Volume 2010, Article ID 539087,23pages doi:10.1155/2010/539087

*Research Article*

**Stability of Nonlinear Autonomous Quadratic** **Discrete Systems in the Critical Case**

**Josef Dibl´ık,**

^{1, 2}**Denys Ya. Khusainov,**

^{3}**Irina V. Grytsay,**

^{3}**and Zden ˘ek ˇSmarda**

^{1}*1**Department of Mathematics, Faculty of Electrical Engineering and Communication,*
*Brno University of Technology, Technick´a 8, 616 00 Brno, Czech Republic*

*2**Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering,*
*Brno University of Technology, Veveˇr´ı331/95, 60200 Brno, Czech Republic*

*3**Department of Complex System Modeling, Faculty of Cybernetics, Taras,*

*Shevchenko National University of Kyiv, Vladimirskaya Str., 64, 01033 Kyiv, Ukraine*

Correspondence should be addressed to Josef Dibl´ık,diblik@feec.vutbr.cz Received 28 January 2010; Accepted 11 May 2010

Academic Editor: Elena Braverman

Copyrightq2010 Josef Dibl´ık 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.

Many processes are mathematically simulated by systems of discrete equations with quadratic
right-hand sides. Their stability is thought of as a very important characterization of the process. In
this paper, the method of Lyapunov functions is used to derive classes of stable quadratic discrete
autonomous systems in a critical case in the presence of a simple eigenvalue*λ*1 of the matrix of
linear terms. In addition to the stability investigation, we also estimate stability domains.

**1. Introduction**

The main results on the stability theory of diﬀerence equations are presented, for example, by Agarwal 1, Agarwal et al. 2, Chetaev 3, Elaydi 4, Halanay and R˘asvan 5, Lakshmikantham and Trigiante6, and Martynjuk7. Instability problems are considered, for example, in8–10by Slyusarchuk. Note that stability and instability results often have a local character and are usually obtained without any estimation of the stability domain, or without investigating the character of instability. Moreover, it should be emphasized that global instability questions have only been discussed for linear systems.

Many processes and phenomena are described by diﬀerential or diﬀerence systems with quadratic nonlinearities. Among others, let us mention epidemic and populations models, models of chemical reactions, and models for describing convection currents in the atmosphere.

The stability of a zero solution of diﬀerence systems

*xk*1 *fxk,* 1.1

where*k* 0,1, . . ., and *x* x1*, . . . , x*_{n}* ^{T}* with diﬀerentiable

*f*f1

*, . . . , f*

_{n}*: R*

^{T}*→ R*

^{n}*, is very often investigated by linearly approximating system1.1in question by using the matrix of linear terms*

^{n}*xk*1 *Axk gxk,* 1.2

where*Af*^{}0,0, . . . ,0is the Jacobian matrix of*f*at0,0, . . . ,0, and*gx fx*−*Ax. This*
approach becomes unsuitable in what is called a critical case, that is, when the spectral radius
of the matrix*ρA *1 because, among all systems1.2, there are classes of stable systems
as well as classes of unstable systems. Concerning this, we formulate the following known
resultssee, e.g., Corollary 4.344, page 222and Theorem 4.384, page 226.

**Theorem 1.1.** 1*IfρA<1, then the zero solution of* 1.2*is exponentially stable.*

2*IfρA 1, then the zero solution of* 1.2*may be stable or unstable.*

3*IfρA>1 andgxisoxas*x → *0, then the zero solution of* 1.2*is unstable.*

In this paper, we consider a particular critical case when there exists a simple
eigenvalue *λ* 1 of the matrix of linear terms and the remaining eigenvalues lie inside a
unit circle centered at origin. The purpose of this paper is to obtain using the method of
Lyapunov functionsconditions for the stability of a zero solution of diﬀerence systems with
quadratic nonlinearities in the above case and derive classes of stable systems. In addition to
the stability investigation, we estimate the stability domains as well. The domains of stability
obtained are also called guaranteed domains of stability. Preliminary results in this direction
were published in11.

* 1.1. Quadratic System and Preliminary Consideration*
In the sequel, the norms used for vectors and matrices are defined as

x

_{n}

*i1*

*x*^{2}_{i}_{1/2}

1.3

for a vector*x* x1*, . . . , x**n** ^{T}*and

A

*λ*maxA^{T}*A*_{1/2}

1.4

for any*m×n*matrix*A. Here and in the sequel,λ*_{max}· or*λ*_{min}·is the maximalor minimal
eigenvalue of the corresponding symmetric and positive-semi- definite matrixsee, e.g.,
12.

Consider a nonlinear autonomous discrete system with a quadratic right-hand side

*x** _{i}*k1

^{n}*s1*

*a*_{is}*x** _{s}*k

^{n}*s,q1*

*b*^{i}_{sq}*x** _{s}*kx

*q*k,

*i*1

*. . . , n,*1.5

where*k*0,1, . . .and the coeﬃcients*a**is*and*b*^{i}* _{sq}*we assume that

*b*

^{i}

_{sq}*b*

^{i}*are constant. As emphasized, for example, in3,7,12, system1.5can be written in a general vector-matrix form*

_{qs}*xk*1 *Axk X** ^{T}*kBxk,

*k*0,1, . . . , 1.6 where

a*A*{a*is*}, *i, s*1,2*. . . , n, is ann*×*n*constant square matrix,

bmatrix*X** ^{T}*{X

_{1}

^{T}*, X*

^{T}_{2}

*, . . . , X*

^{T}*}is*

_{n}*n*×

*n*

^{2}rectangular and all the elements of the

*n*×

*n*matrices

*X*

_{i}*,*

^{T}*i*1, . . . , n, are equal to zero except the

*ith row with entriesx*

*x1*

^{T}*, x*

_{2}

*, . . . , x*

*, that is,*

_{n}*X*^{T}* _{i}*k

⎛

⎜⎜

⎜⎜

⎜⎝

0 0 · · · 0

· · · ·
*x*_{1} *x*_{2} · · · *x*_{n}

· · · · 0 0 · · · 0

⎞

⎟⎟

⎟⎟

⎟⎠*,* 1.7

cmatrix *B** ^{T}* {B1

*, B*

_{2}

*, . . . , B*

*}is*

_{n}*n*

^{2}×

*n*rectangular and the

*n*×

*n*constant matrices

*B*

*{b*

_{i}

^{i}*},*

_{sq}*i, s, q*1, . . . , n, are symmetric.

The stability of the zero solution of1.6depends on the stability of the matrix*A. IfρA<*1,
then the zero solution of1.6is exponentially stable for an arbitrary matrix*B*byTheorem 1.1.

In this case, matrix*B*only impacts on the shape of the stability domain of the equilibrium
state. If the zero solution of1.6is investigated on stability by the second Lyapunov method
and an appropriate Lyapunov function is taken as the quadratic form*Vx * *x*^{T}*Hx*with a
suitable*n*×*n*constant real symmetric positive-definite matrix*H, which is defined below,*
then the first diﬀerenceΔV along the trajectories of1.6equals

ΔVxk *Vxk*1−*V*xk *x** ^{T}*k1Hxk1−

*x*

*kHxk*

^{T}*Axk X** ^{T}*kBxk

_{T}*H*

*Axk X** ^{T}*kBxk

−*x** ^{T}*kHxk

*x** ^{T}*kA

^{T}*x*

*kB*

^{T}

^{T}*Xk*

*H*

*Axk X** ^{T}*kBxk

−*x** ^{T}*kHxk

*x*

*k*

^{T}*A*^{T}*HA*−*H*

*A*^{T}*HX** ^{T}*kB

*B*

^{T}*X*kHA

*B*

^{T}*XkHX*

*kB*

^{T}*xk*

*x*

*k*

^{T}*A*^{T}*HA*−*H*

2B^{T}*XkHAB*^{T}*XkHX** ^{T}*kB

*xk*

1.8

sinceA^{T}*HX** ^{T}*kB

^{T}*B*

^{T}*XkHA.*

Since *ρA* *<* 1, for arbitrary positive-definite symmetric matrix *C, the matrix*
Lyapunov equation

*A*^{T}*HA*−*H*−C 1.9

has a unique solution*H—a positive-definite symmetric matrix*e.g.,4, Theorem 4.30, page
216. We use such matrix*H*to estimate the stability domain. Then, as follows from1.8,

ΔVxk≤ −

*λ*_{min}C−2B · HA · xk − B^{2}· *H · xk*^{2}

· *xk*^{2}*.* 1.10

Analysing1.10, we deduce that the first diﬀerenceΔVxkwill be negative definite if
B^{2}· *H · xk*^{2}2B · HA · xk ≤*λ*_{min}C, 1.11
that is, it will be negative definite in a neighborhood*U** _{δ}* {x∈R

*:x*

^{n}*< δ}*of the steady- state

*xk*≡ 0,

*k*0,1, . . . ,if

*δ*is suﬃciently small. In the case considered, the domain of stability can be described by means of two inequalities. The first inequality 1.11defines a part of the spaceR

*, where the first diﬀerenceΔVxkis negative definite. The second inequality*

^{n}*V*x≤*r,* *x*∈R^{n}*, r >*0, 1.12
describes points inside a level surface. The guaranteed domain of stability is given by
inequality1.12if*r* is taken so small that the domain described by1.12is embedded in
the domain described by inequality1.11.

Considering the investigated critical case, we will deal with a diﬀerent structure of
the right-hand side of the inequality from that seen in 1.10. Namely, we will show that,
unlike the right-hand side of the inequality forΔVxkthat is multiplied byxk^{2}with
dim*xk * *n*in1.10, in the critical case considered, the right-hand side of the inequality
or equalityforΔVxkwill be multiplied only by a termx* _{n−1}*k

^{2}with dim

*x*

*k*

_{n−1}*n*−1

*< n*see2.21in the case

*n*2 and2.69in the general case below.

**2. Main Results**

In this section we derive the classes of the stable systems1.6in a critical case when the
matrix*A*has one simple eigenvalue*λ*1.

**2.1. Instability in One-Dimensional Case**

We start by discussing a simple scalar equation with the eigenvalue of matrix*A*equaling one,
that is,*a*_{11} 1. Then1.6takes the form

*xk*1 *xk bx*^{2}k, *k*0, . . . , 2.1

and it is easy to see that the trivial solution is unstable for an arbitrary*b /*0to show this, we
can apply, e.g., Theorem 1.154, page 29.

This elementary example shows that stability in the case of system 1.6 has an
extraordinary significance and the results on stabilityfor*n /*1lose their meaning for*n*1
when we deal with instability. We show that, if*n /*1 and*B*satisfies certain assumptions, the
zero solution is stable. Moreover, the shape of the guaranteed domain of stability will be
given.

We divide our forthcoming analysis into two parts. In the first one we give an explicit
coeﬃcient criterion in the subcase of*n*2. Then we consider the general*n-dimensional case.*

**2.2. Stability in the General Two-Dimensional Case**

Let*n* 2. Then system1.6with the matrix*A*having a simple eigenvalue*λ* 1 reduces
after linearly transforming the dependent variables if necessaryto

*x*_{1}k1 *ax*_{1}k

*b*^{1}_{11}*x*^{2}_{1}k 2b^{1}_{12}*x*_{1}kx2k *b*^{1}_{22}*x*^{2}_{2}k
*,*
*x*_{2}k1 *x*_{2}k

*b*^{2}_{11}*x*^{2}_{1}k 2b^{2}_{12}*x*_{1}kx2k *b*_{22}^{2} *x*^{2}_{2}k
*.*

2.2

We will assume that|a|*<*1. Define auxiliary numbers as follows:

*αh*
1−*a*^{2}

*,* *β*_{1}*hab*^{1}_{11}*,* *β*_{2}2hab^{1}_{12}*b*_{11}^{2} *,*
*γ*_{1} *h*

*b*^{1}_{11}2

*b*^{2}_{11}2

*,* *γ*_{2} 4h
*b*_{12}^{1} 2

*,* *δ*_{1}2hb^{1}_{11}*b*_{12}^{1} *,*

2.3

where*h*is a positive number.

**Theorem 2.1. Let**handr*be positive numbers. Assume that*|a|*<1 andb*^{2}_{12} *b*^{1}_{22} *b*_{22}^{2} *0. Then*
*the zero solution of system*2.2*is stable in the Lyapunov sense and a guaranteed domain of stability*
*is given by the inequality*

*hx*^{2}_{1}*x*_{2}^{2}≤*r*^{2} 2.4

*ifris taken so small that the domain described by*2.4*is embedded in the domain*

*γ*_{1}*x*^{2}_{1}2δ_{1}*x*_{1}*x*_{2}*γ*_{2}*x*^{2}_{2}2β_{1}*x*_{1}2β_{2}*x*_{2} ≤*α.* 2.5

*If, moreover,b*^{2}_{11}·*b*^{1}_{12}*/0, then a guaranteed domain of stability can be described using inequality*

*hx*^{2}_{1}*x*^{2}_{2}≤r^{∗}^{2} 2.6

*with*

*r*^{∗} min

x1*,x*2

*hx*^{2}_{1}*x*^{2}_{2}*,* 2.7

*where*x1*, x*_{2}*runs over all real solutions of the nonlinear system with unknownsx*_{1}*andx*_{2}*:*

*γ*1*x*^{2}_{1}2δ1*x*1*x*2*γ*2*x*^{2}_{2}2β1*x*12β2*x*2*α,*
*hx*_{1}

*δ*_{1}*x*_{1}*γ*_{2}*x*_{2}*β*_{2}

−*x*_{2}

*γ*_{1}*x*_{1}*δ*_{1}*x*_{2}*β*_{1}
0.

2.8

*Proof. Define*

*B*1

*b*_{11}^{1} *b*^{1}_{12}
*b*_{12}^{1} *b*^{1}_{22}

*,* *B*2

*b*^{2}_{11} *b*^{2}_{12}
*b*^{2}_{12} *b*^{2}_{22}

*,* *x*2*y,* *x*
*x*_{1}

*y*

*.* 2.9

We rewrite system2.2as

*x*1k1 *ax*1k *x** ^{T}*kB1

*xk,*

*yk*1 *yk x** ^{T}*kB2

*xk.*2.10

To investigate the stability of the zero solution, we use, in accordance with the direct
Lyapunov method, an appropriate Lyapunov function*V*. Let a matrix*H, defined as*

*H*

*h h*12

*h*12 *h*22

*,* 2.11

where instead of the entry*h*_{11}we put the number*h, be positive definite. We set*

*V*xk *V*

*x*_{1}k, yk

:*x** ^{T}*kHxk

*x*_{1}k, yk
*h h*_{12}
*h*12 *h*22

*x*_{1}k
*yk*

*hx*^{2}_{1}k 2h12*x*1kyk *h*22*y*^{2}k.

2.12

The first diﬀerence of the function*V* along the trajectories of system2.10equals

ΔVxk *V*xk1−*V*xk

*hx*^{2}_{1}k1 2h_{12}*x*_{1}k1yk1 *h*_{22}*y*^{2}k1

−*hx*^{2}_{1}k−2h12*x*1kyk−*h*22*y*^{2}k
*h*

*ax*_{1}k *x** ^{T}*kB1

*xk*

_{2}2h

_{12}

*ax*_{1}k *x** ^{T}*kB1

*xk*

*yk x** ^{T}*kB2

*xk*

*h*22

*yk x** ^{T}*kB2

*xk*2

−*hx*^{2}_{1}k−2h12*x*1kyk−*h*22*y*^{2}k
*h*

*a*^{2}−1

*x*^{2}_{1}k 2h12a−1x1kyk
*h*

2ax_{1}k

*x** ^{T}*kB1

*xk*

*x** ^{T}*kB1

*xk*

_{2}2h

_{12}

*ax*_{1}k

*x** ^{T}*kB2

*xk*

*yk*

*x*^{T}*B*_{1}*xk*

*x** ^{T}*kB1

*xk*

*x** ^{T}*kB2

*xk*

*h*22

2yk

*x** ^{T}*kB2

*xk*

*x** ^{T}*kB2

*xk*

_{2}

*.*

2.13

It is easy to see thatΔV does not preserve the sign if*h*_{12}*/*0. Therefore, we put*h*_{12} 0 and
ΔVreduces to

ΔVxk *h*
*a*^{2}−1

*x*^{2}_{1}k *h*

2ax1k

*x** ^{T}*kB1

*xk*

*x** ^{T}*kB1

*xk*2

*h*

_{22}

2yk

*x** ^{T}*kB2

*xk*

*x** ^{T}*kB2

*xk*

_{2}

*.*

2.14

In the polynomial ΔV, with respect to *x*1 and *y, we will put together the third-degree*
terms the expression *F*_{3}x1k, yk below and the fourth-degree terms the expression
*F*_{4}x1k, ykbelow. In the computations we use the formulas

*x** ^{T}*kB

*i*

*xk b*

^{i}_{11}

*x*

_{1}

^{2}k 2b

^{i}_{12}

*x*

_{1}kyk

*b*

^{i}_{22}

*y*

^{2}k,

*i*1,2,

*x*

*kB*

^{T}*i*

*xk*2

*b*^{i}_{11}2

*x*^{4}_{1}k 4
*b*^{i}_{12}2

*x*^{2}_{1}ky^{2}k
*b*^{i}_{22}2

*y*^{4}k

4b^{i}_{11}*b*^{i}_{12}*x*_{1}^{3}kyk 2b^{i}_{11}*b*^{i}_{22}*x*^{2}_{1}ky^{2}k 4b^{i}_{12}*b*^{i}_{22}*x*1ky^{3}k, *i*1,2.

2.15

We get

ΔVxk *h*
*a*^{2}−1

*x*^{2}_{1}k *F*_{3}

*x*_{1}k, yk
*F*_{4}

*x*_{1}k, yk

*,* 2.16

where

*F*_{3}

*x*_{1}k, yk

2hab^{1}_{11}*x*_{1}^{3}k 2

2hab^{1}_{12}*h*_{22}*b*_{11}^{2}

*x*^{2}_{1}kyk
2

*hab*^{1}_{22}2h_{22}*b*^{2}_{12}

*x*_{1}ky^{2}k 2h_{22}*b*_{22}^{2} *y*^{3}k,
*F*4

*x*1k, yk

*h*

*b*^{1}_{11}_{2}
*h*22

*b*^{2}_{11}_{2}

*x*^{4}_{1}k 4

*hb*^{1}_{11}*b*^{1}_{12}*h*22*b*^{2}_{11}*b*^{2}_{12}

*x*^{3}_{1}kyk
2

2h

*b*^{1}_{12}2

2h_{22}
*b*^{2}_{12}2

*hb*_{11}^{1} *b*^{1}_{22}*h*_{22}*b*^{2}_{11}*b*_{22}^{2}

*x*^{2}_{1}ky^{2}k
4

*hb*^{1}_{12}*b*_{22}^{1} *h*_{22}*b*^{2}_{12}*b*_{22}^{2}

*x*_{1}ky^{3}k

*h*
*b*^{1}_{22}_{2}

*h*_{22}
*b*^{2}_{22}_{2}

*y*^{4}k.

2.17

Analysing the increment of *V*, we see that, if |a| *<* 1, ΔV will be nonpositive in a small
neighborhood of the zero solution if the multipliers of the terms*x*1*y*^{2},*y*^{3}and*x*1*y*^{3}are equal
to zero and the multiplier of the term*y*^{4}is nonpositive, that is, if

*hab*^{1}_{22}2h22*b*_{12}^{2} 0,
*h*_{22}*b*^{2}_{22}0,
*hb*_{12}^{1} *b*^{1}_{22}*h*22*b*_{12}^{2} *b*^{2}_{22} 0,
*h*

*b*_{22}^{1} _{2}
*h*_{22}

*b*_{22}^{2}_{2}

≤0.

2.18

As long as the Lyapunov function is positive definite,*h >*0 and*h*_{22} *>*0. Therefore, conditions
2.18hold if and only if

*b*^{1}_{22}0, *b*^{2}_{12}0, *b*_{22}^{2} 0. 2.19

Then, system2.2turns into

*x*_{1}k1 *ax*_{1}k

*b*^{1}_{11}*x*^{2}_{1}k 2b_{12}^{1} *x*_{1}kx2k
*,*
*x*2k1 *x*2k *b*^{2}_{11}*x*^{2}_{1}k

2.20

andΔVwithout loss of generality, we put*h*22 1, i.e.,*V*x1*, y hx*^{2}_{1}*y*^{2}into
ΔVxk −

*h*
1−*a*^{2}

−2hab^{1}_{11}*x*1k−2

2hab^{1}_{12}*b*_{11}^{2}

*yk*−
*h*

*b*^{1}_{11}2

*b*^{2}_{11}2

*x*_{1}^{2}k

−4hb^{1}_{11}*b*^{1}_{12}*x*_{1}kyk−4h
*b*_{12}^{1} _{2}

*y*^{2}k

*x*^{2}_{1}k
−

*α*−2β1*x*1k−2β2*yk*−*γ*1*x*^{2}_{1}k−2δ1*x*1kyk−*γ*2*y*^{2}k
*x*^{2}_{1}k.

2.21
The first diﬀerence of the Lyapunov function is nonpositive in a suﬃciently small
neighborhood of the originthis is because*h >* 0, |a|*<* 1, and*α* *h1*−*a*^{2} *>* 0. In other
words, the zero solution is stable in the Lyapunov sense.

Now we will discuss the shape of the guaranteed domain of stability. It can be defined by the inequalities

*γ*_{1}*x*^{2}_{1}2δ_{1}*x*_{1}*yγ*_{2}*y*^{2}2β_{1}*x*_{1}2β_{2}*y*≤*α,*
*hx*^{2}_{1}*y*^{2}≤*r*^{2}*,*

2.22

where *r >* 0. This means that inequalities 2.4 and 2.5 are correct. Both inequalities
geometrically express closed ellipses if*b*^{2}_{11}·*b*^{1}_{12}*/*0. For the second inequality, this is obvious.

For the first one, this follows from the following inequalities:*γ*_{1}*>*0, *γ*_{2}*>*0 and

*γ*_{1}*γ*_{2}−*δ*^{2}_{1}

*h*
*b*^{1}_{11}_{2}

*b*^{2}_{11}_{2}

·

4h
*b*^{1}_{12}_{2}

−4

*hb*^{1}_{11}*b*_{12}^{1} _{2}
4h

*b*^{2}_{11}*b*_{12}^{1} _{2}

*>*0. 2.23

Moreover, for*r* → 0, the ellipse2.4

*hx*^{2}_{1}*y*^{2}≤*r*^{2} 2.24

is contained because it shrinks to the originin the ellipse 2.5, that is, there exists such
*rr*^{∗}that, for*r* ∈0, r^{∗}, the ellipse2.4lies inside the ellipse2.5without any intersection
points and, for*r* *r*^{∗}, there exists at least one common boundary point of both ellipses. Let
us find the value*r*^{∗}. It is characterized by the requirement that the slope coeﬃcients*k*1and
*k*_{2}of both ellipses are the same at the point of contact. Therefore

*k*_{1}−*γ*_{1}*x*_{1}*δ*_{1}*yβ*_{1}

*δ*_{1}*x*_{1}*γ*_{2}*yβ*_{2}*,* *k*_{2}−*hx*_{1}

*y* *,* 2.25

where we assumewithout loss of generalitythat the denominators are nonzero. Thus, we
get a quadratic system of two equations to find the contact pointsx1*, y:*

*γ*_{1}*x*^{2}_{1}2δ_{1}*x*_{1}*yγ*_{2}*y*^{2}2β_{1}*x*_{1}2β_{2}*yα,*
*hx*1

*δ*1*x*1*γ*2*yβ*2

−*y*

*γ*1*x*1*δ*1*yβ*1

0. 2.26

For the corresponding values of*r, we have*

*hx*^{2}_{1}*y*^{2}*r*^{2}*.* 2.27

In accordance with the geometrical meaning of the above quadratic system, we take such a
solutionx1*, y*as a defintion of the minimal positive value of*r*and set*r*^{∗}*r.*

*Example 2.2. Consider a system*

*xk*1 0.5xk *x*^{2}k−4xkyk,

*yk*1 *yk x*^{2}k. 2.28

In our case,*n* 2,*a*0.5 *<*1, and*b*^{1}_{22} *b*^{2}_{12} *b*^{2}_{22} 0. Therefore, byTheorem 2.1, the zero
solution of system2.28is stable in the Lyapunov sense. We will find the guaranteed domain
of stability. We have

*b*^{1}_{11}1, *b*^{1}_{12} −2, *b*^{2}_{11}1, 2.29

and*b*^{2}_{11}·*b*^{1}_{12}−2*/*0. Set*h*2. Then

*αh*
1−*a*^{2}

21−0.25 1.5,
*β*1*hab*_{11}^{1} 2·0.5·11,

*β*_{2}2hab_{12}^{1} *b*^{2}_{11}2·2·0.5·−2 1−3,
*γ*1*h*

*b*_{11}^{1} _{2}

*b*_{11}^{2} _{2}

2·1^{2}1^{2}3,
*γ*_{2}4h

*b*^{1}_{12}_{2}

4·2·−2^{2}32,
*δ*_{1}2hb^{1}_{11}*b*^{1}_{12}2·2·1·−2 −8.

2.30

That is, the guaranteed domain of stability is given by the inequalities

3x^{2}−16xy32y^{2}2x−6y≤1.5, 2.31

2x^{2}*y*^{2}≤*r*^{2} 2.32

if *r* is so small that the domain described by inequality2.32is embedded in the domain
described by inequality2.31. We consider the case when the ellipse2.32is embedded in

−1

−0.5 0 0.5 1

−1.5 −1 −0.5 0 0.5 1 1.5
**Figure 1: Graphical solution of system**2.33and2.34.

the ellipse2.31and the boundaries of both ellipses have only one intersection point. Solving the system2.8, that is, the system

3x^{2}−16xy32y^{2}2x−6y1.5, 2.33

2x

−8x32y−3

−*y*

3x−8y1

0, 2.34

with Mathematica software, we get the solutionsseeFigure 1where the*x-axis is identified*
with the horizontal line and the *y-axis is identified with the vertical line, the blue ellipse*
graphically depicts equation 2.33, and the red hyperbola graphically depicts equation
2.34:

*x, y*

*x*_{1}*, y*_{1}

−1.60766,−0.31220,
*x, y*

*x*_{2}*, y*_{2}

−0.03568,−0.32187,
*x, y*

*x*3*, y*3

0.01952,−0.13664,
*x, y*

*x*_{4}*, y*_{4}

1.10728,0.37750.

2.35

Then, in accordance with2.7,

*r*^{∗} min

*i1,2,3,4*

2x_{i}^{2}*y*_{i}^{2}

2x_{3}^{2}*y*_{3}^{2} *.* 0.1369, 2.36

−0.4

−0.2 0 0.2 0.4

−1.5 −1 −0.5 0 0.5 1
**Figure 2: The guaranteed domain of stability.**

and the guaranteed domain of stability

2x^{2}*y*^{2}≤r^{∗}^{2} 0.1369^{2} 2.37

obtained from2.31,2.32is depicted inFigure 2as an ellipsoidal domain shaded in red
and bounded by the thick red ellipse, with the identification of*x-axis andy-axis being the*
same as before. Here, the domain2.31is bounded by the blue ellipse2.33.

**2.3. Stability in the General**n-Dimensional Case

Consider system1.6inR* ^{n}*. Assume that the matrix

*A*has a simple eigenvalue that is equal to unity with the others lying inside the unit circle. After linearly transforming the dependent variables if necessary, we can assume, without loss of generality, that the matrix

*A*of the linear terms in a block form, that is,

*A*
*A*_{0} *θ*

*θ** ^{T}* 1

*,* *A*0
*a**ij*

*, i, j*1,2, . . . , n−1, 2.38

where*θ* 0,0, . . . ,0* ^{T}*, is then−1-dimensional zero vector and all the eigenvalues of the
matrix

*A*

_{0}lie inside the unit circle. In order to formulate the next result and its proof, we

have to introduce some new definitionsthey copy the ones used inSection 1.1, but we use
dimension or size*n*−1 instead of*n*and note this change as a subscript if necessary:

*x*_{n−1} *x*_{1}*, x*_{2}*, . . . , x*_{n−1}^{T}*,* *yx*_{n}*,*

*B*^{0}_{i}

⎛

⎜⎜

⎜⎜

⎜⎝

*b*_{11}^{i}*b*_{12}* ^{i}* · · ·

*b*

^{i}_{1,n−1}

*b*

_{21}

^{i}*b*

_{22}

*· · ·*

^{i}*b*

^{i}_{2,n−1}

· · · ·
*b*^{i}_{n−1,1}*b*^{i}* _{n−1,2}* · · ·

*b*

^{i}

_{n−1,n−1}⎞

⎟⎟

⎟⎟

⎟⎠*,* *i*1,2, . . . , n, *B*

⎛

⎜⎜

⎝

*b*^{1}_{1n} · · · *b*_{n−1,n}^{1}

· · · ·
*b*^{n−1}_{1n} · · · *b*_{n−1,n}^{n−1}

⎞

⎟⎟

⎠*,*

*B*^{T}

⎛

⎜⎜

⎝

*b*^{1}_{11} · · · *b*^{1}_{1,n−1} *b*_{21}^{1} · · · *b*^{1}_{2,n−1} · · · *b*^{1}* _{n−1,1}* · · ·

*b*

^{1}

_{n−1,n−1}· · · ·
*b*_{11}* ^{n−1}* · · ·

*b*

^{n−1}_{1,n−1}

*b*

^{n−1}_{21}· · ·

*b*

^{n−1}_{2,n−1}· · ·

*b*

^{n−1}*· · ·*

_{n−1,1}*b*

^{n−1}

_{n−1,n−1}⎞

⎟⎟

⎠*.*

2.39

Matrices*B*^{0}* _{i}*,

*i*1,2, . . . , n, are symmetric since

*b*

^{i}

_{sq}*b*

^{i}*,*

_{qs}*i, s, q*1,2, . . . , nseeSection 1.1.

Moreover, we assume that there exists a symmetric positive definiten−1×n−1matrix
*H*such that the symmetric matrix

*CH*−*A*^{T}_{0}*HA*0 2.40

is positive definite. Let*h >*0 be a positive number and

*αλ*_{min}C,
*β*_{1} 1

2

*A*^{T}_{0}*HB*^{T}*,*
*β*2 1

2

*BHA*03A^{T}_{0}*HB** ^{T}*2h

*B*

^{0}

_{n}*T*

*,*
*γ*_{1}

*BHB*^{T}

*hB*^{0}_{n}^{2}*,*

*γ*_{2}4*BHB*^{T}*,*
*δ*_{1}2*B*· *H ·B.*

2.41

**Theorem 2.3. Let**handrbe positive numbers. Assume that

*b*_{nn}^{1} *b*_{nn}^{2} · · ·*b*^{n}* _{nn}*0,

*b*

_{1n}

^{n}*b*

_{2n}

*· · ·*

^{n}*b*

^{n}*0. 2.42*

_{n−1,n}*Then the zero solution of system*1.6*is stable by Lyapunov and the guaranteed domain of stability is*
*described by the inequalities*

*γ*1x^{2}2δ1x*yγ*2*y*^{2}2β1x2β2*y*≤*α,* 2.43

*x*^{T}_{n−1}*Hx*_{n−1}*hy*^{2}≤*r*^{2} 2.44

*ifris so small that the domain described by inequality*2.44*is embedded into the domain described*
*by inequality*2.43.

*Proof. We will perform auxiliary matrix computations. With this in mind, we have defined*
ann−1^{2}×n−1matrix*X*_{n−1}as

*X*^{T}_{n−1}

*X*_{1n−1}^{T}*, X*^{T}_{2n−1}*, . . . , X*_{n−1n−1}^{T}

*,* 2.45

where all the elements of then−1×n−1matrices*X*^{T}* _{in−1}*,

*i*1,2, . . . , n−1 are equal to zero except the row

*i, which equalsx*

^{T}_{n−1}, that is,

*X*^{T}_{in−1}

⎛

⎜⎜

⎜⎜

⎜⎝

0 0 · · · 0

· · · ·
*x*_{1} *x*_{2} · · · *x*_{n−1}

· · · · 0 0 · · · 0

⎞

⎟⎟

⎟⎟

⎟⎠*.* 2.46

Moreover, we define

avectors*Y** _{i}*,

*i*1,2, . . . , n−1, as a rown−1-dimensional vector with coordinates equal to zero except the

*ith element, which equalsx*

*n*, that is,

*Y** _{i}* 0,0, . . . ,0, x

_{n}*,*0, . . . ,0, 2.47

b n−1×n−1zero matrixΘ,

cvectors*b**i* b^{i}_{1n}*, b*^{i}_{2n}*, . . . , b*^{i}_{n−1,n}* ^{T}*,

*i*1,2, . . . , n, dvector

*b*b

^{1}

_{nn}*, b*

^{2}

_{nn}*, . . . , b*

_{nn}

^{n−1}*.*

^{T}It is easy to see that

*X** ^{T}*k

⎛

⎝*X*_{1n−1}* ^{T}* k

*Y*

_{1}

*k · · ·*

^{T}*X*

^{T}*k*

_{n−1n−1}*Y*

_{n−1}*k Θ*

^{T}*θ*

*θ** ^{T}* 0 · · ·

*θ*

*0*

^{T}*x*

^{T}_{n−1}k

*yk*

⎞

⎠*,*

*B*

⎛

⎜⎜

⎜⎜

⎜⎜

⎜⎜

⎝
*B*^{0}_{1} *b*_{1}
*b*^{T}_{1} *b*^{1}_{nn}

· · · ·
*B*_{n}^{0} *b*_{n}*b*^{T}_{n}*b*^{n}_{nn}

⎞

⎟⎟

⎟⎟

⎟⎟

⎟⎟

⎠
*.*

2.48

Now we are able to rewrite system1.6in an equivalent form
*x*_{n−1}k1

*yk*1

*A*_{0} *θ*

*θ** ^{T}* 1

*x*_{n−1}k
*yk*

*X*_{1n−1}* ^{T}* k

*Y*

_{1}

*k · · ·*

^{T}*X*

^{T}*k*

_{n−1n−1}*Y*

_{n−1}*k Θ*

^{T}*θ*

*θ** ^{T}* 0 · · ·

*θ*

*0*

^{T}*x*

^{T}_{n−1}k

*yk*

×

⎛

⎜⎜

⎜⎜

⎜⎝
*B*_{1}^{0} *b*_{1}
*b*^{T}_{1} *b*^{1}_{nn}

· · · ·
*B*_{n}^{0} *b*_{n}*b*^{T}_{n}*b*^{n}_{nn}

⎞

⎟⎟

⎟⎟

⎟⎠

*x*_{n−1}k
*yk*

*A*_{0}*r*11 *r*12

*r*_{21} 1*r*_{22}

*x*_{n−1}k
*yk*

*,*

2.49

where

*r*11 *r*11

*x*_{n−1}k, yk
^{n−1}

*j1*

*X*_{jn−1}* ^{T}* kB

^{0}

_{j}*Y*

_{j}*kb*

^{T}

_{j}

^{T}*B*^{T}*X*_{n−1}k *Byk,*

*r*12*r*12

*x*_{n−1}k, yk
^{n−1}

*j1*

*X*^{T}* _{jn−1}*kb

*j*

*Y*

_{j}*kb*

^{T}

^{j}*nn*

*B*^{T}*x*_{n−1}k *byk,*

*r*_{21}*r*_{21}

*x*_{n−1}k, yk

*x*^{T}_{n−1}kB^{0}_{n}*ykb*^{T}_{n}*,*
*r*_{22} *r*_{22}

*x*_{n−1}k, yk

*x*^{T}_{n−1}kb*n**ykb*_{nn}^{n}*.*

2.50

Before the following computations, for the reader’s convenience, we recall that for then−
1×n−1matricesA,A1, 1×n−1vectors* ,* _{1},n−1×1 vectorsC,C1and 1×1 “matrices”

*m,m*_{1}, we have
A C

* m*

×

A1 C1
1 *m*_{1}

A × A1C × 1 *A*× C1C ×*m*1

× A1*m*× 1 × C1*m*×*m*_{1}

*.* 2.51

To investigate the stability of system1.6, we use the Lyapunov function

*V*

*x*_{n−1}k, yk

*x*_{n−1}* ^{T}* kHxn−1k

*hy*

^{2}k

*x*^{T}_{n−1}k, yk*H θ*
*θ*^{T}*h*

*x*_{n−1}k
*yk*

*,*

2.52

where*HH*_{n−1}is ann−1×n−1constant real symmetric and positive definite matrix.

Let us find the first diﬀerence of the Lyapunov function2.52along the solutions of2.49.

We get

ΔV

*x*_{n−1}k, yk

*x*^{T}_{n−1}k1, yk1*H θ*
*θ*^{T}*h*

*x*_{n−1}k1
*yk*1

−

*x*_{n−1}* ^{T}* k, yk

*H θ*

*θ*

^{T}*h*

*x*_{n−1}k
*yk*

*x*^{T}_{n−1}k, yk*A*0*r*11 *r*12

*r*_{21} 1*r*_{22}
*T*

*H θ*
*θ*^{T}*h*

*A*0*r*11 *r*12

*r*_{21} 1*r*_{22}

*x*_{n−1}k
*yk*

−

*x*_{n−1}* ^{T}* k, yk

*H θ*

*θ*

^{T}*h*

*x*_{n−1}k
*yk*

*x*^{T}_{n−1}k, yk*A*^{T}_{0} *r*_{11}^{T}*r*_{21}^{T}*r*_{12}* ^{T}* 1

*r*

_{22}

*H θ*
*θ*^{T}*h*

×

*A*_{0}*r*_{11} *r*_{12}
*r*_{21} 1*r*_{22}

−
*H θ*
*θ*^{T}*h*

×

*x*_{n−1}k
*yk*

2.53

or, using formula2.51, ΔV

*x*_{n−1}k, yk

*x*_{n−1}* ^{T}* k, yk

*A*

^{T}_{0}

*Hr*

_{11}

^{T}*H*

*hr*

_{21}

^{T}*r*

_{12}

^{T}*H*

*h1r*

_{22}

×

*A*_{0}*r*_{11} *r*_{12}
*r*_{21} 1*r*_{22}

−
*H θ*

*θ*^{T}*h*

×

*x*_{n−1}k
*yk*

*x*_{n−1}* ^{T}* k, yk

×

⎛

⎝

*A*^{T}_{0}*r*_{11}^{T}

*HA*0*r*_{11} *r*_{21}^{T}*hr*_{21}−*H*

*A*^{T}_{0} *r*_{11}^{T}

*Hr*_{12}*r*_{21}^{T}*h1r*_{22}
*r*_{12}^{T}*HA*0*r*_{11} 1*r*_{22}hr21 *r*_{12}^{T}*Hr*_{12} 1*r*_{22}h1*r*_{22}−*h*

⎞

⎠

×

*x*_{n−1}k
*yk*

*.*

2.54 Using formulas2.50, we have

ΔV

*x*_{n−1}k, yk

*x*_{n−1}* ^{T}* k, yk

×
*c*_{11} *c*_{12}
*c*_{21} *c*_{22}

*x*_{n−1}k
*yk*

*,* 2.55

where

*c*11 *c*11

*x*_{n−1}k, yk

*A*0*B*^{T}*X*_{n−1}k *Byk* *T*

*H*

*A*0*B*^{T}*X*_{n−1}k *Byk*
h

*B*_{n}^{0}*T*

*x*_{n−1}k *b*_{n}*yk, x*^{T}_{n−1}kB_{n}^{0}*ykb*^{T}_{n}

−*H,*

*c*_{12} *c*_{12}

*x*_{n−1}k, yk

*A*_{0}*B*^{T}*X*_{n−1}k *Byk* *T*

*H*

*B*^{T}*x*_{n−1}k *byk*
h

1*x*^{T}_{n−1}kb*n**b*^{n}_{nn}*yk*
*B*_{n}^{0}*T*

*x*_{n−1}k *b**n**yk*
*,*
*c*21 *c*21

*x*_{n−1}k, yk
*c*^{T}_{12}*,*

*c*22 *c*22

*x*_{n−1}k, yk

*B*^{T}*x*_{n−1}k *byk* *T*

*H*

*B*^{T}*x*_{n−1}k *byk*
h

1*x*^{T}_{n−1}kb*n**b*_{nn}^{n}*yk*_{2}

−*h.*

2.56

Then ΔV

*x*_{n−1}k, yk

*x*^{T}_{n−1}kc11*x*_{n−1}k 2x^{T}_{n−1}kc12*yk ykc*22*yk*
*x*^{T}_{n−1}

*A*0*B*^{T}*X*_{n−1}k *Byk* *T*

*H*

*A*0*B*^{T}*X*_{n−1}k *Byk*
h

*B*^{0}_{n}_{T}

*x*_{n−1}k *b*_{n}*yk*

*x*^{T}_{n−1}kB_{n}^{0}*ykb*^{T}_{n}

−*H*

*x*_{n−1}k
2x_{n−1}* ^{T}* k

*A*_{0}*B*^{T}*X*_{n−1}k *Byk* *T*

*H*

*B*^{T}*x*_{n−1}k *byk*
h

1*x*_{n−1}* ^{T}* kb

*n*

*b*

^{n}

_{nn}*yk*

*B*

_{n}^{0}

_{T}*x*_{n−1}k *b**n**yk*

*yk*
*yk*

*B*^{T}*x*_{n−1}k *byk*_{T}*H*

*B*^{T}*x*_{n−1}k *byk*
h

1*x*_{n−1}* ^{T}* kb

*n*

*b*

^{n}

_{nn}*yk*

_{2}

−*h*

*yk.*

2.57

After further computation, we get ΔV

*x*_{n−1}k, yk
*x*^{T}_{n−1}k

*A*^{T}_{0}*HA*_{0}−*HA*^{T}_{0}*HB*^{T}*X*_{n−1}k *X*_{n−1}^{T}*BHA*_{0}
*X*_{n−1}* ^{T}* kBHB

^{T}*X*

_{n−1}k

*X*_{n−1}* ^{T}* kBH

*BB*

^{T}*HB*

^{T}*X*

_{n−1}k

*yk*

*B*^{T}*HA*0*A*^{T}_{0}*HB*^{T}

*yk B*^{T}*HBy* ^{2}k *h*
*B*_{n}^{0}_{T}

*x*_{n−1}kx^{T}_{n−1}kB^{0}* _{n}*
2h

*B*_{n}^{0}_{T}

*x*_{n−1}kb^{T}_{n}*yk hb*_{n}*y*^{2}kb^{T}_{n}

*x*_{n−1}k
2x^{T}_{n−1}k

*A*^{T}_{0}*HB*^{T}*x*_{n−1}k *A*^{T}_{0}*Hbyk * *X*_{n−1}* ^{T}* kBH

*B*

^{T}*x*

_{n−1}k

*X*

^{T}_{n−1}kBH

*byk BH*

*B*

^{T}*x*

_{n−1}kyk

*BH*

*by*

^{2}k

*h*

*B*_{n}^{0}*T*

*x*_{n−1}k *b*_{n}*yk*

*hx*_{n−1}* ^{T}* kb

*n*

*B*_{n}^{0}*T*

*x*_{n−1}k *b*_{n}*yk*

hb^{n}_{nn}*yk*
*B*^{0}_{n}_{T}

*x*_{n−1}k *b*_{n}*yk*

*yk*

*x*^{T}_{n−1}k*BH*

*B*^{T}*x*_{n−1}k *byk*
*b*^{T}*H*

*B*^{T}*x*_{n−1}k *byk*

*yk h*

*x*^{T}_{n−1}kb*n*

2

h

*b*_{nn}^{n}*yk*_{2}

2hx^{T}_{n−1}kb*n*2hb_{nn}^{n}*yk *2hx^{T}_{n−1}kb*n**b*^{n}_{nn}*yk*
*y*^{2}k.

2.58