Abstract
We are describing the stable nonautonomous planar dynamic systems with complex coefficients by using the asymptotic solutions (phase functions) of the characteristic (Riccati) equation. In the case of nonautonomous dynamic systems, this approach is more accurate than the eigenvalue method. We are giving a new construction of the energy (Lyapunov) function via phase functions. Using this energy, we are proving new stability and instability theorems in terms of the characteristic function that depends on unknown phase functions. By different choices of the phase functions, we deduce stability theorems in terms of the auxiliary function of coefficients , which is invariant with respect to the lower triangular transformations. We discuss some examples and compare our theorems with the previous results.
MSC: 34D20.
Keywords:
nonautonomous dynamic system; stability; attractivity to the origin; asymptotic stability; asymptotic solutions; characteristic function; Lyapunov function; energy function1 Introduction
We are interested in the behavior of a given solution of the nonlinear planar dynamic system
where are complexvalued functions from , and . Since we are assuming that the solution of (1.1) is given (fixed), system (1.1) may be considered as a linear nonautonomous system with coefficients depending only on a time variable.
Here and further, is the set of k times differentiable functions on , is the set of Lebesgue absolutely integrable functions on , and is the set of functions of bounded variation on .
Dynamic system (1.1) is said to be stable if for any and for any solution of (1.1) there exists such that for all , whenever . Dynamic system (1.1) is said to be attractive (to the origin) if for every solution of (1.1)
Dynamic system (1.1) is asymptotically stable if it is stable and attractive.
A solution of (1.1) is stable if for any there exists such that for all , whenever .
A solution of (1.1) is asymptotically stable (attractive to the origin) if (1.2) is true.
It is wellknown that for a nonautonomous system with the complex eigenvalues , , the classical RouthHurvitz condition of stability , , fails. Indeed, nonautonomous system (1.1) with
is unstable if , although the RouthHurvitz condition is satisfied. Necessary and sufficient conditions of asymptotic stability of this system,
could be found from the explicit solutions
This example shows that the description of stability of nonautonomous dynamic systems in terms of the eigenvalues is not accurate.
The usual method of investigation of asymptotic stability of differential equations is the Lyapunov direct method that uses energy functions and Lyapunov stability theorems [13].
The asymptotic representation of solutions and error estimates in terms of the characteristic function was used in [46] to prove asymptotic stability. In this paper we describe the stable dynamic systems by using energy approach with the use of the characteristic function (see (1.7) below), which is a more accurate tool than the eigenvalues.
The main idea of this paper is to construct the energy function in such a way that the time derivative of this energy is the linear combination of the characteristic functions (see (2.15) below). Using this energy, we prove main stability theorems for twodimensional systems in terms of unknown phase functions (see Theorems 3.13.3).
Theorems 3.13.3 are similar to Lyapunov stability theorems with additional construction of an energy function in terms of the phase functions. Theorems 3.13.3 are applicable to a wide range of nonlinear systems with complexvalued coefficients (see Example 5.2 below or the linear Dirac equation with complex coefficients in [7]) since they have the flexibility in the choice of an energy function.
To show that our theorems are useful, we deduce different versions of stability theorems (old wellknown and some new ones) by using different phase functions as asymptotic solutions of the characteristic equation (see (2.8) below). Moreover, we formulate some of the conditions of stability in terms of the auxiliary function (see (2.10) below), which is invariant with respect to the lower triangular transformations (see Theorem A.1). Note that there is no universal stability theorem in terms of coefficients for nonautonomous system (1.1) since there is no universal formula for an asymptotic solution of the characteristic equation.
As an application (see Example 5.5), we prove the asymptotic stability of the nonlinear Matukuma equation from astrophysics [8,9].
Consider the secondorder linear equation
Define the characteristic (Riccati) equation of (1.6)
where is said to be the characteristic function, and are the phase functions. In Section 6 (see Lemma 6.1) the following lemma is proved.
Lemma 1.1Assume that every solutionof (1.6) approaches zero as, then
whereare solutions of characteristic equation (1.7).
In the proof of Lemma 1.1, it is shown that (1.8) is also a sufficient condition of attractivity of solutions of (1.6) to the origin under additional condition
If the asymptotic behavior of as is known, then the condition of attractivity (1.8) could be clarified. Unfortunately, there is no a simple formula for asymptotic behavior of depending on the behavior of , as . Anyway, under some restrictions, one can obtain stability theorems for (1.6) by considering different asymptotic expansions of .
Assume that for some positive constants , , ,
Theorem 1.2 (Ignatyev [10])
Suppose that the functions, are real, and they satisfy conditions (1.10) and
Then linear equation (1.6) is asymptotically stable.
Condition that is bounded above in (1.10) was removed in [11].
Note that if
then condition (1.8) turns to
and is an integral version of (1.11).
In [12] Ballieu and Peiffer introduced a more general condition than Ignatyev’s one (1.11) for the attractivity (see (1.15), (1.16) below) of a nonlinear secondorder equation.
Theorem 1.3 (PucciSerrin [9], Theorem B)
Suppose that functions, are real, and there exists a nonnegative continuous functionof bounded variation onsuch that
then every bounded solution of the nonlinear equation
In this paper we prove general stability Theorems 3.13.3 in terms of unknown phase functions. Using these theorems we derive the versions of stability theorem of PucciSerrin [9], Smith [13], and some new ones.
2 Energy and some other auxiliary functions
Assuming , consider the following secondorder nonlinear equation associated with system (1.1):
where
Remark 2.1 Note that using equation (1.1), one can eliminate dependence on . Indeed . Similar calculations show that depends only on t, , coefficients , and their derivatives.
Here and further, often we suppress the dependence on t and for simplicity.
Introduce the characteristic function of (2.1) that depends on an unknown phase function :
and the auxiliary function:
where
Define the characteristic (Riccati) function of system (1.1)
Equation is the characteristic equation of system (1.1). For diagonal system (1.1), formulas (2.8) fail (for this case, see (A.23)).
Introduce the auxiliary functions
To explain the motivation for the choice of an energy function for system (1.1) (assuming ), consider a representation of solutions of (1.1) in Euler form (see [6]):
where , , are exact solutions of the characteristic equation , are defined as in (2.7), and
For the case of linear system (1.1), representation (2.11) gives the general solution of (1.1), where , are constants. For a nonlinear system, , depend on a solution . Solving equations (2.11) for , we get
Replacing by arbitrary differentiable functions , we define auxiliary energy functions
Remark 2.2 Although (2.14) are not constants for a nonlinear or nonautonomous system, they are useful for the study of stability. One can expect that for an appropriate choice of these energy functions are approximately conservative expressions for some nonlinear systems that are close to linear.
The derivative of the energy functions (2.14) may be written (see (6.23) below) as a linear combination of the characteristic functions:
From (2.15) it follows that if for any given solution of (1.1) the phase functions satisfy characteristic equation, that is, , , then energy conservation laws , are satisfied.
Otherwise, (2.15) means that the error of asymptotic solutions is measured by the characteristic function.
Define (total) energy function as a nonnegative quadratic form
Remark 2.3 If the phase functions are chosen as
where is an arbitrary differentiable function, then
3 Stability theorems in terms of unknown phase functions
In this section we formulate the main Theorems 3.13.3 of the paper.
Theorem 3.1Suppose that for a solutionof (1.1), we have, and there exist the complexvalued functionsand the real numbers, αsuch that for allwe haveand
Then the solutionof system (1.1) is stable.
Remark 3.1 Since stability conditions (3.1)(3.3) of Theorem 3.1 are given in terms of estimates with constants that depend on solutions of (1.1), system (1.1) is stable if these estimates are satisfied uniformly for all solutions (with constants that do not depend on solutions).
Remark 3.2 Note that for a linear nonautonomus system (1.1) with the choice , , , the error function and conditions (3.1), (3.3) are close to the necessary and sufficient condition of the stability.
Theorem 3.2Suppose that for a solutionof (1.1) , there exist the complexvalued functions, and the real numbers, αsuch that for all, and conditions (3.1), (3.2),
are satisfied withas in (3.4), (3.5).
Then the solutionof system (1.1) is asymptotically stable.
Theorem 3.3Suppose that for a solutionof (1.1), we have, and there exist the complexvalued functionssuch that for allwe have,
Then the solutionof system (1.1) is unstable.
Example 3.1 From Theorem 3.3 it follows that the linear canonical equation
is unstable.
Remark 3.3 If
then , and condition (3.2) is satisfied if .
Otherwise (3.2) is satisfied if , .
Under condition (3.11), condition (3.1) turns to
which is satisfied if
or
Sometimes it is convenient to use other than (3.4) formula for :
Remark 3.4 If , and there exists a function such that
then , . In this case formula (3.5) is simplified
and we get . From Theorem 3.1 it follows that in this case the solution of system (1.1) is asymptotically stable if for some real numbers α, l
are satisfied (see (3.13), (3.6)).
Note that (3.16) is a nonautonomous analogue of the classical asymptotic stability criterion of RouthHurvitz.
If the phase functions are chosen by formula (2.17), then , and
From Theorems 3.13.3 one can deduce stability theorems for secondorder equation (2.1). The attractivity to the origin for the solution of equation (2.1) is valid even by removing condition (3.1) (compare Theorem 3.2 with the following theorem).
Theorem 3.4Suppose that for a given solutionof (2.1), there exist the complexvalued functionssuch that conditions (3.2), (3.6) are satisfied withdefined as
Then the solutionof (2.1) approaches zero as.
Choosing
from Theorem 3.1 (in view of ), we obtain the following theorem.
Theorem 3.5Suppose that for a given solutionof (1.1), , and there exist complexvalued functionssuch that for allwe have,
and (3.6) are satisfied, where,
Then the solutionof system (1.1) is asymptotically stable.
By choosing
we have , and assuming (3.11) we get . From Theorem 3.2 we deduce the following theorem.
Theorem 3.6Suppose that for a given solutionof (1.1), , and there exist complexvalued functionssuch that for allwe have,
and (3.6) are satisfied withis as in (3.5), and:
Then the solutionof system (1.1) is asymptotically stable.
Theorem 3.7Suppose that for a given solutionof (1.1), , there exist complexvalued functionand the real numbers, αsuch that for allwe haveand the conditions
equation (3.3) (or (3.6)) are satisfied, where,
or
Then the solutionof system (1.1) is stable (or asymptotically stable).
Theorem 3.8Suppose that for a solutionof (2.1), , , there exist the real numbers, αand the complexvalued functionsuch that for all, conditions (3.29) and
4 Stability of the planar dynamic systems
From Theorems 3.13.3 one can deduce more useful asymptotic stability theorems in terms of coefficients of (1.1) by choosing the phase functions as asymptotic solutions of the characteristic equation.
Theorem 4.1Suppose that for a solutionof (1.1), we have, and for allthe conditions
and (3.3) (or (3.6)) are satisfied, where,
Then the solutionof system (1.1) is stable (or asymptotically stable).
Theorem 4.2Suppose that for a solutionof (1.1), we have, and for allwe haveand
Then the solutionof system (1.1) is asymptotically stable.
Theorem 4.3Suppose that for a solutionof (1.1), , for some numbers, α, and for all, we have,
and (3.3) (or (3.6)) are satisfied with, where
Then the solutionof system (1.1) is stable (or asymptotically stable).
Example 4.1 From Theorem 4.3 it follows that system (1.1) with
(small damping) is asymptotically stable.
By using JeffreysWentzelKramersBrillouin (JWKB) approximation, we will prove the following theorem.
Theorem 4.4Suppose that for a solutionof (1.1) , for all, the conditions, (4.1),
and (3.3) (or (3.6)) are satisfied, where,
Then the solutionof system (1.1) is stable (or asymptotically stable).
The following theorem is proved by using the HartmanWintner approximation [14].
Theorem 4.5Suppose for a solutionof system (1.1), , there exist the constants, αsuch that and for, we have,
and (3.3) (or (3.6)) are satisfied, where,
Then the solutionof system (1.1) is stable (or asymptotically stable).
Remark 4.1 Note that if and , then ,
In this case, asymptotic stability condition (3.6) is simplified:
Remark 4.2 For the Euler equation with , we have , and the HartmanWintner approximation fails. To consider this case, one may consider the choice with the other phase function that could be found by solving the equation (see (6.56)).
The following theorem is deduced from Theorem 4.1 by taking , , , .
Theorem 4.6Suppose that for a solutionof system (1.1), and for, we haveand
where
5 Stability theorems for the equations with real coefficients
Theorem 5.1Assume that for a solutionof (2.1), the coefficients, are realvalued, for some positive constants, , the conditions
or
are satisfied.
Then the solutionof equation (2.1) is asymptotically stable.
Example 5.1 By Theorem 5.1 the canonical linear equation
is asymptotically stable if one of the following conditions is satisfied:
A region of asymptotic stability of equation (5.3) described in Example 5.1 may be extended to
by using another asymptotic solution of (5.3) (see Example 5.4 or [15,16]).
Theorem 5.2Assume that for a solutionof (2.1), the coefficients, are realvalued, and for,
Then the solutionof equation (2.1) approaches zero as.
Theorem 5.3Assume that for a solutionof (2.1), the coefficients, are realvalued, and for,
Then the solutionof equation (2.1) is asymptotically stable.
Theorem 5.4Suppose that for a solutionof (2.1), the coefficients, are real functions, and condition (5.7) is satisfied. Then the solutionapproaches zero as.
Example 5.2 By Theorem 5.3 the equation
(where β, σ, μ are real numbers and b, k, γ are positive numbers) is asymptotically stable.
Theorem 5.5Assume that for a solutionof (2.1), the coefficients, are real functions and
Then the solutionis asymptotically stable.
Theorem 5.6Suppose that for a solutionof (2.1), the coefficients, are real and condition (5.10) is satisfied. Then the solutionapproaches zero as.
Example 5.3 By Theorem 5.5 the linear equation
is asymptotically stable.
Theorem 5.7Assume that for a solutionof (2.1), the coefficients, are real functions, and for all,
Then the solutionof (2.1) is asymptotically stable.
Example 5.4 From Theorem 5.7 the asymptotic stability of the equation (see also [9,15,16]) follows:
Example 5.5 By Theorem 5.7, the nonlinear Matukuma equation
is asymptotically stable.
Theorem 5.8Suppose that for a solutionof (2.1), the coefficients, are real functions, and the conditions
are satisfied, where
Then the solutionof (2.1) approaches zero as.
Remark 5.1 By taking , , we get , and Theorem 5.8 becomes a version of PucciSerrin Theorem 1.3. In this case, (5.18) is simplified to
Example 5.6 Due to Theorem 5.8, every solution of (1.6) with
Theorem 5.9Suppose that for a solutionof (2.1), the coefficients, are real functions, and for some constant, we have
where
Then the solutionapproaches zero as.
Theorem 5.10Suppose that for a solutionof (2.1), the functions, are real and
Then the solutionof (2.1) approaches zero as.
If
then the attractivity condition (5.25) is simplified
Note that (5.28) is Smith’s [13] necessary and sufficient condition of asymptotic stability of (2.1) in the case of , .
Theorems 5.15.10 are new versions of the stability theorem proved in [15,913,1721] by a different technique of construction of the energy function.
6 Proofs
Lemma 6.1Assume that all the solutions of linear system (1.1) are attractive to the origin, and functionsare solutions of, . Then
Proof of Lemma 6.1 and Lemma 1.1 First, we derive formula (2.8) for the characteristic function. Solving for the first equation of (1.1), we get
To eliminate , we substitute it in the second equation of (1.1) , so we get (2.1): , where P, Q are as in (2.2). From definition (2.5), we get (2.8). Formula (A.22) (see the Appendix) for is proved similarly by elimination of .
The first component of a solution of linear system (1.1) may be represented in the Euler form
where , are solutions of . From we get
Since we are assuming that the solutions , of linear system (1.1) are attractive to the origin, we have
as , that is, (6.1) is satisfied. Note that if additional condition (1.9) is satisfied, then (6.1) is also a sufficient condition of attractivity of solutions of (1.6), since in view of (6.5) as , we have
To prove Lemma 1.1, rewrite equation (1.6) in the form of system (1.1)
which means that
Then (1.8) follows from (6.1). □
Lemma 6.2Ifis a Hermitianmatrix with the entriessuch that
then the matrixis nonnegative (), and for any 2vectoru
Remark 6.1 If
Proof of Lemma 6.2 From the quadratic equation for the real eigenvalues of
we have
Further from
we get
□
Lemma 6.3If there exist the complexvalued functions, and a realvalued functionsuch that
whereis defined in (3.5), then the energy inequality
is satisfied, where the energy functions are defined in a more general form than in (2.14):
Proof of Lemma 6.3 Denoting
we can rewrite energy formula (6.20) in the form
By differentiation, we get
where
By direct calculations
where
Further
where
or
or
or using notation (3.5), we get
By Lemma 6.2 to have the nonnegativity of the matrix N (with the entries ), it is sufficient to show that
The first condition is condition (6.17), and the second condition follows from (6.18) and (6.31):
So, from conditions (6.17), (6.18) it follows ,
or (6.19) by integration. □
Lemma 6.4If the phase functionsare such that (3.1) is satisfied, then
Proof of Lemma 6.4 Introducing the Hermitian matrix with the entries
we have
From condition (3.1) we get
Further, by using Lemma 6.2, we obtain (6.35)
□
Proof of Theorem 3.1 First let us check that under the conditions of Theorem 3.1, Lemma 6.3 is applicable. Condition (6.18) is satisfied by choosing
Condition (6.17) is satisfied as well in view of condition (3.2)
From Lemma 6.3 and Lemma 6.4, we get
where is defined as in (3.13):
Substituting here formula (2.9) for , we get (3.4). Further from (3.3) and (6.43) the boundedness of and the stability follow. □
Proof of Remark 3.2 Note that if for linear system (1.1) , , , , then , , and solutions of (1.1) could be represented in the form (see (6.2))
Solution of (1.1) is bounded and stable if and only if for all and
These exact conditions are close to conditions (3.1), (3.3) of Theorem 3.1 which, under assumption , , turn to (see also (3.13))
□
Proof of Theorem 3.2 From (3.1), (3.2) we get estimate (6.43) as in the proof of Theorem 3.1. Further from (3.6) and (6.43) the boundedness of and as , that is, the asymptotic stability, follow. □
Proof of Theorem 3.3 Choosing
from assumption (3.7), we have and , and
which implies , , , and from (6.25) .
So,
or by integration
where μ is the largest eigenvalue of the nonnegative matrix .
Since both eigenvalues of the matrix K are nonnegative, we have
From this estimate and (3.8), it follows as . □
Proof of Example 3.1 We have
for , and T sufficiently big positive. Choosing
we have
So, conditions (3.7), (3.8) are satisfied, and from Theorem 3.3 it follows that equation (3.10) is unstable. □
Proof of Theorem 3.4 Consider equation (2.1) written in the form
Let us choose
where is defined in (3.5) with . Then the conditions of Lemma 6.3 are satisfied, and we get from Lemma 6.3
where the matrix K is defined in (6.36). Since from (3.11) it follows , by applying Lemma 6.2, we have
or
It means that for equation (2.1) we get
From (3.11) it follows , and we have also
Further, using notation (3.18), (2.6) from (6.52), (6.53), we get
and from (3.6) it follows , . □
Proof of Theorem 3.7 By substitution
functions (2.8), (2.9) may be simplified
Theorem 3.7 follows from Theorem 3.1, Theorem 3.2 by taking a given function and choosing , and phase function as follows (see (6.55)):
Further from (6.56), (3.15)
So, conditions (3.1), (3.2) turn to (3.29), (3.30). From (3.13) we have
or (3.31). □
Proof of Theorem 3.8 Theorem 3.8 follows from Theorem 3.4 applied to the system (6.51). By choosing and θ as in (3.31), in view of , we get (3.33) from (3.6) and (3.18). □
Proof of Theorem 4.1 Theorem 4.1 follows from Theorems 3.1 and 3.2 by choosing, as the approximate solutions of (see (6.56)), the eigenvalue approximation
Condition (3.1) turns to (3.12) (see Remark 3.3), or to (4.2).
In view of (6.64) and
we get from (3.4), (3.5) formulas (4.3), (4.4):
From (4.1) we have , and condition (3.2) is satisfied since from (4.4) we have . □
Proof of Theorem 4.2 Theorem 4.2 follows from Theorems 3.1 and 3.2 by choosing , and the special Riccati equation approximation
By direct calculations,
Condition (3.2) is true, since (3.11) is satisfied (see Remark 3.3). Condition (3.1) with turns to (3.12): or
which follows from (4.5). From (3.15), since , we get
Further from (3.13) we get
and condition (3.6) turns to (4.6). □
Proof of Theorem 4.3 Theorem 4.3 follows from Theorem 3.7 by choosing the linear equation approximation
□
Proof of Example 4.1 Example 4.1 follows from Theorem 4.3. Since , we have
Choosing , by using l’Hospital’s rule, if , , we get
and conditions (4.7), (4.8) with are satisfied.
Asymptotic stability condition (3.6) is satisfied as well:
□
Proof of Theorem 4.4 Theorem 4.4 follows from Theorem 3.1, Theorem 3.2 by choosing , and JWKB approximation:
We have from (6.56), (3.15)
Conditions (3.11) and (3.2) are satisfied. Condition (3.1) turns to (3.12) or (4.11), and from (3.13) we get (4.12)
□
Proof of Theorem 4.5 We deduce Theorem 4.5 from Theorems 3.1 and 3.2 assuming , and by choosing the HartmanWintner approximation [14]
where are solutions of the quadratic equation ,
By calculations,
Denoting
we have
and
From (3.15)
or (4.17)
From (3.4)
From (4.14) we get , and in view of (3.12), condition (3.1) turns to
and it follows from (4.15).
From (4.17) we have , , and condition (3.2) is satisfied.
To prove Remark 4.2, note that if , we have , and from the quadratic equation , we get , or . Further, from the equation , we get and the other phase function . □
Proof of Theorems 5.1, 5.2 Theorem 5.1 follows from Theorem 4.1 applied to system (6.51). Indeed, by substitution , , , , , condition (4.2) of Theorem 4.1 turns to . Further, from condition , we get and (4.1) is satisfied. From (4.4) we get .
By choosing , the conditions of Theorem 4.1 turn to (5.1) (big damping case).
By choosing , the conditions of Theorem 4.1 turn to (5.2) (small damping case).
Theorem 5.2 follows from Theorem 3.4 by choosing
□
Proof of Example 5.1 Since
If , , then , and condition (5.1) of Theorem 5.1 is satisfied:
If (small damping), then , , and condition (5.2) of Theorem 5.1 is satisfied:
If , , then condition (5.1) is satisfied again:
If , , then condition (5.2) is satisfied:
Further, if , , , then in view of (6.87) condition (5.1) is satisfied:
If , , , then in view of (6.86) condition (5.2) is satisfied:
If , then , , and condition (5.1) is satisfied:
If , then , , , condition (5.2) is satisfied:
□
Proof of Theorem 5.3 We deduce Theorem 5.3 from Theorem 3.1 applied to system (6.51), and by substitution , , , , , , ,
From (3.5), (3.13)
Conditions (3.1), (3.6) turn to (5.6), (5.7). If , then (3.2) is satisfied. The case is trivial, since in this case and the functions are exact solutions of (2.1). □
Proof of Theorem 5.4 We deduce Theorem 5.4 from Theorem 3.4 by choosing
From
we get from (3.19)
and from (3.18)
so (3.2) is satisfied if and condition (3.6) turns to (5.7). Case is trivial. □
Proof of Example 5.2 This example follows from Theorem 5.3.
In all these cases, (5.7) is satisfied since . □
Proof of Theorem 5.5 Theorem 5.5 follows from Theorem 4.2 applied to (6.51). □
Proof of Theorem 5.6 We deduce Theorem 5.6 from Theorem 3.4 by choosing
and from (3.19) we get
From (3.18) and (3.6) we get (5.10)
Condition (3.2) is satisfied if . The case is obvious since in that case the exact solutions of (2.1) are , . □
Proof of Example 5.3 This example follows from Theorem 5.5:
□
Proof of Theorem 5.7 Theorem 5.7 follows from Theorem 3.2 applied to (6.51), and by choosing
For this case (3.2) is true, (3.1) turns to (5.13), and (3.6) turns to (5.14). □
Proof of Example 5.4 Example 5.4 follows from Theorem 5.7.
In view of (6.89), we have conditions (5.12), (5.13) of Theorem 5.7 are satisfied if ,
Further, in view of
condition (5.14) or
□
Proof of Example 5.5 Example 5.5 follows from Theorem 5.7. Indeed
Choosing
we get
or
and conditions (5.12), (5.13) are satisfied:
From
condition (5.14) is satisfied since
□
Proof of Theorem 5.8 We deduce Theorem 5.8 from Theorem 3.4 by choosing , the phase from the HartmanWintner approximation
or (5.20)
Condition (3.6) with turns to (5.18). From (3.32) in view of (6.90), we get (5.19):
Condition (3.2) is satisfied in view of Remark 3.3 and . □
Proof of Theorem 5.9 We deduce Theorem 5.9 from Theorem 3.8 by choosing
By calculations
we get (5.22) from (3.29). Further, from (3.33) we get (5.23) since
□
Proof of Theorem 5.10 We deduce Theorem 5.10 from Theorem 3.4 by taking
Further from (3.15)
and (3.6) turns to (5.28). □
Appendix: Some invariants of the planar dynamic systems
By a linear timedependent nonsingular lower triangular transformation
from linear system (1.1) ( does not depend on ), we get another linear system
Define auxiliary functions associated with system (A.2) that depend on phase functions as follows:
where are the phase functions of system (A.2).
Theorem A.1Assume that, , andis a nonsingular lower triangular transformation, and, are solutions of the characteristic equations of linear systems (1.1), (A.2)
with the initial values
Then we have the invariance
Remark A.1 From Theorem A.1 it follows the wellknown result that the function
is invariant of (1.6) with respect to the transformation .
Proof of Theorem A.1 By substitution
we get from (2.8)
where , are defined in (2.10), (A.5).
By direct calculations, from (A.2) we get
and (A.8). Further, we get , or
assuming initial conditions (A.7), we get
So, the solutions , of characteristic equations are connected:
From these expressions, we get (A.9). □
Proof of Remark A.1 Rewrite equation (2.1) in form (6.6). Choosing
we have
where
Remark A.2 There are several characteristic functions of (1.1) depending on the structure of the matrix . Indeed, if , then the characteristic function of (1.1) is given by (2.8). If , but , the characteristic function may be defined by the similar formula
If system (1.1) is diagonal, that is, , then
Competing interests
The author declares that he has no competing interests.
Acknowledgements
This paper is dedicated to my mother Paytsar Hovhannisyan.
The author would like to thank anonymous reviewers for very useful and constructive comments that helped to improve the original manuscript.
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