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.
Keywords:nonautonomous dynamic system; stability; attractivity to the origin; asymptotic stability; asymptotic solutions; characteristic function; Lyapunov function; energy function
where are complex-valued 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.
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.
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 asymptotic representation of solutions and error estimates in terms of the characteristic function was used in [4-6] 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 two-dimensional systems in terms of unknown phase functions (see Theorems 3.1-3.3).
Theorems 3.1-3.3 are similar to Lyapunov stability theorems with additional construction of an energy function in terms of the phase functions. Theorems 3.1-3.3 are applicable to a wide range of nonlinear systems with complex-valued coefficients (see Example 5.2 below or the linear Dirac equation with complex coefficients in ) 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 well-known 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.
Consider the second-order linear equation
Define the characteristic (Riccati) equation of (1.6)
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 .
Theorem 1.2 (Ignatyev )
Then linear equation (1.6) is asymptotically stable.
Condition that is bounded above in (1.10) was removed in .
Note that if
then condition (1.8) turns to
and is an integral version of (1.11).
In  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 second-order equation.
Theorem 1.3 (Pucci-Serrin , Theorem B)
then every bounded solution of the nonlinear equation
In this paper we prove general stability Theorems 3.1-3.3 in terms of unknown phase functions. Using these theorems we derive the versions of stability theorem of Pucci-Serrin , Smith , and some new ones.
2 Energy and some other auxiliary functions
and the auxiliary function:
Define the characteristic (Riccati) function of system (1.1)
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 ):
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
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:
Otherwise, (2.15) means that the error of asymptotic solutions is measured by the characteristic function.
Define (total) energy function as a non-negative quadratic form
Remark 2.3 If the phase functions are chosen as
3 Stability theorems in terms of unknown phase functions
In this section we formulate the main Theorems 3.1-3.3 of the paper.
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).
Example 3.1 From Theorem 3.3 it follows that the linear canonical equation
Remark 3.3 If
Under condition (3.11), condition (3.1) turns to
which is satisfied if
are satisfied (see (3.13), (3.6)).
Note that (3.16) is a nonautonomous analogue of the classical asymptotic stability criterion of Routh-Hurvitz.
From Theorems 3.1-3.3 one can deduce stability theorems for second-order 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).
4 Stability of the planar dynamic systems
From Theorems 3.1-3.3 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.
Example 4.1 From Theorem 4.3 it follows that system (1.1) with
(small damping) is asymptotically stable.
By using Jeffreys-Wentzel-Kramers-Brillouin (JWKB) approximation, we will prove the following theorem.
The following theorem is proved by using the Hartman-Wintner approximation .
In this case, asymptotic stability condition (3.6) is simplified:
Remark 4.2 For the Euler equation with , we have , and the Hartman-Wintner 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)).
5 Stability theorems for the equations with real coefficients
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
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.
Example 5.3 By Theorem 5.5 the linear equation
is asymptotically stable.
Example 5.5 By Theorem 5.7, the nonlinear Matukuma equation
is asymptotically stable.
are satisfied, where
Example 5.6 Due to Theorem 5.8, every solution of (1.6) with
then the attractivity condition (5.25) is simplified
Note that (5.28) is Smith’s  necessary and sufficient condition of asymptotic stability of (2.1) in the case of , .
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
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). □
Remark 6.1 If
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
By direct calculations
or using notation (3.5), we get
The first condition is condition (6.17), and the second condition follows from (6.18) and (6.31):
or (6.19) by integration. □
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
Proof of Theorem 3.3 Choosing
or by integration
Since both eigenvalues of the matrix K are non-negative, we have
Proof of Example 3.1 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
It means that for equation (2.1) we get
Further, using notation (3.18), (2.6) from (6.52), (6.53), we get
Proof of Theorem 3.7 By substitution
functions (2.8), (2.9) may be simplified
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). □
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):
By direct calculations,
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
Asymptotic stability condition (3.6) is satisfied as well:
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 Hartman-Wintner approximation 
and it follows from (4.15).
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 .
Theorem 5.2 follows from Theorem 3.4 by choosing
Proof of Example 5.1 Since
From (3.5), (3.13)
Proof of Theorem 5.4 We deduce Theorem 5.4 from Theorem 3.4 by choosing
we get from (3.19)
and from (3.18)
Proof of Example 5.2 This example follows from Theorem 5.3.
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)
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.
Further, in view of
condition (5.14) or
Proof of Example 5.5 Example 5.5 follows from Theorem 5.7. Indeed
and conditions (5.12), (5.13) are satisfied:
condition (5.14) is satisfied since
Proof of Theorem 5.9 We deduce Theorem 5.9 from Theorem 3.8 by choosing
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 time-dependent non-singular lower triangular transformation
with the initial values
Then we have the invariance
Remark A.1 From Theorem A.1 it follows the well-known result that the function
Proof of Theorem A.1 By substitution
we get from (2.8)
By direct calculations, from (A.2) we get
assuming initial conditions (A.7), we get
From these expressions, we get (A.9). □
Proof of Remark A.1 Rewrite equation (2.1) in form (6.6). Choosing
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
The author declares that he has no competing interests.
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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