### 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

**MSC: **
34D20.

##### Keywords:

nonautonomous dynamic system; stability; attractivity to the origin; asymptotic stability; asymptotic solutions; characteristic function; Lyapunov function; energy function### 1 Introduction

We are interested in the behavior of a given solution

where

Here and further,
*k* times differentiable functions on

Dynamic system (1.1) is said to be stable if for any

Dynamic system (1.1) is asymptotically stable if it is stable and attractive.

A solution

A solution of (1.1)

It is well-known that for a nonautonomous system with the complex eigenvalues

is unstable if

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 [1-3].

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 [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 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

As an application (see Example 5.5), we prove the asymptotic stability of the nonlinear Matukuma equation from astrophysics [8,9].

Consider the second-order linear equation

Define the characteristic (Riccati) equation of (1.6)

where

**Lemma 1.1***Assume that every solution*
*of* (1.6) *approaches zero as*
*then*

*where*
*are 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

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

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 second-order equation.

**Theorem 1.3** (Pucci-Serrin [9], Theorem B)

*Suppose that functions*
*are real*, *and there exists a non*-*negative continuous function*
*of bounded variation on*
*such that*

*then every bounded solution of the nonlinear equation*

*tends to zero as*

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 [9], Smith [13], and some new ones.

### 2 Energy and some other auxiliary functions

Assuming

where

**Remark 2.1** Note that using equation (1.1), one can eliminate dependence
*t*,

Here and further, often we suppress the dependence on *t* and

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

Introduce the auxiliary functions

To explain the motivation for the choice of an energy function for system (1.1) (assuming

where

For the case of linear system (1.1), representation (2.11) gives the general solution
of (1.1), where

Replacing

**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

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

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

where

### 3 Stability theorems in terms of unknown phase functions

In this section we formulate the main Theorems 3.1-3.3 of the paper.

**Theorem 3.1***Suppose that for a solution*
*of* (1.1), *we have*
*and there exist the complex*-*valued functions*
*and the real numbers*
*α**such that for all*
*we have*
*and*

*where*

*Then the solution*
*of 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

**Theorem 3.2***Suppose that for a solution*
*of* (1.1)
*there exist the complex*-*valued functions*
*and the real numbers*
*α**such that for all*
*and conditions* (3.1), (3.2),

*are satisfied with*
*as in* (3.4), (3.5).

*Then the solution*
*of system* (1.1) *is asymptotically stable*.

**Theorem 3.3***Suppose that for a solution*
*of* (1.1), *we have*
*and there exist the complex*-*valued functions*
*such that for all*
*we have*

*where*
*is defined in* (3.5), *and*

*Then the solution*
*of 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

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

then

and we get
*α*, *l*

are satisfied (see (3.13), (3.6)).

Note that (3.16) is a nonautonomous analogue of the classical asymptotic stability criterion of Routh-Hurvitz.

If the phase functions

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).

**Theorem 3.4***Suppose that for a given solution*
*of* (2.1), *there exist the complex*-*valued functions*
*such that conditions* (3.2), (3.6) *are satisfied with*
*defined as*

*Then the solution*
*of* (2.1) *approaches zero as*

Choosing

from Theorem 3.1 (in view of

**Theorem 3.5***Suppose that for a given solution*
*of* (1.1),
*and there exist complex*-*valued functions*
*such that for all*
*we have*

*and* (3.6) *are satisfied*, *where*

*Then the solution*
*of system* (1.1) *is asymptotically stable*.

By choosing

we have

**Theorem 3.6***Suppose that for a given solution*
*of* (1.1),
*and there exist complex*-*valued functions*
*such that for all*
*we have*

*and* (3.6) *are satisfied with*
*is as in* (3.5), *and*

*Then the solution*
*of system* (1.1) *is asymptotically stable*.

**Theorem 3.7***Suppose that for a given solution*
*of* (1.1),
*there exist complex*-*valued function*
*and the real numbers*
*α**such that for all*
*we have*
*and the conditions*

*equation* (3.3) (*or* (3.6)) *are satisfied*, *where*

*or*

*Then the solution*
*of system* (1.1) *is stable* (*or asymptotically stable*).

**Theorem 3.8***Suppose that for a solution*
*of* (2.1),
*there exist the real numbers*
*α**and the complex*-*valued function*
*such that for all*
*conditions* (3.29) *and*

*are satisfied*, *where*
*are given by* (3.29), (3.32).

*Then the solution*
*of equation* (2.1) *approaches zero as*

### 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.

**Theorem 4.1***Suppose that for a solution*
*of* (1.1), *we have*
*and for all*
*the conditions*

*and* (3.3) (*or* (3.6)) *are satisfied*, *where*

*Then the solution*
*of system* (1.1) *is stable* (*or asymptotically stable*).

**Theorem 4.2***Suppose that for a solution*
*of* (1.1), *we have*
*and for all*
*we have*
*and*

*Then the solution*
*of system* (1.1) *is asymptotically stable*.

**Theorem 4.3***Suppose that for a solution*
*of* (1.1),
*for some numbers*
*α*, *and for all*
*we have*

*and* (3.3) (*or* (3.6)) *are satisfied with*
*where*

*Then the solution*
*of 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 Jeffreys-Wentzel-Kramers-Brillouin (JWKB) approximation, we will prove the following theorem.

**Theorem 4.4***Suppose that for a solution*
*of* (1.1)
*for all*
*the conditions*

*and* (3.3) (*or* (3.6)) *are satisfied*, *where*

*Then the solution*
*of system* (1.1) *is stable* (*or asymptotically stable*).

The following theorem is proved by using the Hartman-Wintner approximation [14].

**Theorem 4.5***Suppose for a solution*
*of system* (1.1),
*there exist the constants*
*α**such that and for*
*we have*

*and* (3.3) (*or* (3.6)) *are satisfied*, *where*

*Then the solution*
*of system* (1.1) *is stable* (*or asymptotically stable*).

**Remark 4.1** Note that if

In this case, asymptotic stability condition (3.6) is simplified:

**Remark 4.2** For the Euler equation

The following theorem is deduced from Theorem 4.1 by taking

**Theorem 4.6***Suppose that for a solution*
*of system* (1.1),
*and for*
*we have*
*and*

*where*

*Then the solution*
*of system* (1.1) *is asymptotically stable*.

### 5 Stability theorems for the equations with real coefficients

**Theorem 5.1***Assume that for a solution*
*of* (2.1), *the coefficients*
*are real*-*valued*, *for some positive constants*
*the conditions*

*or*

*are satisfied*.

*Then the solution*
*of 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:

(i)

(ii)

(iii)

(iiii)

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 5.2***Assume that for a solution*
*of* (2.1), *the coefficients*
*are real*-*valued*, *and for*

*Then the solution*
*of equation* (2.1) *approaches zero as*

**Theorem 5.3***Assume that for a solution*
*of* (2.1), *the coefficients*
*are real*-*valued*, *and for*

*Then the solution*
*of equation* (2.1) *is asymptotically stable*.

**Theorem 5.4***Suppose that for a solution*
*of* (2.1), *the coefficients*
*are real functions*, *and condition* (5.7) *is satisfied*. *Then the solution*
*approaches 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 5.5***Assume that for a solution*
*of* (2.1), *the coefficients*
*are real functions and*

*Then the solution*
*is asymptotically stable*.

**Theorem 5.6***Suppose that for a solution*
*of* (2.1), *the coefficients*
*are real and condition* (5.10) *is satisfied*. *Then the solution*
*approaches zero as*

**Example 5.3** By Theorem 5.5 the linear equation

is asymptotically stable.

**Theorem 5.7***Assume that for a solution*
*of* (2.1), *the coefficients*
*are real functions*, *and for all*

*Then the solution*
*of* (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 5.8***Suppose that for a solution*
*of* (2.1), *the coefficients*
*are real functions*, *and the conditions*

*are satisfied*, *where*

*Then the solution*
*of* (2.1) *approaches zero as*

**Remark 5.1** By taking

**Example 5.6** Due to Theorem 5.8, every solution of (1.6) with

approaches zero as

**Theorem 5.9***Suppose that for a solution*
*of* (2.1), *the coefficients*
*are real functions*, *and for some constant*
*we have*

*where*

*Then the solution*
*approaches zero as*

**Theorem 5.10***Suppose that for a solution*
*of* (2.1), *the functions*
*are real and*

*Then the solution*
*of* (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 5.1-5.10 are new versions of the stability theorem proved in [1-5,9-13,17-21] by a different technique of construction of the energy function.

### 6 Proofs

**Lemma 6.1***Assume that all the solutions of linear system* (1.1) *are attractive to the origin*, *and functions*
*are solutions of*
*Then*

*Proof of Lemma 6.1 and Lemma 1.1* First, we derive formula (2.8) for the characteristic function. Solving for

To eliminate
*P*, *Q* are as in (2.2). From definition (2.5), we get (2.8). Formula (A.22) (see the Appendix)
for

The first component of a solution of linear system (1.1) may be represented in the Euler form

where

Since we are assuming that the solutions

as

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 6.2***If*
*is a Hermitian*
*matrix with the entries*
*such that*

*then the matrix*
*is non*-*negative* (
*and for any* 2-*vector**u*

**Remark 6.1** If

then

*Proof of Lemma 6.2* From the quadratic equation for the real eigenvalues of

we have

From

Further from

we get