General solutions of weakly delayed discrete systems in 3D
-
Josef Diblík
,
Hana Boháčková
Abstract
Discrete systems
1 Introduction
The article deals with three-dimensional discrete systems of weakly delayed equations. Weakly delayed systems form a class of systems such that their solutions have a special “merging” property, meaning that some solutions, being defined by different initial data, continue, after an interval, as a single solution. In the case considered, the set of all initial data splits into three mutually different subsets with the initial data of each subset generating solutions that merge into a single one after passing an interval. Then, it is possible to find simple formulas for general solutions, which is the aim of the present article.
Weakly delayed discrete planar systems are investigated in [15], where, using
1.1 Weakly delayed systems
Throughout the article, the following notation is used:
In this article, we investigate discrete systems with constant coefficients and with a single delay
where
The initial problem (1), (2) defines a unique solution of (1) on
The characteristic equation to (1),
is derived in the usual way if we look for a solution to (1) in the form
Definition 1
System (1) is called weakly delayed if, for every
In the sequel, rather than repeating the term weakly delayed system, we will use the abbreviation WD-system, etc. An important property formulated by the below lemma enables regular transformations of WD-systems.
Lemma 1
If the system (1) is a WD-system, then its arbitrary transformation
1.2 Problem under consideration
The article treats WD-systems (1) solving the problem of finding sharp formulas for the general solutions that solely depend on three parameters when a transient interval has been passed. The problem is solved completely, for every possible Jordan form of the matrix
1.3 Structure of the article
After the general solutions are constructed for all possible WD-systems (1), Section 1.4 introduces some preliminaries. Here, the general coefficient condition for a system (1) to be a WD-system is given with some properties of the matrix
1.4 Coefficient criterion for WD-systems and properties of matrix
B
Here, we formulate a general coefficient criterion indicating when system (1) is a WD-system. It directly follows from a detailed analysis of equation (4) (we refer to [34]).
Theorem 1
System (1) is a WD-system if and only if
The following properties are obvious. If (5), (6), and (8) hold, then all eigenvalues of matrix
2 Specific criteria for WD-systems
In this section, we formulate specific criteria for WD-systems derived from Theorem 1. These will depend on the Jordan forms of the matrix
Recall that all eigenvalues of
that can be obtained from (1) by transformation
are, by Lemma 1, again WD-systems. The initial data for (12) derived from (2) are
Remark 1
Now, in the sequel, without loss of generality, we can always assume that the matrix
where
2.1 Jordan canonical forms of
A
and their powers
In this section, we recall all possible Jordan forms of matrix
If (11) has real distinct roots
If (11) has two distinct roots – a single root
if
if
if
if
if
It is easy to see that
where
If the complex conjugate roots are given in the exponential form
and
2.2 Specific criteria
The following formulated criteria are consequences of conditions (5)–(10) in Theorem 1. Their proofs can be done by computing the relevant determinants and analyzing the arising expressions.
2.2.1 Criterion for WD-systems in case (15)
Consider system (1) with the matrix
Theorem 2
System (23) is a WD-system if and only if
2.2.2 Criterion for WD-systems in case (16)
Consider system (1) with the matrix
Theorem 3
System (28) is a WD-system if and only if
2.2.3 Criterion for WD-systems in case (17)
Consider system (1) with the matrix
Theorem 4
System (34) is a WD-system if and only if
2.2.4 Criterion for WD-systems in case (18)
Consider system (1) with the matrix
Theorem 5
System (41) is a WD-system if and only if
2.2.5 Criterion for WD-systems in case (19)
Consider system (1) with the matrix
Theorem 6
System (45) is a WD-system if and only if
2.3 Criterion for WD-systems in case (20)
Consider system (1) with the matrix
Theorem 7
System (51) is a WD-system if and only if
2.3.1 Criterion for WD-systems in case (21)
Consider system (1) with the matrix
Theorem 8
System (58) is a WD-system if and only if
3 Associated nondelayed system
In this section, we will transform system (1) into a higher dimensional nondelayed system to find its general solution. Define new dependent functions
where
where
is a
Let us transform system (66) using a suitable transformation
where
where
The initial data for system (70) are, as it follows from (68), (69),
The solution of initial problems (70) and (72) is given by formula
and the solution of initial problems (66) and (68) is
If the matrix
3.1 Properties of matrices used
Before studying the possible Jordan forms of the matrix
3.1.1 Geometric multiplicities of identical eigenvalues of
A
and
G
Recall some obvious facts. Eigenvalues of matrices
3.1.2 Relationship between the eigenvalues of matrices
A
,
B
, and
A
Theorem 9
Let (1) be a WD system. Then, the set of all the eigenvalues of the matrix
Proof
Let
where
Consider equation
As it follows from (76), equation (78) has exactly
and nonzero roots satisfy
As a consequence of the conclusion of Theorem 9, one can expect that the respective geometric multiplicities of eigenvalues of the matrices
Example 1
Consider weakly delayed system (41) with
Then,
and
Nevertheless, contrary to the property described in Example 1, the following theorem can be proved.
Theorem 10
The geometric multiplicity of the zero eigenvalue of the matrix
Proof
Let us find the geometric multiplicity of the zero eigenvalue of the matrix
where
Now it is clear that the number of linearly independent eigenvectors of matrix
4 Jordan forms of the associated-system matrix
When using formula (73), it is necessary to compute the powers of the matrix
The results of the computations depend on the geometric multiplicity of the zero eigenvalue of matrix
Next we give formulas for powers of the relevant matrix
and
Similarly, instead of
4.1 Jordan forms of matrices by Weyr’s algorithm
For each high-dimensional matrix in Sections 4.3–4.9, its Jordan form will be established using Weyr’s algorithm [4,11,23,35]. We will use only the part of this algorithm described below (following the explanation given in [23]). Let
their ranks
and nullities
where the number
we have
associated with the eigenvalue
The system of eigenvectors and generalized eigenvectors to an eigenvalue of
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4.2 Auxiliary matrices and their powers
Below we use some auxiliary matrices. Define an
and a frequently used in the article matrix
In particular,
4.3 The form and powers of
G
if
A
=
A
1
4.3.1 The case of
m
g
(
B
,
0
)
=
1
In this case, there exists a transition matrix
Then,
Then
4.3.2 The case of
m
g
(
B
,
0
)
=
2
In this case, there exists a transition matrix
and, as the formula for determining the rank
we obtain
To compute the next rank
Let us begin with the determination of an auxiliary rank
We perform the following operations on rows, where each number in round brackets refers to the number of a row in the last matrix:
Since
Therefore,
The characteristic numbers are
and, therefore, Table 1 will have two columns (with the first one consisting of
Reduction of Table 1 if
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Because the roots
Then
4.4 The form and powers of
G
if
A
=
A
2
4.4.1 Preliminaries
Consider an auxiliary matrix
By (30)–(33),
Theorem 11
If
Proof
Let us find
where, see (75),
We obtain
Therefore, taking into account that, by (29),
that is,
4.4.2 The case of
m
g
(
B
,
0
)
=
1
Then,
Then
Then
4.4.3 The case of
m
g
(
B
,
0
)
=
2
To compute
We perform the following operations on rows, where each number in round brackets refers to the number of a row in the last matrix:
By using formula (30) and the fact that all determinants of the second order of
Therefore,
Since
Table 1 will have two columns taking the same form as in Table 3 if
Then
Then
4.5 The form and powers of
G
if
A
=
A
3
4.5.1 Preliminaries
Consider an auxiliary matrix
By (39) and (40), we have
Theorem 12
If
Proof
Let us find
where, see (75),
We obtain
Therefore, taking into account that, by (35),
that is,
4.5.2 The case of
m
g
(
B
,
0
)
=
1
Then,
Then
Then, the associated Jordan form of
Then
4.5.3 The case of
m
g
(
B
,
0
)
=
2
By using formulas (36) and the property that all determinants of the second order of
We perform the following operations on the rows, where each number in round brackets refers to the number of a row in the last matrix:
and then
Now we continue as in Section 4.4.3. Reduction of Table 1 if
Then
Then
4.6 The form and powers of
G
if
A
=
A
4
4.6.1 Preliminaries
Here, we explain why, in this case, the matrix of type
Theorem 13
If
Proof
Let us find
where, see (75),
Immediately from (99) and (100), we obtain
and
In the previous sections, the matrix of linear terms of the system similar to (104)–(106) was denoted by
Now it is clear that, if
4.6.2 The case of
m
g
(
B
,
0
)
=
1
In this case, by Theorem 13, we have
Then,
Then
4.6.3 The case of
m
g
(
B
,
0
)
=
2
In the case considered, by Theorem 13, we have
To compute
We perform the following operations on rows, where each number in round brackets refers to the number of a row in the last matrix:
By using formula (42) and the property
Therefore,
Since
Table 1 will have only two columns and takes the same form as in Table 3 if
Then
4.7 The form and powers of
G
if
A
=
A
5
4.7.1 Preliminaries
Define an auxiliary matrix
Obviously,
Theorem 14
If
Proof
Let us find
where, see (75),
Immediately from (107), we obtain
Therefore, taking into account that, by (47),
that is,
4.7.2 The case of
m
g
(
B
,
0
)
=
1
Then,
Then
Then
Then
4.7.3 The case of
m
g
(
B
,
0
)
=
2
To compute
We perform the following operations on rows, where each number in round brackets refers to the number of a row in the last matrix:
By using the formula (46) and the property
Therefore,
Since
Table 1 (where
Then
Then
Then
4.8 The form and powers of
G
if
A
=
A
6
4.8.1 Preliminaries
Define an auxiliary matrix
Obviously,
Theorem 15
If
Proof
Let us find
where, see (75),
Immediately from (115) and (116), we obtain
Therefore, taking into account that, by (54),
that is,
4.8.2 The case of
m
g
(
B
,
0
)
=
1
Then,
Then
Then
Then
4.8.3 The case of
m
g
(
B
,
0
)
=
2
To compute
We perform the following operations on rows, where each number in round brackets refers to the number of a row in the last matrix:
Using formulas (52) and (53) and the property
Therefore,
Since
Table 1 will only have two columns taking the same form as in Table 3 (where
Then
Then
Then
4.9 The form and powers of
G
if
A
=
A
7
4.9.1 The case of
m
g
(
B
,
0
)
=
1
In this case, we can proceed in much the same way as in Section 4.3. We conclude that there exists a transition matrix
Then
4.9.2 The case of
m
g
(
B
,
0
)
=
2
In this case, we can proceed in much the same way as in Section 4.3.2. By Section 3.1.1 and Theorem 10, we have
To compute
We perform the following operations among rows, where each number in round brackets refers to the number of a row in the last matrix:
where
Therefore,
Since
Table 1 takes the form of Table 3 (constructed for
Then
5 Formulas for the general solution of problem (1), (2)
The initial data for system (1) are defined by (2). The solution of the initial problem (66), (68), that is, the problem
is (we refer to formulas (69)–(74))
where the relation between
Using an auxiliary matrix
formula (65) (i.e.,
where the initial data
Theorem 16
Let the Jordan form of A be
where
If
Theorem 17
Let the Jordan form of A be
where
if
if
if
Theorem 18
Let the Jordan form of A be
where
if
if
if
Theorem 19
Let the Jordan form of A be
where
If
Theorem 20
Let the Jordan form of A be
where
if
if
if
if
if
Theorem 21
Let the Jordan form of A be
where
if
if
if
if
if
Theorem 22
Let the Jordan form of A be
where
If
6 Initial values defining the merging of solutions
If
More exactly, if
Similarly, if
In the following theorem, formulas for computing the above-mentioned three initial values (158) through initial values
Define vectors
Theorem 23
Initial data (158) are defined by formulas
where
and
Proof
The initial data (158) are defined as suitable linear combinations of
Multiplying (160) successively by vectors
From Theorem 23, formula (159), we conclude that there exist linear combinations involving initial values
Let real numbers
where
where
where
Theorem 24
7 Examples
The results of the article will be now illustrated by three examples. In the first one, computations are performed in detail while in the remaining ones, parts of computations are omitted.
Example 2
Let system (1) be reduced to system (23), where
that is, to the system
Conditions (24)–(27) are fulfilled, system (164)–(166) is a WD-system and
and
The general solution (164)–(166) is expressed by the formula:
For
The general solution (167) depends only on parameters
must be identical for
or if
we obtain, by (168)–(170), the same values
We conclude that the solutions of system (164)–(166) defined by different initial data (171) and (172) are identical for
Example 3
Let system (1) be reduced to system (41), where
that is, to the system
Conditions (42)–(44) are fulfilled, system (173)–(175) is a WD-system and
and
Then, the general solution of the system (173)–(175) is expressed by the formula
where
The aforementioned general solution of system (173)–(175) depends on only three initial data
Now, we will demonstrate (in accordance with Theorem 24, case
or
give the same values
and the solutions of system (173)–(175) defined by (176) and (177) are identical for
Example 4
Let system (1) be reduced to system (51), where
that is, to the system
Conditions (52)–(57) are fulfilled, system (178)–(180) is a WD-system and
and
and
Then, the general solution of the system (178)–(180) is expressed by the formula
where
The aforementioned general solution of system (178)–(180) depends on only three initial data
Now, we will demonstrate (in accordance with Theorem 24, case
and
give the same new initial values
Initial data (181) and (182) define two solutions of the system (178)–(180) that are identical for
8 Remarks, conclusions, and open problems
Since Section 2, Remark 1, we have assumed, without loss of generality, that the matrix
defined by initial data (14), that is,
then can be transformed into solutions of system (1) by the substitution (13), that is, by the formula
where (see (124))
and the initial data
The article considers WD-systems (1) and their explicit general solutions are found by transforming them into nondelayed discrete higher dimensional systems. Simple formulas are found that describe general solutions depending on three arbitrary parameters only (rather than on the given
The property of the geometric multiplicities described in Section 3.1, the properties of auxiliary matrices
Discrete planar WD-systems are investigated in [15,16,20]. The criteria formulated in Theorems 2–7 (i.e., the cases of
Papers [7–10] consider mathematical models of endocrine regulation applying the so-called finite-dimension (FD) reducible differential delayed systems
Let us compare FD-reducible systems and WD-systems. FD-reducible systems are characterized by there existing a constant matrix
It describes a linearized model of the Mach number dynamics concerning the time-optimal control of a high-speed closed-circuit wind tunnel. In system (183)–(185),
In [15,16], a concept of conditional stability for planar WD-systems is formulated with relevant results proved. Generalizing this concept to three-dimensional discrete WD-systems may be a challenge for further research. Another problem is to construct nondelayed discrete systems having, on an interval, general solutions coinciding with general solutions of WD-systems. For a planar system, such results can be found in [15]. In addition, as our approach gives exact results, it might be usefull to write a computer program detecting WD-systems and producing explicit forms of general solutions. Another important challenge is to consider WD-systems of arbitrary dimension, developing the coefficient criteria and giving exact formulas for their solutions. Also, in the future, it might be useful to apply the known concrete formulas to representing the solutions of delayed systems of discrete equations, such as those derived and used, for instance, in papers [21,25,28,29,32], to obtain concrete formulas representing general solutions for higher dimensional discrete delayed systems. Attention deserves a generalization of the results for the case of the independent variable changing over a time-scale (for time-scales theory we refer to [3]) and as well as an analysis of connections between the solutions of WD-systems and the discrete variants of systems obtained by modifying the Euler-Poincaré equations [2]. Let us briefly comment on the recent papers [13,17,18]. In [13], the asymptotic behavior of solutions of fractional discrete equations is studied by the retract apparatus while, in [17], this technique is applied to studying the asymptotic behavior of solutions of Emden-Fowler second-order discrete equations. An original geometrical method is developed in [18] to investigate the behavior of solutions of complex differential equations in a neighborhood of a singular point. All the results derived in the aforementioned three articles have qualitative character and do not give exact formulas describing solutions as the present paper does. Nevertheless, it is an open question whether, for some particular cases of fractional discrete equations considered in [13], it will be possible to develop a WD-system classification and derive exact formulas for their solutions. Finally, for rudiments of theory of discrete equations, we refer to [1,5,19,24,30,33].
Acknowledgments
The authors greatly appreciate the work of the anonymous referees and the Editor, whose comments and suggestions have helped to improve this article in many aspects.
-
Funding information: The research of M. Růžičková has been supported by the Polish Ministry of Science and Higher Education under a subsidy for maintaining the research potential of the Faculty of Mathematics, University of Białystok, Poland.
-
Author contributions: All authors contributed equally to the writing of this article. All authors read and approved the final manuscript.
-
Conflict of interest: The authors state no conflict of interest.
-
Data availability statement: Data sharing not applicable to this article as no data sets were generated or analyzed during the current study.
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- Research Articles
- Incompressible limit for the compressible viscoelastic fluids in critical space
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- Intervals of bifurcation points for semilinear elliptic problems
- On multiplicity of solutions to nonlinear Dirac equation with local super-quadratic growth
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