxk(n + 1)
= β1[a11x1(n) + a12x2(n) + · · · + a1kxk(n)]
+ β2[a21x1(n) + a22x2(n) + · · · + a2kxk(n)]
+ · · · + βk[ak1x1(n) + ak2x2(n) + · · · + akkxk(n)]
= β1x1(n) + β2x2(n) + · · · + βkxk(n).
Equating the coefficients corresponding to xj(n) (j = 1, 2, . . . , k), we
obtain the system of linear homogeneous algebraic equations for βj (j =
1, . . . , k),
a1jβ1 + a2jβ2 + · · · + akjβk = βj , (2.5)
or, in the matrix form,
(AT − Ik)β = 0,
where β = (β1, . . . , βk)
T . Since the equation det(AT − λIk) = 0 has the
root λ = 1, the determinant of system (2.5) is equal to zero. Therefore,
this system has a solution in which not all constants are equal to zero.
To be definite, we assume that βk 6= 0. Then we can use the variable
y instead of the variable xk. Other variables xj (j = 1, . . . , k − 1) are
preserved without any change. Denoting
cji = aji −
βi
βk
ajk, cj =
ajk
βk
(i, j = 1, 2, . . . , k − 1),
we transform Eqs. (2.2) to the form
xj(n + 1) = cj1x1(n) + cj2x2(n) + · · · + cj,k−1xk−1(n) + cjy(n)
(j = 1, . . . , k − 1), (2.6)
y(n + 1) = y(n), (2.7)
where cji and cj are constants.
The characteristic equation of system (2.6), (2.7) is reduced to two
equations:
λ − 1 = 0
and
det(C − λIk−1) = 0, (2.8)
where C = (cij)
k−1
i,j=1. Since the characteristic equation is invariant with
486 On the stability of solutions...
respect to linear transformations and, in this case, has k − 1 roots whose
moduli are less than unity, Eq. (2.8) has k − 1 roots, and their moduli
are less than unity. We denote
xj = yj + ljy (j = 1, . . . , k − 1), (2.9)
where lj (j = 1, . . . , k − 1) are constants which we choose to be such
that the right-hand sides of system (2.6) do not contain y(n). In these
designations with regard for (2.7), system (2.6) takes the form:
yj(n + 1) = cj1y1(n) + cj2y2(n) + · · · + cj,k−1yk−1(n)
+ [cj1l1 + cj2l2 + · · · + (cjj − 1)lj + · · · + cj,k−1lk−1 + cj ]y(n)
(j = 1, . . . , k − 1).
We choose the constants lj to be such that
cj1l1 + cj2l2 + · · ·+(cjj −1)lj + · · ·+ cj,k−1lk−1 = −cj (j = 1, . . . , k−1).
(2.10)
Unity is not a root of the characteristic equation (2.8); hence the determi-
nant of system (2.10) is not equal to zero. Therefore, this system has the
unique solution (l1, . . . , lk−1). As a result of change (2.9), system (2.6),
(2.7) transforms to the form
yj(n + 1) = cj1y1(n) + cj2y2(n) + · · · + cj,k−1yk−1(n)
(j = 1, . . . , k − 1),
y(n + 1) = y(n),
and nonlinear system (2.1) takes the form
yj(n + 1) = cj1y1(n) + cj2y2(n) + · · · + cj,k−1yk−1(n)
+ Yj(y1(n), . . . , yk−1(n), y(n)) (j = 1, . . . , k − 1),
y(n + 1) = y(n) + Y (y1(n), . . . , yk−1(n), y(n)),
(2.11)
where Yj (j = 1, . . . , k − 1) and Y are holomorphic functions of y1, . . . ,
yk−1, y. Their expansions in power series lack constant and first-degree
terms:
Yj(y1, y2, . . . , yk−1, y) =
∞
∑
i1+i2+···+ik−1+ik=2
v
(j)
i1,i2,...,ik−1,ik
yi1
1 yi2
2 . . . y
ik−1
k−1 yik
(j = 1, . . . , k − 1),
Y (y1, y2, . . . , yk−1, y) =
∞
∑
i1+i2+···+ik−1+ik=2
vi1,i2,...,ik−1,ikyi1
1 yi2
2 . . . y
ik−1
k−1 yik .
A. O. Ignatyev 487
By virtue of (2.9), it is clear that the stability problem for the trivial
solution of system (2.1) is equivalent to that for the zero solution of
system (2.11). Further, the form of (2.11) will be basic for studying the
stability of the zero solution in the case where this problem can be solved
with the use of terms of the first and second orders.
Theorem 2.1. If the function Y is such that the coefficient v0,0,...,0,2 is
not equal to zero, then the solution
y1 = 0, y2 = 0, . . . , yk−1 = 0, y = 0
of system (2.11) is unstable.
Proof. Let
V1(y1, . . . , yk−1) =
∑
s1+s2+···+sk−1=2
Bs1,s2,...,sk−1
ys1
1 ys2
2 . . . y
sk−1
k−1
be a quadratic form such that
V1(c11y1 + · · · + c1,k−1yk−1, . . . , ck−1,1y1 + · · · + ck−1,k−1yk−1)
− V1(y1, . . . , yk−1) = y2
1 + y2
2 + · · · + y2
k−1. (2.12)
Since the moduli of all eigenvalues of the matrix C = (cij)
k−1
I,j=1 are less
than unity, such quadratic form is unique according to [13, Theorem 4.30]
and negative definite. Consider the Lyapunov function
V (y1, . . . , yk−1, y) = V1(y1, . . . , yk−1) + αy, (2.13)
where α = const. Let us find ∆V :
∆V
∣
∣
∣
∣
(2.11)
=
∑
s1+···+sk−1=2
Bs1,...,sk−1
{[c11y1 + · · ·+c1,k−1yk−1 +Y1(y1, . . . , yk−1, y)]s1
· · · × [ck−1,1y1 + · · · + ck−1,k−1yk−1 + Yk−1(y1, . . . , yk−1, y)]sk−1
− ys1
1 . . . y
sk−1
k−1 } + αY (y1, . . . , yk−1, y).
Taking (2.12) into account, ∆V can be written in the form
∆V
∣
∣
∣
∣
(2.11)
= W (y1, . . . , yk−1, y) + W∗(y1, . . . , yk−1, y),
where
488 On the stability of solutions...
W = (y2
1 + y2
2 + · · · + y2
k−1) + αv0,0,...,0,2y
2
+ α(v2,0,...,0y
2
1 + v1,1,...,0y1y2 + · · · + v1,0,...,1,0y1yk−1
+ v1,0,...,0,1y1y + v0,2,...,0y
2
2 + · · · + v0,0,...,1,1yk−1y),
and W∗ is a holomorphic function whose Maclaurin series begins with
terms of the third power in y1, . . . , yk−1, y. We choose the sign of α such
that αv0,...,0,2 > 0. We now show that α can be chosen so small that the
quadratic form W is positive definite. To do this, we show that α can be
chosen so that the principal minors of the matrix
1 + αv2,0,...,0
1
2αv1,1,...,0 . . . 1
2αv1,0,...,1,0
1
2αv1,0,...,0,1
1
2αv1,1,...,0 1 + αv0,2,...,0 . . . 1
2αv0,1,...,1,0
1
2αv0,1,...,0,1
1
2αv1,0,1,...,0
1
2αv0,1,1,...,0 . . . 1
2αv0,0,1,...,1,0
1
2αv0,0,1,...,0,1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2αv1,0,...,1,0
1
2αv0,1,...,1,0 . . . 1 + αv0,...,0,2,0
1
2αv0,...,0,1,1
1
2αv1,0,...,0,1
1
2αv0,1,...,0,1 . . . 1
2αv0,0,...,1,1
1
2αv0,0,...,0,2
are positive. In fact, any principal minor ∆s of this matrix is a continuous
function of α: ∆s = ∆s(α). Note that ∆s(0) = 1 for s = 1, 2, . . . , k − 1.
Thus, there exists α∗ > 0 such that, for |α| < α∗, we have ∆s(α) ≥
1
2 (s = 1, 2, . . . , k−1). We now prove that the inequality ∆k > 0 holds for
sufficiently small |α|. To do this, we expand ∆k in terms of the elements
of the last row. We obtain ∆k = 1
2αv0,0,...,0,2∆k−1 +α2∆∗, where ∆∗ is a
polynomial in α and vi1,i2,...,ik (i1 + i2 + · · ·+ ik = 2, ij ≥ 0). Hence, we
have ∆k > 0 for sufficiently small |α|. That is, the quadratic form W is
positive definite for α, whose absolute value is small enough and whose
sign coincides with the sign of v0,0,...,2. Therefore, the sum W +W∗ is also
positive definite in a sufficiently small neighborhood of the origin. At the
same time, the function V of form (2.13) is alternating. Hence, the zero
solution of system (2.11) is unstable. The proof is completed.
Thus, in the case where v0,0,...,2 6= 0, the stability problem has been
solved independently of the terms, whose degrees are higher than two.
Consider now the case where v0,0,...,2 = 0. We transform system (2.11) to
the form where v
(j)
0,0,...,2 = 0 (j = 1, 2, . . . , k − 1). We denote
yj = ξj + mjy
2 (j = 1, 2, . . . , k − 1), (2.14)
where mj are constants. In these designations, system (2.11) takes the
A. O. Ignatyev 489
form
ξj(n + 1) = cj1ξ1(n) + cj2ξ2(n) + · · · + cj,k−1ξk−1(n)
+ y2(n)(cj1m1 + cj2m2 + · · · + cj,k−1mk−1)
+ Yj(ξ1(n) + m1y
2(n), . . . , ξk−1(n) + mk−1y
2(n), y(n))
− mj [y
2(n) + 2y(n)Y (ξ1(n) + m1y
2(n), . . . , ξk−1(n) + mk−1y
2(n), y(n))
+ Y 2(ξ1(n) + m1y
2(n), . . . , ξk−1(n) + mk−1y
2(n), y(n))], (2.15)
y(n + 1) = y(n) + Y (ξ1(n) + m1y
2(n), . . . , ξk−1(n) + mk−1y
2(n), y(n)).
(2.16)
We choose constants m1, . . . , mk−1 to be such that the coefficients cor-
responding to y2(n) on the right-hand sides of system (2.15) are equal
to zero. Equating the corresponding coefficients to zero, we obtain the
system of linear algebraic equations for m1, . . . , mk−1:
cj1m1 + cj2m2 + · · ·+ cj,k−1mk−1 = mj −v
(j)
0,0,...,2 (j = 1, 2, . . . , k−1).
This system has a unique solution, because unity is not an eigenvalue of
the matrix C. Substituting the obtained values m1, . . . , mk−1 to (2.15)
and (2.16), we get the system
ξj(n + 1) = cj1ξ1(n) + cj2ξ2 + · · · + cj,k−1ξk−1(n)
+ Ξj(ξ1(n), . . . , ξk−1(n), y(n))
(j = 1, . . . , k − 1), (2.17)
y(n + 1) = y(n) + Y∗(ξ1(n), . . . , ξk−1(n), y(n)), (2.18)
where
Ξj(ξ1, . . . , ξk−1, y) = Yj(ξ1 + m1y
2, . . . , ξk−1 + mk−1y
2, y)
− 2mjyY (ξ1 + m1y
2, . . . , ξk−1 + mk−1y
2, y)
− mjY
2(ξ1 + m1y
2, . . . , ξk−1 + mk−1y
2, y) − v
(j)
0,0,...,2y
2,
Y∗(ξ1, . . . , ξk−1, y) = Y (ξ1 + m1y
2, . . . , ξk−1 + mk−1y
2, y).
Expansions of Ξj and Y∗ in power series begin with terms of the
second degree, and the coefficients corresponding to y2 in expansions of
Ξj and Y∗ are equal to zero. System (2.17) and (2.18) will be basic in
our further investigation of the stability of the zero solution
490 On the stability of solutions...
ξ1 = 0, ξ2 = 0, . . . , ξk−1 = 0, y = 0. (2.19)
By Ξ
(0)
j (y) (j = 1, . . . , k − 1) and Y
(0)
∗ (y), we denote, respectively,
the sum of all terms in the functions Ξj and Y∗ which do not include
ξ1, . . . , ξk−1, so that
Ξ
(0)
j (y) = Ξj(0, . . . , 0, y) = hjy
3 +
∞
∑
s=4
h
(s)
j ys,
Y
(0)
∗ (y) = Y∗(0, . . . , 0, y) = hy3 +
∞
∑
s=4
h(s)ys,
where h, hj , h
(s), h
(s)
j (j = 1, . . . , k − 1; s = 4, 5, . . . ) are constants.
Theorem 2.2. Solution (2.19) of system (2.17), (2.18) is asymptotically
stable for h < 0 and unstable for h > 0.
Proof. We now show that there exists a Lyapunov function V such that
it depends on ξ1, . . . , ξk−1, y, and ∆V is positive definite. Consider the
system of linear equations
ξj(n + 1) = cj1ξ(n) + cj2ξ2(n) + · · · + cj,k−1ξk−1(n) (j = 1, . . . , k − 1).
(2.20)
Let W =
∑
i1+···+ik−1=2 wi1,...,ik−1
ξi1
1 . . . ξ
ik−1
k−1 be a quadratic form of the
variables ξ1, . . . , ξk−1 and such that
∆W
∣
∣
∣
∣
(2.20)
= ξ2
1 + · · · + ξ2
k−1. (2.21)
Since all eigenvalues of the matrix C are inside of the unit disk, the
form W satisfying (2.21) exists and is unique and negative definite [13,
Theorem 4.30].
If the functions Ξj (j = 1, . . . , k − 1) do not depend on y, then a
variation ∆W of the function W along system (2.17), i.e. the expression
∑
i1+···+ik−1=2
wi1,··· ,ik−1
{[c11ξ1 + c12ξ2 + · · · + c1,k−1ξk−1 + Ξ1]
i1
× · · · [ck−1,1ξ1 + · · · + ck−1,k−1ξk−1 + Ξk−1]
ik−1 − ξi1
1 · · · ξ
ik−1
k−1 }, (2.22)
is a positive definite function of the variables ξ1, . . . , ξk−1 for sufficiently
small ξ1, . . . , ξk−1.
A. O. Ignatyev 491
On the other hand, if the function Y∗ does not depend on ξ1, . . . , ξk−1
(i.e. if Y∗ = Y
(0)
∗ ), then the variation of 1
2hy2 is equal to
∆
(1
2
hy2
)
=
1
2
h
[
2yY
(0)
∗ + Y
(0)
∗
2]
= h2y4 + hh(4)y5 + o(y5), (2.23)
and this variation is a positive definite function of y for sufficiently small
|y|. Therefore, under these conditions, the variation of the function V1 =
1
2hy2 +W (ξ1, . . . , ξk−1) along the total system (2.17), (2.18) is a positive
definite function of all variables ξ1, . . . , ξk−1, y in some neighborhood of
the origin. In view of (2.21) and (2.23), this variation can be represented
in the form
(h2 + g1)y
4 + ξ2
1 + · · · + ξ2
k−1 +
k−1
∑
i,j=1
g
(1)
ij ξiξj , (2.24)
where g1 is a holomorphic function of the variable y, vanishing for y = 0,
and g
(1)
ij are holomorphic functions of the variables ξ1, . . . , ξk−1, vanishing
for ξ1 = · · · = ξk−1 = 0. But since the functions Ξj (j = 1, . . . , k −
1) include y, and the function Y∗ includes ξ1, . . . , ξk−1, the variation of
the function V1 along system (2.17), (2.18) is not, in general, positive
definite. In this variation, there appear the terms breaking the positive
definiteness.
Note that expression (2.24) remains positive definite if the function
g1 contains not only the variable y, but also the variables ξ1, . . . , ξk−1,
and the functions g
(1)
ij contain not only variables ξ1, . . . , ξk−1, but also
the variable y. It is only important that the functions g1 and g
(1)
ij vanish
for ξ1 = · · · = ξk−1 = y = 0. Taking into account this fact, we write
the second variation of the function V1 along system (2.17), (2.18) in the
form
∆V1 = ∆
(1
2
hy2
)
+ ∆W = hyY∗ +
1
2
hY 2
∗
+
∑
i1+···+ik−1=2
wi1,...,ik−1
{[c11ξ1 + c12ξ2 + · · · + c1,k−1ξk−1 + Ξ1]
i1
× · · · [ck−1,1ξ1 + · · · + ck−1,k−1ξk−1 + Ξk−1]
ik−1 − ξi1
1 · · · ξ
ik−1
k−1 }
= [h2 + g1(ξ1, . . . , ξk−1, y)]y4 + ξ2
1 + · · · + ξ2
k−1
+
k−1
∑
i,j=1
g
(1)
ij (ξ1, . . . , ξk−1, y)ξiξj + Q(ξ1, . . . , ξk−1, y), (2.25)
492 On the stability of solutions...
where the functions g1 and g
(1)
ij (i, j = 1, . . . , k− 1) vanish for ξ1 = · · · =
ξk−1 = y = 0, and Q is the sum of all terms which can be included
neither to the expression
g1(ξ1, . . . , ξk−1, y)y4 (2.26)
nor to the expression
k−1
∑
i,j=1
g
(1)
ij (ξ1, . . . , ξk−1, y)ξiξj . (2.27)
All terms which are included into Q can be divided into four follow-
ing groups: the terms free of ξ1, . . . , ξk−1, the terms linear with respect
to ξ1, . . . , ξk−1, the terms quadratic with respect to ξ1, . . . , ξk−1, and the
terms having degree higher than two with respect to ξ1, . . . , ξk−1. It
is evident that all terms of the last group can be included into expres-
sion (2.27); therefore, we consider only three first groups of terms.
All terms free of ξ1, . . . , ξk−1 are obviously included in expressions
(2.23) (where they have been written explicitly) and in
∑
i1+···+ik−1=2
wi1,...,ik−1
Ξ
(0)
1
i1
· · ·Ξ
(0)
k−1
ik−1
(where there are summands of the sixth and higher degrees with respect
to y). All these summands can be included into expression (2.26). Hence,
the function Q does not include the terms free of ξ1, . . . , ξk−1.
Terms linear with respect to ξ1, . . . , ξk−1 are included into expres-
sion (2.25) both by means of summands from hyY∗ + 1
2hY 2
∗ and from
(2.22). If these terms have order not less than the fourth one with re-
spect to y, then it is clear that they can be included into expression
(2.26). Thus, the function Q has only those terms linear with respect to
ξ1, . . . , ξk−1 which have degrees two and three with respect to y.
Finally, consider the terms quadratic with respect to ξ1, . . . , ξk−1. If
these terms have the total degree higher than two, then they can be
included into expression (2.27); and, therefore, they are not included in
the function Q. All quadratic terms with respect to ξ1, . . . , ξk−1 having
the second degree (i.e. the terms with constant coefficients) are included
into the expression
∑
i1+···+ik−1=2
wi1,··· ,ik−1
{[c11ξ1 + c12ξ2 + · · · + c1,k−1ξk−1]
i1
A. O. Ignatyev 493
× · · · [ck−1,1ξ1 + · · · + ck−1,k−1ξk−1]
ik−1 − ξi1
1 . . . ξ
ik−1
k−1 }
= ξ2
1 + · · · + ξ2
k−1
and, hence, are not included into the function Q.
Thus, the function Q has the form
Q = y2Q2(ξ1, . . . , ξk−1) + y3Q3(ξ1, . . . , ξk−1), (2.28)
where Q2 and Q3 are linear forms with respect to ξ1, . . . , ξk−1:
Q2 = q
(2)
1 ξ1 + q
(2)
2 ξ2 + · · · + q
(2)
k−1ξk−1,
Q3 = q
(3)
1 ξ1 + q
(3)
2 ξ2 + · · · + q
(3)
k−1ξk−1.
The presence of summand (2.28) in (2.25) breaks the positive defi-
niteness of ∆V1. To get rid of the summand y2Q2(ξ1, . . . , ξk−1), we add
the summand y2P2(ξ1, . . . , ξk−1) = y2(p
(2)
1 ξ1 + p
(2)
2 ξ2 + · · · + p
(2)
k−1ξk−1)
to the function V1. Here, p
(2)
j (j = 1, . . . , k − 1) are constants. In other
words, consider the function
V2 =
1
2
hy2 + W (ξ1, . . . , ξk−1) + y2P2(ξ1, . . . , ξk−1) (2.29)
instead of the function V1. The term y2P2(ξ1, . . . , ξk−1) brings the fol-
lowing summands to ∆V2:
∆(y2P2(ξ1, . . . , ξk−1)) = [y2 + 2yY∗(ξ1, . . . , ξk−1, y) + Y 2
∗ (ξ1, . . . , ξk−1, y)]
×
k−1
∑
j=1
p
(2)
j [cj,1ξ1 + cj,2ξ2 + · · · + cj,k−1ξk−1 + Ξj(ξ1, . . . , ξk−1, y)]
−y2[p
(2)
1 ξ1 + p
(2)
2 ξ2 + · · · + p
(2)
k−1ξk−1]
= y2
k−1
∑
j=1
p
(2)
j (cj1ξ1 + cj2ξ2 + · · · + cj,k−1ξk−1 − ξj)
+G(ξ1, . . . , ξk−1, y),
where the function G is the sum of summands every of which can be
included either to expression (2.26) or to (2.27). Let us choose constants
p
(2)
1 , . . . , p
(2)
k−1 such that the equality
k−1
∑
j=1
p
(2)
j (cj1ξ1 + cj2ξ2 + · · · + cj,k−1ξk−1 − ξj) = −
k−1
∑
j=1
q
(2)
j ξj (2.30)
494 On the stability of solutions...
holds. To do this, let us equate the coefficients corresponding to ξj (j =
1, . . . , k − 1) on the right- and left-hand sides of equality (2.30). We
obtain the system of linear equations for p
(2)
j (j = 1, . . . , k − 1):
c1jp
(2)
1 + c2jp
(2)
2 + · · · + (cjj − 1)p
(2)
j + · · · + ck−1,jp
(2)
k−1 = −q
(2)
j
(j = 1, . . . , k − 1). (2.31)
The determinant of this system is not equal to zero, because all eigen-
values of C are inside the unit disk. Therefore, system (2.31) has the
unique solution. Substituting the obtained values p
(2)
1 , . . . , p
(2)
k−1 in the
expression P2(ξ1, . . . , ξk−1), we get
∆V2 = [h2 + g2(ξ1, . . . , ξk−1, y)]y4 + (ξ2
1 + · · · + ξ2
k−1)
+
k−1
∑
i,j=1
g
(2)
ij (ξ1, . . . , ξk−1, y)ξiξj + y3Q3(ξ1, . . . , ξk−1), (2.32)
where g2 and g
(2)
ij are functions vanishing for ξ1 = ξ2 = · · · = ξk−1 = y =
0.
Similarly, we can show that it is possible to get rid of the summand
y3Q3(ξ1, . . . , ξk−1) in expression (2.32). To do this, all we need is to add
the summand
y3P3(ξ1, . . . , ξk−1) = y3(p
(3)
1 ξ1 + p
(3)
2 ξ2 + · · · + p
(3)
k−1ξk−1)
to the function V2, where p
(3)
j (j = 1, . . . , k − 1) are constants. In other
words, consider the function
V =
1
2
hy2 + W (ξ1, . . . , ξk−1) + y2P2(ξ1, . . . , ξk−1) + y3P3(ξ1, . . . , ξk−1)
(2.33)
instead of the function V2. Its variation along system (2.17), (2.18) is
equal to
∆V = [h2 + g(ξ1, . . . , ξk−1, y)]y4 + (ξ2
1 + · · · + ξ2
k−1)
+
k−1
∑
i,j=1
gij(ξ1, . . . , ξk−1, y)ξiξj , (2.34)
where g and gij are functions vanishing for ξ1 = ξ2 = · · · = ξk−1 = y = 0.
It follows from (2.34) that ∆V is positive definite in a sufficiently
small neighborhood of the origin, and the function V of form (2.33) is
A. O. Ignatyev 495
negative definite for h < 0 and changes its sign for h > 0. Hence,
according to Theorems B and C, we can conclude that solution (2.19) of
system (2.17), (2.18) is asymptotically stable for h < 0 and unstable for
h > 0. This completes the proof.
Remark 2.1. Obviously, substitutions (2.3), (2.9), and (2.14) are such
that the investigation of the stability of solution (2.19) of system (2.17),
(2.18) is equivalent to the investigation of the stability of the zero solution
of system (2.1).
Remark 2.2. In Theorems 2.1 and 2.2, there are the conditions under
which the problem of the stability of the zero solution of system (2.1) can
be solved in the critical case where one eigenvalue of the linearized system
is equal to unity. The obtained criteria do not depend on nonlinear terms
with degrees of smallness more than three. If we obtain h = 0, then the
stability problem cannot be solved by terms of the first, second, and
third degrees of smallness in the expansions of the right-hand sides of
the system of difference equations. To solve this problem in this case, it
is necessary to consider also the terms of higher degrees.
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Contact information
Aleksandr
Olegovich Ignatyev
Institute of Applied Mathematics
and Mechanics of NASU,
74, R. Luxemburg Str.,
Donetsk 83114,
Ukraine
E-Mail: ignat@iamm.ac.donetsk.ua,
aoignat@mail.ru
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| id | nasplib_isofts_kiev_ua-123456789-10948 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1810-3200 |
| language | Russian |
| last_indexed | 2025-11-29T03:24:09Z |
| publishDate | 2008 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Игнатьев, А.О. 2010-08-10T10:14:36Z 2010-08-10T10:14:36Z 2008 Об устойчивости решений систем разностных уравнений в одном критическом случае / А.О. Игнатьев // Український математичний вісник. — 2008. — Т. 5, № 4. — С. 488-506. — Бібліогр.: 33 назв. — рос. 1810-3200 https://nasplib.isofts.kiev.ua/handle/123456789/10948 Рассматривается система нелинейных разностных уравнений, допускающая нулевое решение. С помощью метода функций Ляпунова изучается его устойчивость. Наряду с полной системой уравнений рассматривается линеаризованная система разностных уравнений. Известно, что если все корни характеристического уравнения линеаризованной системы по модулю меньше единицы, то нулевое решение полной системы асимптотически устойчиво. Если хотя бы один из корней характеристического уравнения по модулю больше единицы, то нулевое решение полной системы неустойчиво. В случае, когда часть корней характеристического уравнения по модулю меньше единицы, а часть равна единице, задача устойчивости не решается рассмотрением лишь линейных членов, и для ее решения нужно привлечь нелинейные слагаемые. Такой случай называется критическим. В настоящей работе рассмотрен критический случай одного корня, равного единице, когда задача устойчивости решается членами до третьего порядка малости в разложении правых частей исходных уравнений в ряды Маклорена. A system of nonlinear difference equations which admits the zero solution is considered. Its stability is studied by means of Lyapunov’s direct method. Side by side with this system, a linearized system of difference equations is also considered. It is well known that if all roots of the characteristic equation of a linearized system lie within the unit circle on the complex plane, then the zero solution of the original full system is asymptotically stable. If at least one eigenvalue lies outside the unit disk, then the zero solution of the original system is unstable. In the case where the moduli of some eigenvalues are equal to unity, and the moduli of others are less than unity, the stability problem cannot be solved by considering only the linear terms. To solve this problem, it is necessary to use the terms of higher orders in expansions of the righthand sides of the original system of difference equations in Maclaurin series. Such case is called a critical one. In this paper, we consider the critical case where one eigenvalue is equal to unity, and the stability problem can be solved by involving terms up to the third order in expansions of the right-hand sides of the initial equations in Maclaurin series. ru Інститут прикладної математики і механіки НАН України Об устойчивости решений систем разностных уравнений в одном критическом случае On the stability of solutions of systems of difference equations in a critical case Article published earlier |
| spellingShingle | Об устойчивости решений систем разностных уравнений в одном критическом случае Игнатьев, А.О. |
| title | Об устойчивости решений систем разностных уравнений в одном критическом случае |
| title_alt | On the stability of solutions of systems of difference equations in a critical case |
| title_full | Об устойчивости решений систем разностных уравнений в одном критическом случае |
| title_fullStr | Об устойчивости решений систем разностных уравнений в одном критическом случае |
| title_full_unstemmed | Об устойчивости решений систем разностных уравнений в одном критическом случае |
| title_short | Об устойчивости решений систем разностных уравнений в одном критическом случае |
| title_sort | об устойчивости решений систем разностных уравнений в одном критическом случае |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/10948 |
| work_keys_str_mv | AT ignatʹevao obustoičivostirešeniisistemraznostnyhuravneniivodnomkritičeskomslučae AT ignatʹevao onthestabilityofsolutionsofsystemsofdifferenceequationsinacriticalcase |