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Abstract and Applied Analysis
Volumeย 2011ย (2011), Article IDย 154916, 23 pages
http://dx.doi.org/10.1155/2011/154916
Research Article

Instable Trivial Solution of Autonomous Differential Systems with Quadratic Right-Hand Sides in a Cone

1Department of Complex System Modeling, Faculty of Cybernetics, Taras Shevchenko National University of Kyiv, Vladimirskaya Str. 64, 01033 Kyiv, Ukraine
2Department of Mathematics, Faculty of Electrical Engineering and Communication, Technickรก 8, Brno University of Technology, 61600 Brno, Czech Republic
3Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Veveล™รญ 331/95, Brno University of Technology, 60200 Brno, Czech Republic

Received 5 October 2010; Accepted 2 November 2010

Academic Editor: Miroslavaย Rลฏลพiฤkovรก

Copyright ยฉ 2011 D. Ya. Khusainov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The present investigation deals with global instability of a general ๐‘›-dimensional system of ordinary differential equations with quadratic right-hand sides. The global instability of the zero solution in a given cone is proved by Chetaev's method, assuming that the matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts. The sufficient conditions for global instability obtained are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. In the proof, a result is used on the positivity of a general third-degree polynomial in two variables to estimate the sign of the full derivative of an appropriate function in a cone.

1. Introduction

Recently, there has been a rapidly growing interest in investigating the instability conditions of differential systems. The number of papers dealing with instability problems is rather low compared with the huge quantity of papers in which the stability of the motion of differential systems is investigated. The first results on the instability of zero solution of differential systems were obtained in a general form by Lyapunov [1] and Chetaev [2].

Further investigation on the instability of solutions of systems was carried out to weaken the conditions of the Lyapunov and Chetaev theorems for special-form systems. Some results are presented, for example, in [3โ€“10], but instability problems are analysed only locally. For example, in [7], a linear system of ordinary differential equations in the matrix form is considered, and conditions such that the corresponding forms (of the second and the third power) have fixed sign in some cone of the space โ„๐‘› are derived. To investigate this property another problem inverse to the known Lyapunov problem for the construction of Lyapunov functions is solved.

In the present paper, instability solutions of systems with quadratic right-hand sides is investigated in a cone dealing with a general ๐‘›-dimensional system with quadratic right-hand sides. We assume that the matrix of linear terms has a simple positive eigenvalue and the remaining eigenvalues have negative real parts.

Unlike the previous investigations, we prove the global instability of the zero solution in a given cone and the conditions for global instability are formulated by inequalities involving norms and eigenvalues of auxiliary matrices. The main tool is the method of Chetaev and application of a suitable Chetaev-type function. A novelty in the proof of the main result (Theorem 3.1) is the utilization of a general third-order polynomial inequality of two variables to estimate the sign of the full derivative of an appropriate function along the trajectories of a given system in a cone.

In the sequel, the norms used for vectors and matrices are defined as ๎ƒฉโ€–๐‘ฅโ€–=๐‘›๎“๐‘–=1๐‘ฅ2๐‘–๎ƒช1/2,(1.1) for a vector ๐‘ฅ=(๐‘ฅ1,โ€ฆ,๐‘ฅ๐‘›)๐‘‡ and โ€–๎€ท๐œ†โ„ฑโ€–=max๎€ทโ„ฑ๐‘‡โ„ฑ๎€ธ๎€ธ1/2,(1.2) for any ๐‘šร—๐‘› matrix โ„ฑ. Here and throughout the paper, ๐œ†max(โ‹…) (or ๐œ†min(โ‹…)) is the maximal (or minimal) eigenvalue of the corresponding symmetric and positive-semidefinite matrix โ„ฑ๐‘‡โ„ฑ (see, e.g., [11]).

In this paper, we consider the instability of the trivial solution of a nonlinear autonomous differential system with quadratic right-hand sidesฬ‡๐‘ฅ๐‘–=๐‘›๎“๐‘ =1๐‘Ž๐‘–๐‘ ๐‘ฅ๐‘ +๐‘›๎“๐‘ ,๐‘ž=1๐‘๐‘–๐‘ ๐‘ž๐‘ฅ๐‘ ๐‘ฅ๐‘ž,๐‘–=1,โ€ฆ,๐‘›,(1.3) where coefficients ๐‘Ž๐‘–๐‘  and ๐‘๐‘–๐‘ ๐‘ž are constants. Without loss of generality, throughout this paper we assume๐‘๐‘–๐‘ ๐‘ž=๐‘๐‘–๐‘ž๐‘ .(1.4) As emphasized, for example, in [2, 10โ€“12], system (1.3) can be written in a general vector-matrix formฬ‡๐‘ฅ=๐ด๐‘ฅ+๐‘‹๐‘‡๐ต๐‘ฅ,(1.5) where ๐ด is an ๐‘›ร—๐‘› constant square matrix, matrix ๐‘‹๐‘‡ is an ๐‘›ร—๐‘›2 rectangular matrix ๐‘‹๐‘‡=๎€ฝ๐‘‹๐‘‡1,๐‘‹๐‘‡2,โ€ฆ,๐‘‹๐‘‡๐‘›๎€พ,(1.6) where the entries of the ๐‘›ร—๐‘› square matrices ๐‘‹๐‘–, ๐‘–=1,โ€ฆ,๐‘› are equal to zero except the ๐‘–th row with entries ๐‘ฅ๐‘‡=(๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›), that is, ๐‘‹๐‘‡๐‘–=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘ฅ00โ‹ฏ0โ‹ฏโ‹ฏโ‹ฏโ‹ฏ00โ‹ฏ01๐‘ฅ2โ‹ฏ๐‘ฅ๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 00โ‹ฏ0โ‹ฏโ‹ฏโ‹ฏโ‹ฏ00โ‹ฏ0,(1.7) and ๐ต is a rectangular ๐‘›2ร—๐‘› matrix such that ๐ต๐‘‡=๎€ฝ๐ต1,๐ต2,โ€ฆ,๐ต๐‘›๎€พ,(1.8) where matrices ๐ต๐‘–={๐‘๐‘–๐‘ ๐‘ž}, ๐‘–,๐‘ ,๐‘ž=1,โ€ฆ,๐‘›, that is, matrices ๐ต๐‘–=โŽ›โŽœโŽœโŽœโŽœโŽ๐‘๐‘–11๐‘๐‘–12โ‹ฏ๐‘๐‘–1๐‘›๐‘๐‘–21๐‘๐‘–22โ‹ฏ๐‘๐‘–2๐‘›๐‘โ‹ฏโ‹ฏโ‹ฏโ‹ฏ๐‘–๐‘›1๐‘๐‘–๐‘›2โ‹ฏ๐‘๐‘–๐‘›๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽ (1.9) are ๐‘›ร—๐‘› constant and symmetric. Representation (1.5) permits an investigation of differential systems with quadratic right-hand sides by methods of matrix analysis. Such approach was previously used, for example, in [13].

If matrix ๐ด admits one simple positive eigenvalue, the system (1.5) can be transformed, using a suitable linear transformation of the dependent variables, to the same form (1.5) but with the matrix ๐ด having the form๎‚ต๐ด๐ด=0๐œƒ๐œƒ๐‘‡๐œ†๎‚ถ,(1.10) where ๐ด0 is an (๐‘›โˆ’1)ร—(๐‘›โˆ’1) constant matrix, ๐œƒ=(0,0,โ€ฆ,0)๐‘‡ is the (๐‘›โˆ’1)-dimensional zero vector and ๐œ†>0. With regard to this fact, we do not introduce new notations for the coefficients ๐‘๐‘–๐‘ ๐‘ž, ๐‘–,๐‘ ,๐‘ž=1,2,โ€ฆ,๐‘› in (1.5), assuming throughout the paper that ๐ด in (1.5) has the form (1.10), preserving the old notations ๐‘Ž๐‘–๐‘— for entries of matrix ๐ด0. This means that we assume that ๐ด={๐‘Ž๐‘–๐‘ }, ๐‘–,๐‘ =1,2,โ€ฆ,๐‘› with ๐‘Ž๐‘›๐‘ =๐‘Ž๐‘ ๐‘›=0 for ๐‘ =1,2,โ€ฆ,๐‘›โˆ’1 and ๐‘Ž๐‘›๐‘›=๐œ†, and ๐ด0={๐‘Ž๐‘–๐‘ }, ๐‘–,๐‘ =1,2,โ€ฆ,๐‘›โˆ’1.

We will give criteria of the instability of a trivial solution of the system (1.5) if the matrix ๐ด of linear terms is defined by (1.10).

2. Preliminaries

In this part we collect the necessary material-the definition of a cone, auxiliary Chetaev-type results on instability in a cone and, finally, a third degree polynomial inequality, which will be used to estimate the sign of the full derivative of a Chetaev-type function along the trajectories of system (1.5).

2.1. Instability of the Zero Solution of Systems of Differential Equations in a Cone

We consider an autonomous system of differential equationsฬ‡๐‘ฅ=๐‘“(๐‘ฅ),(2.1) where ๐‘“โˆถโ„๐‘›โ†’โ„๐‘› satisfies a local Lipschitz condition and ๐‘“(0)=0, that is, (2.1) admits the trivial solution. We will consider solutions of (2.1) determined by points (๐‘ฅ,๐‘ก)=(๐‘ฅ0,0) where ๐‘ฅ0โˆˆโ„๐‘›. The symbol ๐‘ฅ(๐‘ฅ0,๐‘ก) denotes the solution ๐‘ฅ=๐‘ฅ(๐‘ก) of (2.1), satisfying initial condition ๐‘ฅ(0)=๐‘ฅ0.

Definition 2.1. The zero solution ๐‘ฅโ‰ก0 of (2.1) is called unstable if there exists ๐œ€>0 such that, for arbitrary ๐›ฟ>0, there exists an ๐‘ฅ0โˆˆโ„๐‘› with โ€–๐‘ฅ0โ€–<๐›ฟ and ๐‘‡โ‰ฅ0 such that โ€–๐‘ฅ(๐‘ฅ0,๐‘‡)โ€–โ‰ฅ๐œ€.

Definition 2.2. A set ๐พโŠ‚๐‘…๐‘› is called a cone if ๐›ผ๐‘ฅโˆˆ๐พ for arbitrary ๐‘ฅโˆˆ๐พ and ๐›ผ>0.

Definition 2.3. A cone ๐พ is said to be a global cone of instability for (2.1) if ๐‘ฅ(๐‘ฅ0,๐‘ก)โˆˆ๐พ for arbitrary ๐‘ฅ0โˆˆ๐พ and ๐‘กโ‰ฅ0 and lim๐‘กโ†’โˆžโ€–๐‘ฅ(๐‘ฅ0,๐‘ก)โ€–=โˆž.

Definition 2.4. The zero solution ๐‘ฅโ‰ก0 of (2.1) is said to be globally unstable in a cone ๐พ if ๐พ is a global cone of instability for (2.1).

Now, we prove results analogous to the classical Chetaev theorem (see, e.g., [2]) on instability in a form suitable for our analysis. As usual, if ๐’ฎ is a set, then ๐œ•๐’ฎ denotes its boundary and ๐’ฎ its closure, that is, ๐’ฎโˆถ=๐’ฎโˆช๐œ•๐’ฎ.

Theorem 2.5. Let ๐‘‰โˆถโ„๐‘›โ†’โ„, ๐‘‰(0,โ€ฆ,0)=0 be a continuously differentiable function. Assume that the set ๐พ={๐‘ฅโˆˆ๐‘…๐‘›โˆถ๐‘‰(๐‘ฅ)>0}(2.2) is a cone. If the full derivative of ๐‘‰ along the trajectories of (2.1) is positive for every ๐‘ฅโˆˆ๐พ, that is, if ฬ‡๐‘‰(๐‘ฅ)โˆถ=grad๐‘‡๐‘‰(๐‘ฅ)๐‘“(๐‘ฅ)>0,๐‘ฅโˆˆ๐พ,(2.3) then ๐พ is a global cone of instability for the system (2.1).

Proof. Let ๐œ€ be a positive number. We define a neighborhood of the origin ๐‘ˆ๐œ€โˆถ={๐‘ฅโˆˆ๐‘…๐‘›โˆถโ€–๐‘ฅโ€–<๐œ€},(2.4) and a constant ๐‘€๐œ€โˆถ=max๐‘ฅโˆˆ๐‘ˆ๐œ€โˆฉ๐พ๐‘‰(๐‘ฅ).(2.5) Moreover, define a set ๐‘Š๐›ฟ๎‚†โˆถ=๐‘ฅโˆˆ๐‘ˆ๐œ€โˆฉ๎‚‡๐พ,๐‘‰(๐‘ฅ)โ‰ฅ๐›ฟ,(2.6) where ๐›ฟ is a positive number such that ๐›ฟ<๐‘€๐œ€. Then, ๐‘Š๐›ฟโ‰ โˆ….
Let ๐‘ฅ0โˆˆ๐‘Š๐›ฟโˆฉ๐พ, then ๐‘‰(๐‘ฅ0)=๐›ฟ1โˆˆ[๐›ฟ,๐‘€๐œ€]. We show that there exists a ๐‘ก=๐‘ก๐‘‡=๐‘ก๐‘‡(๐œ€,๐‘ฅ0) such that ๐‘ฅ(๐‘ฅ0,๐‘ก๐‘‡)โˆ‰๐‘ˆ๐œ€ and ๐‘ฅ(๐‘ฅ0,๐‘ก๐‘‡)โˆˆ๐พ.
Suppose to the contrary that this is not true and ๐‘ฅ(๐‘ฅ0,๐‘ก)โˆˆ๐‘ˆ๐œ€ for all ๐‘กโ‰ฅ0. Since ฬ‡๐‘‰(๐‘ฅ)>0, the function ๐‘‰ is increasing along the solutions of (2.1). Thus ๐‘ฅ(๐‘ฅ0,๐‘ก) remains in ๐พ. Due to the compactness of ๐‘Š๐›ฟ, there exists a positive value ๐›ฝ such that for ๐‘ฅ(๐‘ฅ0,๐‘ก)โˆˆ๐‘Š๐›ฟ๐‘‘๐‘‰๎€ท๐‘ฅ๎€ท๐‘ฅ๐‘‘๐‘ก0,๐‘ก๎€ธ๎€ธ=grad๐‘‡๐‘‰๎€ท๐‘ฅ๎€ท๐‘ฅ0๐‘“๎€ท๐‘ฅ๎€ท๐‘ฅ,๐‘ก๎€ธ๎€ธ0,๐‘ก๎€ธ๎€ธ>๐›ฝ.(2.7) Integrating this inequality over the interval [0,๐‘ก], we get ๐‘‰๎€ท๐‘ฅ๎€ท๐‘ฅ0๎€ท๐‘ฅ,๐‘ก๎€ธ๎€ธโˆ’๐‘‰0๎€ธ๎€ท๐‘ฅ๎€ท๐‘ฅ=๐‘‰0,๐‘ก๎€ธ๎€ธโˆ’๐›ฟ1>๐›ฝ๐‘ก.(2.8) Then there exists a ๐‘ก=๐‘ก๐‘‡=๐‘ก๐‘‡(๐œ€,๐‘ฅ0) satisfying ๐‘ก๐‘‡>๎€ท๐‘€๐œ€โˆ’๐›ฟ1๎€ธ๐›ฝ,(2.9) such that ๐‘‰(๐‘ฅ(๐‘ฅ0,๐‘ก๐‘‡))>๐‘€๐œ€ and, consequently, ๐‘ฅ(๐‘ฅ0,๐‘ก๐‘‡)โˆ‰๐‘ˆ๐œ€. This is contrary to our supposition. Since ๐œ€>0 is arbitrary, we have lim๐‘กโ†’โˆžโ€–โ€–๐‘ฅ๎€ท๐‘ฅ0๎€ธโ€–โ€–,๐‘ก=โˆž,(2.10) that is, the zero solution is globally unstable, and ๐พ is a global cone of instability.

Theorem 2.6. Let ๐‘‰โˆถโ„๐‘›โ†’โ„ be a continuously differentiable function and let ๐‘†,๐‘โˆถโ„๐‘›โ†’โ„, ๐‘(0,โ€ฆ,0)=0 be continuous functions such that ๐‘‰=๐‘†โ‹…๐‘. Assume that the set ๐พ1={๐‘ฅโˆˆ๐‘…๐‘›โˆถ๐‘(๐‘ฅ)>0}(2.11) is a cone, and ๐‘†(๐‘ฅ)>0 for any ๐‘ฅโˆˆ๐พ1. If the full derivative (2.3) of ๐‘‰ along the trajectories of (2.1) is positive for every ๐‘ฅโˆˆ๐พ1, that is, if ฬ‡๐‘‰(๐‘ฅ)>0 for every ๐‘ฅโˆˆ๐พ1, then ๐พ1 is a global cone of instability for the system (2.1).

Proof. The proof is a modification of the proof of Theorem 2.5. Let ๐œ€ be a positive number. We define a neighborhood ๐‘ˆ๐œ€ of the origin by formula (2.4) and a constant ๐‘€๐œ€โˆถ=max๐‘ฅโˆˆ๐‘ˆ๐œ€โˆฉ๐พ1๐‘‰(๐‘ฅ).(2.12) Moreover, define a set ๐‘Š๐›ฟ๎‚†โˆถ=๐‘ฅโˆˆ๐‘ˆ๐œ€โˆฉ๐พ1๎‚‡,๐‘‰(๐‘ฅ)โ‰ฅ๐›ฟ,(2.13) where ๐›ฟ is a positive number such that ๐›ฟ<๐‘€๐œ€. Then ๐‘Š๐›ฟโ‰ โˆ….
Let ๐‘ฅ0โˆˆ๐‘Š๐›ฟโˆฉ๐พ1. Then ๐‘‰(๐‘ฅ0)=๐›ฟ1โˆˆ[๐›ฟ,๐‘€๐œ€]. We show that there exists a ๐‘ก=๐‘ก๐‘‡=๐‘ก๐‘‡(๐œ€,๐‘ฅ0) such that ๐‘ฅ(๐‘ฅ0,๐‘ก๐‘‡)โˆ‰๐‘ˆ๐œ€ and ๐‘ฅ(๐‘ฅ0,๐‘ก๐‘‡)โˆˆ๐พ1.
Suppose to the contrary that this is not true and ๐‘ฅ(๐‘ฅ0,๐‘ก)โˆˆ๐‘ˆ๐œ€ for all ๐‘กโ‰ฅ0. Since ฬ‡๐‘‰(๐‘ฅ)>0, the function ๐‘‰ is increasing along the solutions of (2.1). Due to the compactness of ๐‘Š๐›ฟ, there exists a positive value ๐›ฝ such that for ๐‘ฅ(๐‘ฅ0,๐‘ก)โˆˆ๐‘Š๐›ฟ๐‘‘๐‘‰๎€ท๐‘ฅ๎€ท๐‘ฅ๐‘‘๐‘ก0,๐‘ก๎€ธ๎€ธ=grad๐‘‡๐‘‰๎€ท๐‘ฅ๎€ท๐‘ฅ0๐‘“๎€ท๐‘ฅ๎€ท๐‘ฅ,๐‘ก๎€ธ๎€ธ0,๐‘ก๎€ธ๎€ธ>๐›ฝ.(2.14) Integrating this inequality over interval [0,๐‘ก], we get ๐‘‰๎€ท๐‘ฅ๎€ท๐‘ฅ0๎€ท๐‘ฅ,๐‘ก๎€ธ๎€ธโˆ’๐‘‰0๎€ธ๎€ท๐‘ฅ๎€ท๐‘ฅ=๐‘‰0,๐‘ก๎€ธ๎€ธโˆ’๐›ฟ1๎€ท๐‘ฅ๎€ท๐‘ฅ=๐‘†0๐‘๎€ท๐‘ฅ๎€ท๐‘ฅ,๐‘ก๎€ธ๎€ธ0,๐‘ก๎€ธ๎€ธโˆ’๐›ฟ1>๐›ฝ๐‘ก.(2.15) Since ๐‘†(๐‘ฅ(๐‘ฅ0,๐‘ก))>0, the inequality ๐‘๎€ท๐‘ฅ๎€ท๐‘ฅ0>๐›ฟ,๐‘ก๎€ธ๎€ธ1+๐›ฝ๐‘ก๐‘†๎€ท๐‘ฅ๎€ท๐‘ฅ0,๐‘ก๎€ธ๎€ธ>0(2.16) is an easy consequence of (2.15). Thus ๐‘ฅ(๐‘ฅ0,๐‘ก) remains in ๐พ1. Apart from this, (2.15) also implies the existence of a ๐‘ก=๐‘ก๐‘‡=๐‘ก๐‘‡(๐œ€,๐‘ฅ0) satisfying ๐‘ก๐‘‡>๎€ท๐‘€๐œ€โˆ’๐›ฟ1๎€ธ๐›ฝ,(2.17) such that ๐‘‰(๐‘ฅ(๐‘ฅ0,๐‘ก๐‘‡))>๐‘€๐œ€. Consequently, ๐‘ฅ(๐‘ฅ0,๐‘ก๐‘‡)โˆ‰๐‘ˆ๐œ€. This is contrary to our supposition. Since ๐œ€>0 is arbitrary, we have lim๐‘กโ†’โˆžโ€–โ€–๐‘ฅ๎€ท๐‘ฅ0๎€ธโ€–โ€–,๐‘ก=โˆž,(2.18) that is, the zero solution is globally unstable and ๐พ1 is a global cone of instability.

Definition 2.7. A function ๐‘‰ satisfying all the properties indicated in Theorem 2.5 is called a Chetaev function for the system (2.1). A function ๐‘‰ satisfying all the properties indicated in Theorem 2.6 is called a Chetaev-type function for the system (2.1).

2.2. Auxiliary Inequality

Our results will be formulated in terms of global cones of instability. These will be derived using an auxiliary inequality valid in a given cone. Let (๐‘ฅ,๐‘ฆ)โˆˆโ„2 and let ๐‘˜ be a positive number. We define a cone ๎€ฝ๐’ฆโˆถ=(๐‘ฅ,๐‘ฆ)โˆˆโ„2๎€พ.โˆถ๐‘ฆ>๐‘˜|๐‘ฅ|(2.19)

Lemma 2.8. Let ๐‘Ž, ๐‘, ๐‘, ๐‘‘, and ๐‘˜ be given constants such that ๐‘>0, ๐‘‘>0, ๐‘˜>0, and |๐‘|โ‰ค๐‘˜๐‘‘. Assume, moreover, either |๐‘Ž|โ‰ค๐‘˜๐‘,(2.20) or โŽงโŽชโŽจโŽชโŽฉ๎ƒŽ|๐‘Ž|>๐‘˜๐‘,(2.21)|๐‘|โ‰ ๐‘˜๐‘‘,๐‘˜โ‰ฅmax||||๐‘Ž+๐‘˜๐‘,๎ƒŽ๐‘+๐‘˜๐‘‘||||๐‘Žโˆ’๐‘˜๐‘||||โŽซโŽชโŽฌโŽชโŽญ,๐‘โˆ’๐‘˜๐‘‘(2.22) then ๐‘Ž๐‘ฅ3+๐‘๐‘ฅ2๐‘ฆ+๐‘๐‘ฅ๐‘ฆ2+๐‘‘๐‘ฆ3>0,(2.23) for every (๐‘ฅ,๐‘ฆ)โˆˆ๐’ฆ.

Proof. We partition ๐’ฆ into two disjoint cones ๐’ฆ1๎€ฝโˆถ=(๐‘ฅ,๐‘ฆ)โˆˆโ„2๎€พ,๐’ฆโˆถ๐‘ฆ>๐‘˜|๐‘ฅ|,๐‘ฅ>02๎€ฝโˆถ=(๐‘ฅ,๐‘ฆ)โˆˆโ„2๎€พ,โˆถ๐‘ฆ>๐‘˜|๐‘ฅ|,๐‘ฅโ‰ค0(2.24) and rewrite (2.23) as ๐‘ฅ๎€ท๐‘Ž๐‘ฅ2+๐‘๐‘ฆ2๎€ธ๎€ท+๐‘ฆ๐‘๐‘ฅ2+๐‘‘๐‘ฆ2๎€ธ>0.(2.25) We prove the validity of (2.23) in each of the two cones separately.
The case of the cone ๐’ฆ1. Suppose that (2.20) holds. Estimating the left-hand side of (2.25), we get๐‘ฅ๎€ท๐‘Ž๐‘ฅ2+๐‘๐‘ฆ2๎€ธ๎€ท+๐‘ฆ๐‘๐‘ฅ2+๐‘‘๐‘ฆ2๎€ธ๎€ท>๐‘ฅ๐‘Ž๐‘ฅ2+๐‘๐‘ฆ2๎€ธ๎€ท+๐‘˜๐‘ฅ๐‘๐‘ฅ2+๐‘‘๐‘ฆ2๎€ธ๎€บ๐‘ฅ=๐‘ฅ2(๐‘Ž+๐‘˜๐‘)+๐‘ฆ2๎€ป(๐‘+๐‘˜๐‘‘)>0,(2.26) and (2.23) holds.
If inequalities (2.21) and (2.22) are valid, then, estimating the left-hand side of (2.25), we get๐‘ฅ๎€ท๐‘Ž๐‘ฅ2+๐‘๐‘ฆ2๎€ธ๎€ท+๐‘ฆ๐‘๐‘ฅ2+๐‘‘๐‘ฆ2๎€ธ๎€ท>๐‘ฅ๐‘Ž๐‘ฅ2+๐‘๐‘ฆ2๎€ธ๎€ท+๐‘˜๐‘ฅ๐‘๐‘ฅ2+๐‘‘๐‘ฆ2๎€ธ๎€บ๐‘ฅ=๐‘ฅ2(๐‘Ž+๐‘˜๐‘)+๐‘ฆ2๎€ป๎€บโˆ’||||๐‘ฅ(๐‘+๐‘˜๐‘‘)โ‰ฅ๐‘ฅ๐‘Ž+๐‘˜๐‘2+(๐‘+๐‘˜๐‘‘)๐‘ฆ2๎€ป=๎‚ธ๐‘ฆ(๐‘+๐‘˜๐‘‘)๐‘ฅ2โˆ’||||๐‘Ž+๐‘˜๐‘๐‘ฅ๐‘+๐‘˜๐‘‘2๎‚นโŽกโŽขโŽขโŽฃ๎ƒŽ=(๐‘+๐‘˜๐‘‘)๐‘ฅ๐‘ฆโˆ’||||๐‘Ž+๐‘˜๐‘๐‘ฅโŽคโŽฅโŽฅโŽฆโŽกโŽขโŽขโŽฃ๎ƒŽ๐‘+๐‘˜๐‘‘๐‘ฆ+||||๐‘Ž+๐‘˜๐‘๐‘ฅโŽคโŽฅโŽฅโŽฆ๐‘+๐‘˜๐‘‘=(๐‘+๐‘˜๐‘‘)๐‘ฅ2โŽกโŽขโŽขโŽฃ๎ƒŽ๐‘˜โˆ’||||๐‘Ž+๐‘˜๐‘โŽคโŽฅโŽฅโŽฆโŽกโŽขโŽขโŽฃ๎ƒŽ๐‘+๐‘˜๐‘‘๐‘˜+||||๐‘Ž+๐‘˜๐‘โŽคโŽฅโŽฅโŽฆ๐‘+๐‘˜๐‘‘โ‰ฅ0,(2.27) and (2.23) holds again.
The case of the cone ๐’ฆ2. Suppose that (2.20) hold, then, estimating the left-hand side of (2.25), we get ๐‘ฅ๎€ท๐‘Ž๐‘ฅ2+๐‘๐‘ฆ2๎€ธ๎€ท+๐‘ฆ๐‘๐‘ฅ2+๐‘‘๐‘ฆ2๎€ธ๎€ท=โˆ’|๐‘ฅ|๐‘Ž๐‘ฅ2+๐‘๐‘ฆ2๎€ธ๎€ท+๐‘ฆ๐‘๐‘ฅ2+๐‘‘๐‘ฆ2๎€ธ๎€ท>โˆ’|๐‘ฅ|๐‘Ž๐‘ฅ2+๐‘๐‘ฆ2๎€ธ๎€ท+๐‘˜|๐‘ฅ|๐‘๐‘ฅ2+๐‘‘๐‘ฆ2๎€ธ๎€บ=โˆ’|๐‘ฅ|(๐‘Žโˆ’๐‘˜๐‘)๐‘ฅ2+(๐‘โˆ’๐‘˜๐‘‘)๐‘ฆ2๎€ปโ‰ฅ0,(2.28) and (2.23) holds.
If inequalities (2.21) and (2.22) are valid, then the estimation of (2.25) implies (we use (2.28))๐‘ฅ๎€ท๐‘Ž๐‘ฅ2+๐‘๐‘ฆ2๎€ธ๎€ท+๐‘ฆ๐‘๐‘ฅ2+๐‘‘๐‘ฆ2๎€ธ๎€บ>โˆ’|๐‘ฅ|(๐‘Žโˆ’๐‘˜๐‘)๐‘ฅ2+(๐‘โˆ’๐‘˜๐‘‘)๐‘ฆ2๎€ป=||||๎‚ธ๐‘ฆ๐‘โˆ’๐‘˜๐‘‘|๐‘ฅ|2โˆ’๐‘Žโˆ’๐‘˜๐‘||||๐‘ฅ๐‘โˆ’๐‘˜๐‘‘2๎‚น=โŽงโŽชโŽชโŽชโŽจโŽชโŽชโŽชโŽฉ||||๎ƒฌ๎ƒŽโ‰ฅ0if๐‘Žโˆ’๐‘˜๐‘<0,๐‘โˆ’๐‘˜๐‘‘|๐‘ฅ|๐‘ฆโˆ’๐‘Žโˆ’๐‘˜๐‘||||๐‘ฅ๎ƒŽ๐‘โˆ’๐‘˜๐‘‘๎ƒญ๎ƒฌ๐‘ฆ+๐‘Žโˆ’๐‘˜๐‘||||๐‘ฅ๎ƒญโ‰ฅ||||๐‘ฅ๐‘โˆ’๐‘˜๐‘‘๐‘โˆ’๐‘˜๐‘‘2๎ƒฌ๎ƒŽ๐‘˜+๐‘Žโˆ’๐‘˜๐‘||||๎ƒŽ๐‘โˆ’๐‘˜๐‘‘๎ƒญ๎ƒฌ๐‘˜โˆ’๐‘Žโˆ’๐‘˜๐‘||||๎ƒญ๐‘โˆ’๐‘˜๐‘‘โ‰ฅ0if๐‘Žโˆ’๐‘˜๐‘>0.(2.29) Hence, (2.23) holds again.

3. Global Cone of Instability

In this part we derive a result on the instability of system (1.5) in a cone. In order to properly formulate the results, we have to define some auxiliary vectors and matrices (some definitions copy the previous ones used in Introduction, but with a dimension of ๐‘›โˆ’1 rather than ๐‘›). We denote ๐‘ฅ(๐‘›โˆ’1)=๎€ท๐‘ฅ1,๐‘ฅ2,โ€ฆ,๐‘ฅ๐‘›โˆ’1๎€ธ๐‘‡,๐‘๐‘–=๎‚€๐‘๐‘–1๐‘›,๐‘๐‘–2๐‘›,โ€ฆ,๐‘๐‘–๐‘›โˆ’1,๐‘›๎‚๐‘‡ฬƒ๎€ท๐‘,๐‘–=1,2,โ€ฆ,๐‘›,๐‘=1๐‘›๐‘›,๐‘2๐‘›๐‘›,โ€ฆ,๐‘๐‘›โˆ’1๐‘›๐‘›๎€ธ๐‘‡.(3.1) Apart from this, we define symmetric (๐‘›โˆ’1)ร—(๐‘›โˆ’1) matrices ๐ต0๐‘–=๎€ฝ๐‘๐‘–๐‘ ๐‘ž๎€พ,๐‘–=1,2,โ€ฆ,๐‘›,๐‘ ,๐‘ž=1,2,โ€ฆ,๐‘›โˆ’1,(3.2) that is, ๐ต0๐‘–=โŽ›โŽœโŽœโŽœโŽœโŽ๐‘๐‘–11๐‘๐‘–12โ‹ฏ๐‘๐‘–1,๐‘›โˆ’1๐‘๐‘–21๐‘๐‘–22โ‹ฏ๐‘๐‘–2,๐‘›โˆ’1๐‘โ‹ฏโ‹ฏโ‹ฏโ‹ฏ๐‘–๐‘›โˆ’1,1๐‘๐‘–๐‘›โˆ’1,2โ‹ฏ๐‘๐‘–๐‘›โˆ’1,๐‘›โˆ’1โŽžโŽŸโŽŸโŽŸโŽŸโŽ ,๎‚โŽ›โŽœโŽœโŽœโŽœโŽ๐‘๐ต=11๐‘›๐‘12๐‘›โ‹ฏ๐‘1๐‘›โˆ’1,๐‘›๐‘21๐‘›๐‘22๐‘›โ‹ฏ๐‘2๐‘›โˆ’1,๐‘›๐‘โ‹ฏโ‹ฏโ‹ฏโ‹ฏ๐‘›โˆ’11๐‘›๐‘๐‘›โˆ’12๐‘›โ‹ฏ๐‘๐‘›โˆ’1๐‘›โˆ’1,๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽ .(3.3) Finally, we define an (๐‘›โˆ’1)ร—(๐‘›โˆ’1)2 matrix ๐ต๐‘‡=๎‚†๐ต๐‘‡1,๐ต๐‘‡2,โ€ฆ,๐ต๐‘‡๐‘›โˆ’1๎‚‡,(3.4) where (๐‘›โˆ’1)ร—(๐‘›โˆ’1) matrices ๐ต๐‘‡๐‘–, ๐‘–=1,2,โ€ฆ,๐‘›โˆ’1 are defined as ๐ต๐‘‡๐‘–=โŽ›โŽœโŽœโŽœโŽœโŽ๐‘1๐‘–1๐‘1๐‘–2โ‹ฏ๐‘1๐‘–,๐‘›โˆ’1๐‘2๐‘–1๐‘2๐‘–2โ‹ฏ๐‘2๐‘–,๐‘›โˆ’1๐‘โ‹ฏโ‹ฏโ‹ฏโ‹ฏ๐‘›โˆ’1๐‘–1๐‘๐‘›โˆ’1๐‘–2โ‹ฏ๐‘๐‘›โˆ’1๐‘–,๐‘›โˆ’1โŽžโŽŸโŽŸโŽŸโŽŸโŽ .(3.5) We consider a matrix equation๐ด๐‘‡0๐ป+๐ป๐ด0=โˆ’๐ถ,(3.6) where ๐ป and ๐ถ are (๐‘›โˆ’1)ร—(๐‘›โˆ’1) matrices. It is well-known (see, e.g., [14]) that, for a given positive definite symmetric matrix ๐ถ, (3.6) can be solved for a positive definite symmetric matrix ๐ป if and only if the matrix ๐ด0 is asymptotically stable.

Theorem 3.1 (Main result). Assume that the matrix ๐ด0 is asymptotically stable, ๐‘๐‘›๐‘›๐‘›>0 and โ„Ž is a positive number. Let ๐ถ be an (๐‘›โˆ’1)ร—(๐‘›โˆ’1) positive definite symmetric matrix and ๐ป be a related (๐‘›โˆ’1)ร—(๐‘›โˆ’1) positive definite symmetric matrix solving equation (3.6). Assume that the matrix ๎‚๐ต(โˆ’๐ป๐‘‡โˆ’๎‚๐ต๐ป+โ„Ž(๐ต0๐‘›)๐‘‡) is positive definite, โ€–โ€–2โ„Ž๐‘๐‘›ฬƒ๐‘โ€–โ€–โ‰คโˆšโˆ’๐ป๐œ†min(๐ป)โ„Žโ‹…๐‘๐‘›๐‘›๐‘›,(3.7) and, in addition, one of the following conditions is valid:
either โ€–โ€–๐ป๐ต๐‘‡โ€–โ€–โ‰ค๎‚™๐œ†min(๐ป)โ„Žโ‹…๐œ†min๎‚€๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚(3.8) or โ€–โ€–๐ป๐ต๐‘‡โ€–โ€–>๎‚™๐œ†min(๐ป)โ„Žโ‹…๐œ†min๎‚€๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚,(3.9) a strong inequality holds in (3.7), and ๎‚™๐œ†min(๐ป)โ„Ž๎‚†โˆšโ‰ฅmax๐’ฏ1,โˆš๐’ฏ2๎‚‡,(3.10) where ๐’ฏ1=โ€–โ€–๐ป๐ต๐‘‡โ€–โ€–โˆ’โˆš๐œ†min(๐ป)/โ„Žโ‹…๐œ†min๎‚€๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚โˆ’โ€–โ€–2โ„Ž๐‘๐‘›ฬƒ๐‘โ€–โ€–+โˆšโˆ’๐ป๐œ†min(๐ป)โ„Žโ‹…๐‘๐‘›๐‘›๐‘›,๐’ฏ2=โ€–โ€–๐ป๐ต๐‘‡โ€–โ€–+โˆš๐œ†min(๐ป)/โ„Žโ‹…๐œ†min๎‚€๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚โ€–โ€–2โ„Ž๐‘๐‘›ฬƒ๐‘โ€–โ€–+โˆšโˆ’๐ป๐œ†min(๐ป)โ„Žโ‹…๐‘๐‘›๐‘›๐‘›.(3.11) Then the set ๐‘ฅ๐พโˆถ=๎‚†๎‚€๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚โˆถโˆšโ„Ž๐‘ฅ๐‘›>๎”๐‘ฅ๐‘‡(๐‘›โˆ’1)๐ป๐‘ฅ(๐‘›โˆ’1)๎‚‡(3.12) is a global cone of instability for the system (1.5).

Proof. First we make auxiliary computations. For the reader's convenience, we recall that, for two (๐‘›โˆ’1)ร—(๐‘›โˆ’1) matrices ๐’œ, ๐’œ1, two 1ร—(๐‘›โˆ’1) vectors โ„“, โ„“1, two (๐‘›โˆ’1)ร—1 vectors ๐’ž, ๐’ž1 and two 1ร—1 โ€œmatrices" ๐‘š, ๐‘š1, the multiplicative rule ๎‚ต๐’œ๐’œ๐’žโ„“๐‘š๎‚ถ๎‚ต1๐’ž1โ„“1๐‘š1๎‚ถ=๎‚ต๐’œ๐’œ1+๐’žโ„“1๐ด๐’ž1+๐’ž๐‘š1โ„“๐’œ1+๐‘šโ„“1โ„“๐’ž1+๐‘š๐‘š1๎‚ถ(3.13) holds. This rule can be modified easily for the case of arbitrary rectangular matrices under the condition that all the products are well defined.
We will rewrite system (1.5) in an equivalent form, suitable for further investigation. With this in mind, we define an (๐‘›โˆ’1)2ร—(๐‘›โˆ’1) matrix ๐‘‹(๐‘›โˆ’1) as ๐‘‹๐‘‡(๐‘›โˆ’1)=๎‚€๐‘‹๐‘‡1(๐‘›โˆ’1),๐‘‹๐‘‡2(๐‘›โˆ’1),โ€ฆ,๐‘‹๐‘‡๐‘›โˆ’1(๐‘›โˆ’1)๎‚,(3.14) where all the elements of the (๐‘›โˆ’1)ร—(๐‘›โˆ’1) matrices ๐‘‹๐‘‡๐‘–(๐‘›โˆ’1), ๐‘–=1,2,โ€ฆ,๐‘›โˆ’1 are equal to zero except the ๐‘–th row, which equals ๐‘ฅ๐‘‡(๐‘›โˆ’1), that is, ๐‘‹๐‘‡๐‘–(๐‘›โˆ’1)=โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘ฅ00โ‹ฏ0โ‹ฏโ‹ฏโ‹ฏโ‹ฏ00โ‹ฏ01๐‘ฅ2โ‹ฏ๐‘ฅ๐‘›โˆ’1โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ 00โ‹ฏ0โ‹ฏโ‹ฏโ‹ฏโ‹ฏ00โ‹ฏ0.(3.15) Moreover, we define 1ร—(๐‘›โˆ’1) vectors ๐‘Œ๐‘–, ๐‘–=1,2,โ€ฆ,๐‘›โˆ’1 with components equal to zero except the ๐‘–th element, which equals ๐‘ฅ๐‘›, that is, ๐‘Œ๐‘–=๎€ท0,โ€ฆ,0,๐‘ฅ๐‘›๎€ธ,0,โ€ฆ,0,(3.16) and (๐‘›โˆ’1)ร—(๐‘›โˆ’1) zero matrix ฮ˜.
It is easy to see that matrices ๐‘‹๐‘‡ and ๐ต in (1.5) can be expressed as ๐‘‹๐‘‡=๎ƒฉ๐‘‹๐‘‡1(๐‘›โˆ’1)๐‘Œ๐‘‡1โ‹ฏ๐‘‹๐‘‡๐‘›โˆ’1(๐‘›โˆ’1)๐‘Œ๐‘‡๐‘›โˆ’1๐œƒฮ˜๐œƒ๐‘‡0โ‹ฏ๐œƒ๐‘‡0๐‘ฅ๐‘‡(๐‘›โˆ’1)๐‘ฅ๐‘›๎ƒช,โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐ต๐ต=01๐‘1๐‘๐‘‡1๐‘1๐‘›๐‘›๐ตโ‹ฏโ‹ฏ0๐‘›๐‘๐‘›๐‘๐‘‡๐‘›๐‘๐‘›๐‘›๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ .(3.17) Now we are able to rewrite the system (1.5) under the above assumption regarding the representation of the matrix ๐ด in the form (1.10) in an equivalent form ๎‚ตฬ‡๐‘ฅ(๐‘›โˆ’1)ฬ‡๐‘ฅ๐‘›๎‚ถ=๎‚ต๐ด0๐œƒ๐œƒ๐‘‡๐œ†๐‘ฅ๎‚ถ๎‚ต(๐‘›โˆ’1)๐‘ฅ๐‘›๎‚ถ+๎ƒฉ๐‘‹๐‘‡1(๐‘›โˆ’1)๐‘Œ๐‘‡1โ‹ฏ๐‘‹๐‘‡๐‘›โˆ’1(๐‘›โˆ’1)๐‘Œ๐‘‡๐‘›โˆ’1๐œƒฮ˜๐œƒ๐‘‡0โ‹ฏ๐œƒ๐‘‡0๐‘ฅ๐‘‡(๐‘›โˆ’1)๐‘ฅ๐‘›๎ƒชร—โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐ต01๐‘1๐‘T1๐‘1๐‘›๐‘›๐ตโ‹ฏโ‹ฏ0๐‘›๐‘๐‘›๐‘๐‘‡๐‘›๐‘๐‘›๐‘›๐‘›โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ๎‚ต๐‘ฅ(๐‘›โˆ’1)๐‘ฅ๐‘›๎‚ถ.(3.18) Finally, since the equalities ๐‘›โˆ’1๎“๐‘—=1๐‘‹๐‘‡๐‘—(๐‘›โˆ’1)๐ต0๐‘—=๐ต๐‘‡๐‘‹(๐‘›โˆ’1),๐‘›โˆ’1๎“๐‘—=1๐‘Œ๐‘‡๐‘—๐‘๐‘‡๐‘—=๎‚๐ต๐‘ฅ๐‘›,๐‘›โˆ’1๎“๐‘—=1๐‘‹๐‘‡๐‘—(๐‘›โˆ’1)๐‘๐‘—=๎‚๐ต๐‘ฅ(๐‘›โˆ’1),๐‘›โˆ’1๎“๐‘—=1๐‘Œ๐‘‡๐‘—๐‘๐‘—๐‘›๐‘›=ฬƒ๐‘๐‘ฅ๐‘›(3.19) can be verified easily using (3.13), we have ๎‚ตฬ‡๐‘ฅ(๐‘›โˆ’1)ฬ‡๐‘ฅ๐‘›๎‚ถ=โŽ›โŽœโŽœโŽ๐ด0+๐‘Ÿ11๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐‘Ÿ12๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐‘Ÿ21๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐œ†+๐‘Ÿ22๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚โŽžโŽŸโŽŸโŽ ๎‚ต๐‘ฅ(๐‘›โˆ’1)๐‘ฅ๐‘›๎‚ถ,(3.20) where ๐‘Ÿ11๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚=๐‘›โˆ’1๎“๐‘—=1๎‚ƒ๐‘‹๐‘‡๐‘—(๐‘›โˆ’1)๐ต0๐‘—+๐‘Œ๐‘‡๐‘—๐‘๐‘‡๐‘—๎‚„=๐ต๐‘‡๐‘‹(๐‘›โˆ’1)+๎‚๐ต๐‘ฅ๐‘›,๐‘Ÿ12๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚=๐‘›โˆ’1๎“๐‘—=1๎‚ƒ๐‘‹๐‘‡๐‘—(๐‘›โˆ’1)๐‘๐‘—+๐‘Œ๐‘‡๐‘—๐‘๐‘—๐‘›๐‘›๎‚„=๎‚๐ต๐‘ฅ(๐‘›โˆ’1)+ฬƒ๐‘๐‘ฅ๐‘›,๐‘Ÿ21๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚=๐‘ฅ๐‘‡(๐‘›โˆ’1)๐ต0๐‘›+๐‘ฅ๐‘›๐‘๐‘‡๐‘›,๐‘Ÿ22๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚=๐‘ฅ๐‘‡(๐‘›โˆ’1)๐‘๐‘›+๐‘ฅ๐‘›๐‘๐‘›๐‘›๐‘›.(3.21) The remaining part of the proof is based on Theorem 2.6 with a Chetaev-type function ๐‘‰=๐‘†โ‹…๐‘ and with suitable functions ๐‘† and ๐‘. Such functions we define as ๐‘‰๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚=๎€ท๐‘ฅ๐‘‡(๐‘›โˆ’1)๐‘ฅ๐‘›๎€ธ๎‚ต๐œƒโˆ’๐ป๐œƒ๐‘‡โ„Ž๐‘ฅ๎‚ถ๎‚ต(๐‘›โˆ’1)๐‘ฅ๐‘›๎‚ถ,(3.22) that is, ๐‘‰๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚=โˆ’๐‘ฅ๐‘‡(๐‘›โˆ’1)๐ป๐‘ฅ(๐‘›โˆ’1)+โ„Ž๐‘ฅ2๐‘›,๐‘†๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚=๎”๐‘ฅ๐‘‡(๐‘›โˆ’1)๐ป๐‘ฅ(๐‘›โˆ’1)+โˆšโ„Ž๐‘ฅ๐‘›,๐‘๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๎”=โˆ’๐‘ฅ๐‘‡(๐‘›โˆ’1)๐ป๐‘ฅ(๐‘›โˆ’1)+โˆšโ„Ž๐‘ฅ๐‘›.(3.23) We will verify the necessary properties. Obviously, ๐‘‰=๐‘†โ‹…๐‘, the set ๐พ1๐‘ฅโˆถ=๎‚†๎‚€๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚โˆˆโ„๐‘›๎€ท๐‘ฅโˆถ๐‘(๐‘›โˆ’1),๐‘ฅ๐‘›๎€ธ๎‚‡=๐‘ฅ>0๎‚†๎‚€๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚โˆˆโ„๐‘›โˆถโˆšโ„Ž๐‘ฅ๐‘›>๎”๐‘ฅ๐‘‡(๐‘›โˆ’1)๐ป๐‘ฅ(๐‘›โˆ’1)๎‚‡(3.24) is a cone and ๐‘†(๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›)>0 for every (๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›)โˆˆ๐พ1.
The full derivative of ๐‘‰ (in the form (3.22)) along the trajectories of the system (1.5) (we use its transformed form (3.20)) equalsฬ‡๐‘‰๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚=๎€ทฬ‡๐‘ฅ๐‘‡(๐‘›โˆ’1)ฬ‡๐‘ฅ๐‘›๎€ธ๎‚ต๐œƒโˆ’๐ป๐œƒ๐‘‡โ„Ž๐‘ฅ๎‚ถ๎‚ต(๐‘›โˆ’1)๐‘ฅ๐‘›๎‚ถ+๎€ท๐‘ฅ๐‘‡(๐‘›โˆ’1)๐‘ฅ๐‘›๎€ธ๎‚ต๐œƒโˆ’๐ป๐œƒ๐‘‡โ„Ž๎‚ถ๎‚ตฬ‡๐‘ฅ(๐‘›โˆ’1)ฬ‡๐‘ฅ๐‘›๎‚ถ=๎€ท๐‘ฅ๐‘‡(๐‘›โˆ’1)๐‘ฅ๐‘›๎€ธโŽ›โŽœโŽœโŽ๐ด๐‘‡0+๐‘Ÿ๐‘‡11๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐‘Ÿ๐‘‡21๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐‘Ÿ๐‘‡12๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐œ†+๐‘Ÿ22๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚โŽžโŽŸโŽŸโŽ ๎‚ต๐œƒโˆ’๐ป๐œƒ๐‘‡โ„Ž๐‘ฅ๎‚ถ๎‚ต(๐‘›โˆ’1)๐‘ฅ๐‘›๎‚ถ+๎€ท๐‘ฅ๐‘‡(๐‘›โˆ’1)๐‘ฅ๐‘›๎€ธ๎‚ต๐œƒโˆ’๐ป๐œƒ๐‘‡โ„Ž๎‚ถโŽ›โŽœโŽœโŽ๐ด0+๐‘Ÿ11๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐‘Ÿ12๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐‘Ÿ21๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐œ†+๐‘Ÿ22๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚โŽžโŽŸโŽŸโŽ ร—๎‚ต๐‘ฅ(๐‘›โˆ’1)๐‘ฅ๐‘›๎‚ถ.(3.25) Using formula (3.13), we get ฬ‡๐‘‰๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚=๎€ท๐‘ฅ๐‘‡(๐‘›โˆ’1)๐‘ฅ๐‘›๎€ธโŽ›โŽœโŽœโŽ๐‘11๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐‘12๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐‘21๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐‘22๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚โŽžโŽŸโŽŸโŽ ๎‚ต๐‘ฅ(๐‘›โˆ’1)๐‘ฅ๐‘›๎‚ถ,(3.26) where ๐‘11๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๎‚ƒ๐ด=โˆ’0+๐‘Ÿ11(๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›)๎‚„๐‘‡๎‚ƒ๐ด๐ปโˆ’๐ป0+๐‘Ÿ11๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›,๐‘๎‚๎‚„12๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚=โ„Ž๐‘Ÿ๐‘‡21๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚โˆ’๐ป๐‘Ÿ12๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚,๐‘21๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚=โ„Ž๐‘Ÿ21๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚โˆ’๐‘Ÿ๐‘‡12๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐ป=๐‘๐‘‡12๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚,๐‘22๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๎‚ƒ=2โ„Ž๐œ†+๐‘Ÿ22๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›.๎‚๎‚„(3.27) We reduce these formulas using (3.21). Then, ๐‘11๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๎€ท๐ด=โˆ’๐‘‡0๐ป+๐ป๐ด0๎€ธโˆ’๎‚€๐ต๐‘‡๐‘‹(๐‘›โˆ’1)+๎‚๐ต๐‘ฅ๐‘›๎‚๐‘‡๎‚€๐ปโˆ’๐ป๐ต๐‘‡๐‘‹(๐‘›โˆ’1)+๎‚๐ต๐‘ฅ๐‘›๎‚,๐‘12๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๎‚€๐‘ฅ=โ„Ž๐‘‡(๐‘›โˆ’1)๐ต0๐‘›+๐‘ฅ๐‘›๐‘๐‘‡๐‘›๎‚๐‘‡๎‚€๎‚โˆ’๐ป๐ต๐‘ฅ(๐‘›โˆ’1)+ฬƒ๐‘๐‘ฅ๐‘›๎‚,๐‘21๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๎‚€๐‘ฅ=โ„Ž๐‘‡(๐‘›โˆ’1)๐ต0๐‘›+๐‘ฅ๐‘›๐‘๐‘‡๐‘›๎‚โˆ’๎‚€๎‚๐ต๐‘ฅ(๐‘›โˆ’1)+ฬƒ๐‘๐‘ฅ๐‘›๎‚๐‘‡๐‘๐ป,22๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๎‚€=2โ„Ž๐œ†+๐‘ฅ๐‘‡(๐‘›โˆ’1)๐‘๐‘›+๐‘ฅ๐‘›๐‘๐‘›๐‘›๐‘›๎‚.(3.28) The derivative (3.26) turns into ฬ‡๐‘‰๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚=๐‘ฅ๐‘‡(๐‘›โˆ’1)๐‘11๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐‘ฅ(๐‘›โˆ’1)+๐‘ฅ๐‘‡(๐‘›โˆ’1)๐‘12๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐‘ฅ๐‘›+๐‘ฅ๐‘›๐‘21๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐‘ฅ(๐‘›โˆ’1)+๐‘ฅ๐‘›๐‘22๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚๐‘ฅ๐‘›=๐‘ฅ๐‘‡(๐‘›โˆ’1)๎‚ธโˆ’๎€ท๐ด๐‘‡0๐ป+๐ป๐ด0๎€ธโˆ’๎‚€๐ต๐‘‡๐‘‹(๐‘›โˆ’1)+๎‚๐ต๐‘ฅ๐‘›๎‚๐‘‡๎‚€๐ปโˆ’๐ป๐ต๐‘‡๐‘‹(๐‘›โˆ’1)+๎‚๐ต๐‘ฅ๐‘›๎‚๎‚น๐‘ฅ(๐‘›โˆ’1)+๐‘ฅ๐‘‡(๐‘›โˆ’1)๎‚ธโ„Ž๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1)๐ต0๐‘›+๐‘ฅ๐‘›๐‘๐‘‡๐‘›๎‚๐‘‡๎‚€๎‚โˆ’๐ป๐ต๐‘ฅ(๐‘›โˆ’1)+ฬƒ๐‘๐‘ฅ๐‘›๎‚๎‚น๐‘ฅ๐‘›+๐‘ฅ๐‘›๎‚ธโ„Ž๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1)๐ต0๐‘›+๐‘ฅ๐‘›๐‘๐‘‡๐‘›๎‚โˆ’๎‚€๎‚๐ต๐‘ฅ(๐‘›โˆ’1)+ฬƒ๐‘๐‘ฅ๐‘›๎‚๐‘‡๐ป๎‚น๐‘ฅ(๐‘›โˆ’1)+๐‘ฅ๐‘›๎‚ƒ๎‚€2โ„Ž๐œ†+๐‘ฅ๐‘‡(๐‘›โˆ’1)๐‘๐‘›+๐‘ฅ๐‘›๐‘๐‘›๐‘›๐‘›๐‘ฅ๎‚๎‚„๐‘›=โˆ’๐‘ฅ๐‘‡(๐‘›โˆ’1)๎€ท๐ด๐‘‡0๐ป+๐ป๐ด0๎€ธ๐‘ฅ(๐‘›โˆ’1)+2โ„Ž๐œ†๐‘ฅ2๐‘›โˆ’๐‘ฅ๐‘‡(๐‘›โˆ’1)๎‚ต๎‚€๐ต๐‘‡๐‘‹(๐‘›โˆ’1)๎‚๐‘‡๐ป+๐ป๐ต๐‘‡๐‘‹(๐‘›โˆ’1)๎‚ถ๐‘ฅ(๐‘›โˆ’1)โˆ’๐‘ฅ๐‘‡(๐‘›โˆ’1)๎‚ต๎‚€๎‚๐ต๐‘ฅ๐‘›๎‚๐‘‡๎‚๐ป+๐ป๐ต๐‘ฅ๐‘›๎‚ถ๐‘ฅ(๐‘›โˆ’1)+๐‘ฅ๐‘‡(๐‘›โˆ’1)๎‚€๎€ท๐ต2โ„Ž0๐‘›๎€ธ๐‘‡๎‚๎‚๎‚๐‘ฅโˆ’๐ป๐ตโˆ’๐ต๐ป(๐‘›โˆ’1)๐‘ฅ๐‘›+2๐‘ฅ๐‘‡(๐‘›โˆ’1)๎€ทโ„Ž๐‘๐‘›ฬƒ๐‘๎€ธ๐‘ฅโˆ’๐ป2๐‘›๎‚€๐‘ฅ+2โ„Ž๐‘‡(๐‘›โˆ’1)๐‘๐‘›+๐‘ฅ๐‘›๐‘๐‘›๐‘›๐‘›๎‚๐‘ฅ2๐‘›.(3.29) Finally, using (3.6), we get ฬ‡๐‘‰๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚=๐‘ฅ๐‘‡(๐‘›โˆ’1)๐ถ๐‘ฅ(๐‘›โˆ’1)+2โ„Ž๐œ†๐‘ฅ2๐‘›โˆ’2๐‘ฅ๐‘‡(๐‘›โˆ’1)๐ป๐ต๐‘‡๐‘‹(๐‘›โˆ’1)๐‘ฅ(๐‘›โˆ’1)+2๐‘ฅ๐‘‡(๐‘›โˆ’1)๎‚ƒ๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚„๐‘ฅ(๐‘›โˆ’1)๐‘ฅ๐‘›+2๐‘ฅ๐‘‡(๐‘›โˆ’1)๎€ท2โ„Ž๐‘๐‘›ฬƒ๐‘๎€ธ๐‘ฅโˆ’๐ป2๐‘›+2โ„Ž๐‘๐‘›๐‘›๐‘›๐‘ฅ3๐‘›.(3.30) Let us find the conditions for the positivity of ฬ‡๐‘‰(๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›) in the cone ๐พ1. We use (3.30). If (๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›)โˆˆ๐พ1, then ๐‘ฅ๐‘›โ‰ฅ0 and ฬ‡๐‘‰๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚โ‰ฅ๐‘ฅ๐‘‡(๐‘›โˆ’1)๐ถ๐‘ฅ(๐‘›โˆ’1)+2โ„Ž๐œ†๐‘ฅ2๐‘›โ€–โ€–๐ปโˆ’2๐ต๐‘‡โ€–โ€–โ‹…โ€–โ€–๐‘ฅ(๐‘›โˆ’1)โ€–โ€–3+2๐œ†min๎‚€๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚โ‹…โ€–โ€–๐‘ฅ(๐‘›โˆ’1)โ€–โ€–2โ‹…๐‘ฅ๐‘›โ€–โ€–โˆ’22โ„Ž๐‘๐‘›ฬƒ๐‘โ€–โ€–โ‹…โ€–โ€–๐‘ฅโˆ’๐ป(๐‘›โˆ’1)โ€–โ€–โ‹…๐‘ฅ2๐‘›+2โ„Ž๐‘๐‘›๐‘›๐‘›๐‘ฅ3๐‘›.(3.31) We set โ€–โ€–๐ป๐‘Ž=โˆ’2๐ต๐‘‡โ€–โ€–,๐‘=2๐œ†min๎‚€๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚,โ€–โ€–๐‘=โˆ’22โ„Ž๐‘๐‘›ฬƒ๐‘โ€–โ€–,โˆ’๐ป๐‘‘=2โ„Ž๐‘๐‘›๐‘›๐‘›.(3.32) If ๐‘Žโ€–โ€–๐‘ฅ(๐‘›โˆ’1)โ€–โ€–3โ€–โ€–๐‘ฅ+๐‘(๐‘›โˆ’1)โ€–โ€–2โ‹…๐‘ฅ๐‘›โ€–โ€–๐‘ฅ+๐‘(๐‘›โˆ’1)โ€–โ€–โ‹…๐‘ฅ2๐‘›+๐‘‘๐‘ฅ3๐‘›>0(3.33) in ๐พ1, then ฬ‡๐‘‰(๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›)>0 since ๐ถ is a positive definite matrix and ๐‘ฅ๐‘‡(๐‘›โˆ’1)๐ถ๐‘ฅ(๐‘›โˆ’1)+2โ„Ž๐œ†๐‘ฅ2๐‘›โ‰ฅ๐œ†minโ€–โ€–๐‘ฅ(๐ถ)(๐‘›โˆ’1)โ€–โ€–2+2โ„Ž๐œ†๐‘ฅ2๐‘›>0.(3.34) If (๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›)โˆˆ๐พ1, then ๐‘ฅ๐‘›>๎ƒŽ๐‘ฅ๐‘‡(๐‘›โˆ’1)๐ป๐‘ฅ(๐‘›โˆ’1)โ„Žโ‰ฅ๎‚™๐œ†min(๐ป)โ„Žโ‹…โ€–โ€–๐‘ฅ(๐‘›โˆ’1)โ€–โ€–,๐พ(3.35)1โŠ‚๐’ฆโˆ—๎ƒฏ๎‚€๐‘ฅโˆถ=๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚โˆˆโ„๐‘›โˆถ๐‘ฅ๐‘›>๎‚™๐œ†min(๐ป)โ„Žโ‹…โ€–โ€–๐‘ฅ(๐‘›โˆ’1)โ€–โ€–๎ƒฐ.(3.36) Now, we use Lemma 2.8 with ๐’ฆ=๐’ฆโˆ—, ๐‘ฆ=๐‘ฅ๐‘›, ๐‘ฅ=โ€–๐‘ฅ(๐‘›โˆ’1)โ€–, with coefficients ๐‘Ž, ๐‘, ๐‘, and ๐‘‘ defined by formula (3.32) and with โˆš๐‘˜โˆถ=๐œ†min(๐ป)/โ„Ž.
Obviously |๐‘|โ‰ค๐‘˜๐‘‘ because, due to (3.7), inequality โ€–โ€–2โ„Ž๐‘๐‘›ฬƒ๐‘โ€–โ€–โ‰คโˆšโˆ’๐ป๐œ†min(๐ป)โ„Žโ‹…๐‘๐‘›๐‘›๐‘›(3.37) holds. Moreover, |๐‘Ž|โ‰ค๐‘˜๐‘ if (3.8) holds, that is, if โ€–โ€–๐ป๐ต๐‘‡โ€–โ€–โ‰ค๎‚™๐œ†min(๐ป)โ„Žโ‹…๐œ†min๎‚€๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚.(3.38) Further, |๐‘Ž|>๐‘˜๐‘ if (3.9) holds, that is, if โ€–โ€–๐ป๐ต๐‘‡โ€–โ€–>๎‚™๐œ†min(๐ป)โ„Žโ‹…๐œ†min๎‚€๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๐ต๐ป+โ„Ž๐‘›๎€ท๐ต0๐‘›๎€ธ๐‘‡๎‚,(3.39) and (2.22) holds due to (4.10) and the condition |๐‘|โ‰ ๐‘˜๐‘‘. Thus the assumptions of Lemma 2.8 are true, the inequality (3.33) holds in the cone ๐’ฆโˆ— and, due to embedding (3.36), in the cone ๐พ1 as well.
All the assumptions of Theorem 2.6 are fulfilled with regard to system (1.5) and the theorem is proved, because ๐พ1=๐พ.

Remark 3.2. We will focus our attention to Lemma 2.8 about the positivity of a third-degree polynomial in two variables in the cone ๐’ฆ. We used it to estimate the derivative ฬ‡๐‘‰ expressed by formula (3.30). Obviously, there are other possibilities of estimating its sign. Let us demonstrate one of them. Let us, for example, estimate the right-hand side of (3.31) in the cone ๐พ1 using inequality (3.35), then ฬ‡๐‘‰๎‚€๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›๎‚โ‰ฅ๐‘ฅ๐‘‡(๐‘›โˆ’1)๐ถ๐‘ฅ(๐‘›โˆ’1)+2โ„Ž๐œ†๐‘ฅ2๐‘›โ€–โ€–๐ปโˆ’2๐ต๐‘‡โ€–โ€–โ‹…โ€–โ€–๐‘ฅ(๐‘›โˆ’1)โ€–โ€–3+2๐œ†min๎‚€๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚โ‹…โ€–โ€–๐‘ฅ(๐‘›โˆ’1)โ€–โ€–2โ‹…๐‘ฅ๐‘›โ€–โ€–โˆ’22โ„Ž๐‘๐‘›ฬƒ๐‘โ€–โ€–โ‹…โ€–โ€–๐‘ฅโˆ’๐ป(๐‘›โˆ’1)โ€–โ€–โ‹…๐‘ฅ2๐‘›+2โ„Ž๐‘๐‘›๐‘›๐‘›๐‘ฅ3๐‘›โ‰ฅ๐œ†minโ€–โ€–๐‘ฅ(๐ถ)(๐‘›โˆ’1)โ€–โ€–2+2โ„Ž๐œ†๐‘ฅ2๐‘›โ€–โ€–๐ปโˆ’2๐ต๐‘‡โ€–โ€–โ‹…โ€–โ€–๐‘ฅ(๐‘›โˆ’1)โ€–โ€–3๎‚™+2๐œ†min(๐ป)โ„Žโ‹…๐œ†min๎‚€๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚โ‹…โ€–โ€–๐‘ฅ(๐‘›โˆ’1)โ€–โ€–3โ€–โ€–โˆ’22โ„Ž๐‘๐‘›ฬƒ๐‘โ€–โ€–โ‹…โ€–โ€–๐‘ฅโˆ’๐ป(๐‘›โˆ’1)โ€–โ€–โ‹…๐‘ฅ2๐‘›๎‚™+2๐œ†min(๐ป)โ„Žโ‹…โ€–โ€–๐‘ฅ(๐‘›โˆ’1)โ€–โ€–โ‹…โ„Ž๐‘๐‘›๐‘›๐‘›โ‹…๐‘ฅ2๐‘›,(3.40) and the positivity of ฬ‡๐‘‰(๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›) will be guaranteed if โ€–โ€–๐ป๐ต๐‘‡โ€–โ€–โ‰ค๎‚™๐œ†min(๐ป)โ„Žโ‹…๐œ†min๎‚€๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚,โ€–โ€–2โ„Ž๐‘๐‘›ฬƒ๐‘โ€–โ€–โ‰คโˆšโˆ’๐ป๐œ†min(๐ป)โ„Žโ‹…๐‘๐‘›๐‘›๐‘›.(3.41) We see that this approach produces only one set of inequalities for the positivity of ฬ‡๐‘‰(๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›), namely the case when (3.7) and (3.8) holds. Unfortunately, using such approach, we are not able to detect the second case (3.7) and (3.9) when ฬ‡๐‘‰(๐‘ฅ๐‘‡(๐‘›โˆ’1),๐‘ฅ๐‘›) is positive. This demonstrates the advantage of detailed estimates using the above third-degree polynomial in two variables.

4. Planar Case

Now we consider a particular case of the system (1.5) for ๐‘›=2. This means that, in accordance with (1.5) and (1.10), we consider a systemฬ‡๐‘ฅ1(๐‘ก)=๐‘Ž๐‘ฅ1(๐‘ก)+๐‘111๐‘ฅ21(๐‘ก)+2๐‘112๐‘ฅ1(๐‘ก)๐‘ฅ2(๐‘ก)+๐‘122๐‘ฅ22(๐‘ก),ฬ‡๐‘ฅ2(๐‘ก)=๐œ†๐‘ฅ2(๐‘ก)+๐‘211๐‘ฅ21(๐‘ก)+2๐‘212๐‘ฅ1(๐‘ก)๐‘ฅ2(๐‘ก)+๐‘222๐‘ฅ22(๐‘ก),(4.1) where ๐‘Ž<0 and ๐œ†>0. The solution of matrix equation (3.6) for ๐ด0=(๐‘Ž), ๐ป=(โ„Ž11), and ๐ถ=(๐‘) with ๐‘>0, that is,๎€ท๐‘Žโ„Ž11๎€ธ+๎€ทโ„Ž11๐‘Ž๎€ธ=โˆ’(๐‘)(4.2) gives ๎€ทโ„Ž๐ป=11๎€ธ=๎‚€โˆ’๐‘๎‚,2๐‘Ž(4.3) with โ„Ž11=โˆ’๐‘/2๐‘Ž>0. The set ๐พ defined by (3.12) where โ„Ž>0 and ๐‘ฅ(๐‘›โˆ’1)=๐‘ฅ1 reduces to๎‚ป๎€ท๐‘ฅ๐พ=1,๐‘ฅ2๎€ธโˆถ๐‘ฅ2>๎‚™๐‘โ‹…||๐‘ฅ2|๐‘Ž|โ„Ž1||๎‚ผ.(4.4) Now, from Theorem 3.1, we will deduce sufficient conditions indicating ๐พ being a global cone of instability for system (4.1). In our particular case, we have ๐‘๐‘–=๎€ท๐‘๐‘–12๎€ธฬƒ๎€ท๐‘,๐‘–=1,2,๐‘=122๎€ธ,๐ต0๐‘–=๎€ท๐‘๐‘–11๎€ธ๎‚๎€ท๐‘,๐‘–=1,2,๐ต=112๎€ธ,๐ต๐‘‡=๎€ท๐‘111๎€ธ=๐ต01.(4.5) Now, we compute all necessary expressions used in Theorem 3.1. We have ๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚€โˆ’๐‘=โˆ’๎‚๎€ท๐‘2๐‘Ž112๎€ธโˆ’๎€ท๐‘112๎€ธ๎‚€โˆ’๐‘๎‚๎€ท๐‘2๐‘Ž+โ„Ž211๎€ธ=๎‚ตโ„Ž๐‘211โˆ’๐‘๐‘|๐‘Ž|112๎‚ถ,โ€–โ€–2โ„Ž๐‘๐‘›ฬƒ๐‘โ€–โ€–=||||โˆ’๐ป2โ„Ž๐‘212โˆ’๐‘๐‘2|๐‘Ž|122||||,โˆš๐œ†min๎‚™(๐ป)โ„Ž=๐‘โ„Ž,๎‚™2|๐‘Ž|๐œ†min(๐ป)โ„Ž=๎‚™๐‘,โ€–โ€–๐ป2|๐‘Ž|โ„Ž๐ต๐‘‡โ€–โ€–=||||๐‘๐‘2|๐‘Ž|111||||=๐‘||๐‘2|๐‘Ž|111||,๐œ†min๎‚€๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚=โ„Ž๐‘211โˆ’๐‘๐‘|๐‘Ž|112,๐’ฏ1=โ€–โ€–๐ป๐ต๐‘‡โ€–โ€–โˆ’โˆš๐œ†min(๐ป)/โ„Žโ‹…๐œ†min๎‚€๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚โˆ’โ€–โ€–2โ„Ž๐‘๐‘›ฬƒ๐‘โ€–โ€–+โˆšโˆ’๐ป๐œ†min(๐ป)โ„Žโ‹…๐‘๐‘›๐‘›๐‘›=(||๐‘๐‘/2|๐‘Ž|)111||โˆ’โˆš๎€ท๐‘/2|๐‘Ž|โ„Žโ‹…โ„Ž๐‘211โˆ’(๐‘/|๐‘Ž|)๐‘112๎€ธโˆ’||2โ„Ž๐‘212โˆ’(๐‘/2|๐‘Ž|)๐‘122||+โˆš๐‘โ„Ž/2|๐‘Ž|โ‹…๐‘222,๐’ฏ2=โ€–โ€–๐ป๐ต๐‘‡โ€–โ€–+โˆš๐œ†min(๐ป)/โ„Žโ‹…๐œ†min๎‚€๎‚๐ตโˆ’๐ป๐‘‡โˆ’๎‚๎€ท๐ต๐ต๐ป+โ„Ž0๐‘›๎€ธ๐‘‡๎‚โ€–โ€–2โ„Ž๐‘๐‘›ฬƒ๐‘โ€–โ€–+โˆšโˆ’๐ป๐œ†min(๐ป)โ„Žโ‹…๐‘๐‘›๐‘›๐‘›=(||๐‘๐‘/2|๐‘Ž|)111||+โˆš(๎€ท๐‘/2|๐‘Ž|โ„Ž)โ‹…โ„Ž๐‘211โˆ’(๐‘/|๐‘Ž|)๐‘112๎€ธ||2โ„Ž๐‘212โˆ’(๐‘/2|๐‘Ž|)๐‘122||+โˆš(๐‘โ„Ž/2|๐‘Ž|)โ‹…๐‘222.(4.6)

Theorem 4.1 (Planar Case). Assume that ๐‘Ž<0, ๐‘222>0, โ„Ž>0, ๐‘>0 and โ„Ž๐‘211|๐‘Ž|>๐‘๐‘112. Let ||||2โ„Ž๐‘212โˆ’๐‘๐‘2|๐‘Ž|122||||โ‰ค๎‚™๐‘โ„Ž2|๐‘Ž|โ‹…๐‘222,(4.7) and, in addition, one of the following conditions is valid:
either ๐‘||๐‘2|๐‘Ž|111||โ‰ค๎‚™๐‘โ‹…๎‚ต2|๐‘Ž|โ„Žโ„Ž๐‘211โˆ’๐‘๐‘|๐‘Ž|112๎‚ถ(4.8) or ๐‘||๐‘2|๐‘Ž|111||>๎‚™๐‘โ‹…๎‚ต2|๐‘Ž|โ„Žโ„Ž๐‘211โˆ’๐‘๐‘|๐‘Ž|112๎‚ถ,(4.9) strong inequality holds in (4.7), and ๎‚™๐‘๎‚†โˆš2|๐‘Ž|โ„Žโ‰ฅmax๐’ฏ1,โˆš๐’ฏ2๎‚‡,(4.10) where ๐’ฏ1 and ๐’ฏ1 are defined by (4.6). Then the set ๐พ defined by (4.4) is a global cone of instability for the system (4.1).

It is easy to see that the choice โ„Ž=1, ๐‘=|๐‘Ž| significantly simplifies all assumptions. Therefore we give such a particular case of Theorem 4.1.

Corollary 4.2 (Planar Case). Assume that ๐‘Ž<0, ๐‘222>0 and ๐‘211>๐‘112. Let |||2๐‘212โˆ’12๐‘122|||โ‰ค1โˆš2โ‹…๐‘222,(4.11) and, in addition, one of the following conditions is valid:
either 12||๐‘111||โ‰ค1โˆš2โ‹…๎€ท๐‘211โˆ’๐‘112๎€ธ(4.12) or 12||๐‘111||>1โˆš2โ‹…๎€ท๐‘211โˆ’๐‘112๎€ธ,(4.13) strong inequality holds in (4.11), and 1โˆš2๎‚†โˆšโ‰ฅmax๐’ฏ1,โˆš๐’ฏ2๎‚‡,(4.14) where ๐’ฏ1=||๐‘(1/2)111||โˆ’๎‚€โˆš1/2๎‚โ‹…๎€ท๐‘211โˆ’๐‘112๎€ธโˆ’||2๐‘212โˆ’(1/2)๐‘122||+๎‚€โˆš1/2๎‚โ‹…๐‘222,๐’ฏ2=||๐‘(1/2)111||+๎‚€โˆš1/2๎‚โ‹…๎€ท๐‘211โˆ’๐‘112๎€ธ||2๐‘212โˆ’(1/2)๐‘122||+๎‚€โˆš1/2๎‚โ‹…๐‘222,(4.15) Then the set ๎ƒฏ๎€ท๐‘ฅ๐พ=1,๐‘ฅ2๎€ธโˆถ๐‘ฅ2>1โˆš2โ‹…||๐‘ฅ1||๎ƒฐ(4.16) is a global cone of instability for the system (4.1).

Example 4.3. The set ๐พ defined by (4.16) is a global cone of instability for the system ฬ‡๐‘ฅ1(๐‘ก)=๐‘Ž๐‘ฅ1(๐‘ก)+๐‘ฅ21โˆš(๐‘ก)+22๐‘ฅ1(๐‘ก)๐‘ฅ2(๐‘ก)+๐‘ฅ22(๐‘ก),ฬ‡๐‘ฅ2(๐‘ก)=๐œ†๐‘ฅ2โˆš(๐‘ก)+22๐‘ฅ21(๐‘ก)+2๐‘ฅ1(๐‘ก)๐‘ฅ2โˆš(๐‘ก)+22๐‘ฅ22(๐‘ก),(4.17) where ๐‘Ž<0 and ๐œ†>0 since inequalities (4.11) and (4.12) in Corollary 4.2 hold.

Example 4.4. The set ๐พ defined by (4.16) is a global cone of instability for the system ฬ‡๐‘ฅ1(๐‘ก)=๐‘Ž๐‘ฅ1(๐‘ก)+4๐‘ฅ21โˆš(๐‘ก)+22๐‘ฅ1(๐‘ก)๐‘ฅ2(๐‘ก)+๐‘ฅ22(๐‘ก),ฬ‡๐‘ฅ2(๐‘ก)=๐œ†๐‘ฅ2โˆš(๐‘ก)+22๐‘ฅ21(๐‘ก)+2๐‘ฅ1(๐‘ก)๐‘ฅ2โˆš(๐‘ก)+202๐‘ฅ22(๐‘ก),(4.18) where ๐‘Ž<0 and ๐œ†>0 since inequalities (4.11), (4.13), (4.14) in Corollary 4.2 hold.

Acknowledgments

This research was supported by Grants nos. P201/11/0768 and P201/10/1032 of Czech Grant Agency, and by the Council of Czech Government nos. MSM 0021630503, MSM 0021630519, and MSM 0021630529, and by Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication.

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