\def\mh{\mathbreak}

\centerline{\bf C\,H\,A\,P\,T\,E\,R\ \ \ I}

\bigskip

\centerline{\bf ESTIMATES OF  LYAPUNOV EXPONENTS}
\centerline{\bf AND THE HAUSDORFF DIMENSION}
\centerline{\bf OF ATTRACTORS}

\bigskip

\bigskip

Let us consider a system
$$
\frac{dx}{dt}=A(t)x,\quad x\in\br^n,
$$
where $A(t)$ a real $n\times n$-matrix. Let $X(t)$ be its
fundamental matrix with initial data $X(0)=I$, where $I$ is an
unit matrix.

Further we will denote by $\br^n$ an Euclidean $n$-space and by
$\bc^n$ a unitary  one.

Let us recall that the square roots of the eigenvalues of matrix
$X(t)^*X(t)$ are called the singular numbers
$\rho_1(t)\geq\mh
\geq\ldots\geq\rho_n(t)$ of the matrix $X(t)$. Let us
also recall the following definition of Lyapunov  exponents
$\nu_j$ [24,25]:
$$
\nu_j=\ln\varlimsup_{t\to+\infty} \rho_j(t)^{1/t}.
$$

The estimates of the Lyapunov exponents, developed earlier in the
framework of the classical theory of linear systems [1,7,12,30],
recently obtained a new motivation: they are used to estimate the
dimensions of attractors [24,25,46].

First, we estimate from above the value $\nu_1$, generalizing
the classical Wazewski ine\-qu\-a\-li\-ty [1,8,12] and making more precise
the estimates of solutions obtained earlier by the
Gronwall lemma [8,12].

Let $L$ be a linear operator in the space of continuous functions
given on $[0,+\infty)$. By $(Lv)(t)$ denote the value of the
function $Lv(\cdot)$ at the point $t$.

We now assume that $L$ has the following property of
monoto\-ni\-ci\-ty. If  $v_1(t)\leq v_2(t)$,
$\ \forall\,t\geq0$, then
$\big(Lv_1\big)(t)\leq\mh\leq
\big(Lv_2\big)(t)$, $\ \forall\,t\geq0$.
We also suppose that if $\ v_1(t)\equiv v_2(t)$ on the interval
$[0,\tau]$, then  $\big(Lv_1\big)(\tau)=\big(Lv_2\big)(\tau)$.

Suppose that for all function $v(t)\geq0$, $\ \forall\,t\geq0$ and the
number $C\geq0$ the following ine\-qu\-a\-li\-ty

\vskip-2mm
$$
v(t)\leq C+\big(Lv\big)(t), \qquad\forall\,t\geq0
\eqno(1)
$$
is satisfied.

\medskip

{\bf Lemma 1.}
{\it
If for some number $q\in(0,1)$
$$
\big(L1\big)(t)\leq q,\qquad \forall\,t\geq0,
\eqno(2)
$$
then
$$
v(t)\leq\frac{C}{1-q},\qquad \forall\,t\geq0.
\eqno(3)
$$
}

\medskip

{\bf Proof.}
Inequality (3)  results from the following relation
$$
\max_{[0,T]}v(t)\leq C+q\max_{[0,T]}v(t),
$$
which in turn results from (1) and (2).

We now suppose that for the continuous nonnegative functions
$v(t)$, $\om(t,\tau)$, the continuous function $\,p(t)$,
and the number $\,C\geq0$ we have
$$
v(t)\leq Ce^{p(t)}+\intl_o^t\om(t,\tau)v(\tau)\,d\tau,
\qquad \forall\,t\geq0.
\eqno(4)
$$

\medskip

{\bf Lemma 2.}
{\it
If for some number $q\in(0,1)$
$$
\intl_0^t\om(t,\tau)\exp\big(p(\tau)-p(t)\big)\,d\tau\leq q,\qquad
\forall\, t\geq0,
\eqno(5)
$$
then
$$
v(t)\leq\frac{C}{1-q}e^{p(t)},\qquad\forall\,t\geq0.
\eqno(6)
$$
}

\medskip

{\bf Proof.}
Let us write ine\-qu\-a\-li\-ty (4) as follows
$$
\big[e^{-p(t)}v(t)\big]\leq C+\intl_0^t\om(t,\tau)
e^{p(\tau)-p(t)}\big[e^{-p(\tau)}v(\tau)\big]\,d\tau.
$$
Applying Lemma 1 to this ine\-qu\-a\-li\-ty and noting that
$$
\big(L1\big)(t)=\intl_0^t\om(t,\tau)
e^{p(\tau)-p(t)}d\tau,
$$
we get
$$
\big[e^{-p(t)}v(t)\big]\leq\frac C{1-q},
$$
which proves the lemma.

\smallskip

Let us name one of the simplest applications of Lem-\linebreak
ma 2.

Consider the system
$$
\dot x_i=a_ix_i+\sum_{j=1}^nb_{ij}(t)x_j,
\eqno(7)
$$
where $a_i$ are complex numbers; $b_{ij}(t)$ are complex-valued
continuous functions , $i=1,\ldots,n$, $\ t\in{\bold R}^1$.
We denote
$$
\al_i=Rea_i,\qquad b_i(t)=\max\limits_j\big|b_{ij}(t)\big|.
$$

\medskip

{\bf Theorem 1.}
{\it
Let for certain numbers $\la$ and $q\in(0,1)$ the inequalities
$\la\geq\max\limits_i\al_i$,
$$
\intl_0^te^{\la(\tau-t)}\sum\limits_{i=1}^nb_i(\tau)
e^{\al_i(t-\tau)}d\tau\leq q,\quad \forall\ t\geq0
\eqno(8)
$$
be true.

Then
$$
\sum_{i=1}^n\big|x_i(t)\big|\leq\frac{e^{\la t}}{1-q}
\sum_{i=1}^n\big|x_i(0)\big|,\quad\forall\ t\geq0.
\eqno(9)
$$
}

\medskip

{\bf Proof.}
We write the solution of system (7) in the form:
$$
x_i(t)=e^{a_it}\bigg(x_i(0)+
\intl_0^te^{-a_i\tau}\sum_{j=1}^nb_{ij}(\tau)x_j(\tau)\,d\tau\bigg).
$$
Hence it follows that
$$
\eqalign{
& \sum_{i=1}^n|x_i(t)|{\leq} e^{\la t}\sum_{i=1}^n
|x_i(0)|+\cr
& +\intl_0^t\bigg(\sum_{i=1}^nb_i(\tau)e^{\al_i(t{-}\tau)}
\bigg)\bigg(\sum_{j=1}^n|x_j(\tau)|\bigg)d\tau.
}
$$
Thus, ine\-qu\-a\-li\-ty (4) with $p(t)=\la t,$
$$
v(t)=\sum_{i=1}^n\big|x_i(t)\big|,
$$

$$
C=\sum_{i=1}^n \big|x_i(0)\big|,\qquad
\om(t,\tau)=\sum_{i=1}^nb_i(\tau)e^{\al_i(t-\tau)}
$$
is valid.

From (8) it follows that condition (5) is satis\-fi\-ed.
The\-re\-fo\-re, ine\-qu\-a\-li\-ty (6) coincides with estimate (9).

We also denote $\beta_i=\sup\limits_{t\geq0}b_i(t)$.

\medskip

{\bf Corollary 1.}
{\it
Suppose that for some number $\la$ we have
$$
\la\geq\max\limits_i\al_i,\qquad q:=\sum_{i=1}^n
\frac{\beta_i}{\la-\al_i}<1.
$$

Then estimate (9) is true.
}

\medskip

It can be noted that in the case when for some $i$  and $j$
$\ \al_i\neq\al_j$ and $b_{ij}(t)\equiv b_{ik}(t)$,
$\forall\,i,j,k$ the estimate of an exponential growth of the solutions
of linear systems (9) is always more precise than the estimate
obtained by the Gronwall lemma [8,12]. Besides we can easily see
that estimate (9) is invariant with respect to nonsingular linear
substitutions in system (7).

We now formulate a certain generalized Wazewski  ine\-qu\-a\-li\-ty
[12,18].

Consider the system
$$
\dot x_i=A_i(t)x_i+\sum_{j=1}^n B_{ij}(t)x_j,
\eqno(10)
$$
where $A_i(t)$ are continuous $n_i\times n_i$--matrices,
$B_{ij}(t)$ are continuous $n_i\times n_j$--matrices.
By $\mu_i(t)$ we denote the maximal eigenvalue of the matrix
$A_i(t)+A_i(t)^*$. Suppose that for the continuous nonnegative
functions $b_i(t)$ the following estimate
$$
\eqalign{
& 2\sum_{j=1}^n x_i^*B_{ij}(t)x_j\leq b_i(t)\sum_{j=1}^n
|x_j|^2,\cr
& \forall\,x_i\in{\bold R}^{n_i},\ \forall\,i=1,\ldots,n
}
$$
is true. Let us also denote
$$
p_i(t)=\intl_0^t\mu_i(\tau)\,d\tau.
$$

\medskip

{\bf Theorem 2.}
{\it
If for a continuous function $p(t)$ such taht $p(t)\geq p_i(t)$,
$\ \forall\,t\geq0$ and a number $q\in(0,1)$
$$
\eqalign{
& \intl_0^t\exp\big(p(\tau)-p(t)\big)\sum_{i=1}^nb_i(\tau)\exp
\big(p_i(t)-p_i(\tau)\big)\,d\tau\leq q,\cr
& \forall\ t\geq0,
} \eqno(11)
$$
then
$$
\sum_{i=1}^n\big|x_i(t)\big|^2\leq\frac{e^{p(t)}}{1-q}
\sum_{i=1}^n\big|x_i(0)\big|^2. \eqno(12)
$$
}

For $i=1$, $\,b_1(t)\equiv0$ we have the Wazewski ine\-qu\-a\-li\-ty.

{\bf Proof.}
From equation (10) the following ine\-qu\-a\-li\-ty
$$
\big|x_i(t)\big|^2{\leq} e^{p_i(t)}\bigg(\big|x_i(0)\big|^2{+}
\intl_0^te^{-p_i(\tau)}b_i(\tau)\sum_{j=1}^n
\big|x_j(\tau)\big|^2d\tau\bigg)
$$
can be obtained. Hence
$$
\eqalign{ &
\ds \sum_{i=1}^n\big|x_i(t)\big|^2\leq e^{p(t)}\sum_{i=1}^n
\big|x_i(0)\big|^2+\cr    &
\ds +\intl_0^t\left(\sum_{i=1}^ne^{p_i(t)-
p_i(\tau)}b_i(\tau)\right)\left(\sum_{j=1}^n\big|x_j(\tau)\big|^2
\right)\,d\tau.
}
$$
Thus ine\-qu\-a\-li\-ty (4) with
$$
v(t)=\sum_{i=1}^n\big|x_i(t)\big|^2,\quad C=v(0),
$$

$$
\om(t,\tau)=\sum_{i=1}^n b_i(\tau)e^{p_i(t)-p_i(\tau)}
$$
is valid. From (11) using Lemma 2 we obtain
estimate (12).

\medskip

{\it Example.}
It will be shown that Theorem 2 can give a more precise
estimate than the Wazewski ine\-qu\-a\-li\-ty. Consider the following
system
$$
\eqalign{ &
\ds \dot x_1=2x_1,\cr    &
\ds \dot x_2=b(t)x_1,
}
$$
where the continuous function $b(t)$ satisfies the condition
$0{<}\ga{\leq}|b(t)|\leq\beta<4$ $\ \forall\,\,t\geq0$. Here
$\ga$ and $\beta$ are some numbers. In this case we get $\mu_1^{}=4$,
$\,\mu_2^{}=0$.  Then for $p(t)=4t$ condition (11) can be written
in the form:
$$
\frac{\beta}4<1.
$$
Thus, estimate (12) take the form
$$
x_1(t)^2+x_2(t)^2\leq\frac{e^{4t}}{1-\frac{\beta}4}
\big(x_1(0)^2+x_2(0)^2\big).
$$
By the Wazewski ine\-qu\-a\-li\-ty we get the following estimate
$$
x_1(t)^2+x_2(t)^2{\leq}\exp\intl_0^t\big(2{+}
\sqrt{4{+}b(\tau)^2}\,\big)d\tau\,\big(x_1(0)^2{+}
x_2(0)^2\big).
$$
It is clear that for sufficiently large $t$ the above estimate is less
precise than estimate (12).

\smallskip

As an example, consider the following Lorenz system [11]
$$
\dot X=-dX+dY,\quad \dot Y=(r-Z)X-Y,\quad
$$

$$
\dot Z=-bZ+XY, \eqno(13)
$$
where $d$, $r$, $b$ are positive numbers.
Setting $X=\kappa\sg$ we write the matrix of the system,
linearized along a certain trajectory $\sg(t)$, $\,Y(t)$, $\,Z(t)$:
$$
A(t)=\left(\matrix
-d& d/\kappa& 0\cr
 \kappa\big(r-Z(t)\big)& -1& -\kappa\sg(t)\cr
 \kappa Y(t)& \kappa\sg(t)& -b
\endmatrix\right).
$$
In order to obtain the estimates of the Hausdorff dimension of attractors
and the global stability of system (13) it is important to
estimate the maximal Lyapunov exponents of the linear system such
that its matrix is as follows
$$
-A(t)^*=\left(\matrix
d& -\kappa\big(r-Z(t)\big)& -\kappa Y(t)\cr
  -d/\kappa& 1& -\kappa\sg(t)\cr
0& \kappa\sg(t)&  b
\endmatrix\right).
$$
For this purpose this system is represented as (10) with
$$
A_1(t)=d,\quad B_{11}(t)=0,\quad
$$

$$
B_{12}(t)=\big(-\kappa\big(r-Z(t)\big),-\kappa Y(t)\big),
$$

$$
A_2(t)=\left(\matrix
1& -\kappa\sg(t)\cr
\kappa\sg(t)& 1
\endmatrix\right),
$$
$$
B_{21}(t)=\left(\matrix
-d/\kappa\cr
0
\endmatrix\right),\quad B_{22}(t)=\left(\matrix
0& 0\cr
0&  b-1
\endmatrix\right),
$$
$$
\eqalign{
& b_1(t)=\kappa\sqrt{\big(r-Z(t)\big)^2+Y(t)^2},\cr
& b_2(t)=\max\left(\frac d{\kappa},2(b-1)\right).
}
$$
Further, we suppose that $d/\kappa\geq 2(b-1)$. It is clear that
$\mu_1(t)=2d$, $\ \mu_2(t)=2$. Therefore condition (11)
for $p(t)=\mh=
\la t$ takes the form:
$$
\eqalign{
& \frac d{\kappa(\la{-}2)}{+}\kappa e^{(2d-\la)t}\!\intl_0^t
e^{(\la-2d)\tau}\sqrt{\big(r-Z(\tau)\big)^2{+}Y(\tau)^2}d\tau{\leq}\cr
& \leq q.
}
$$
Assume that on the attractor of system (13) we have
$$
\eqalign{
& (\la-2d)e^{(2d-\la)t}\intl_0^te^{(\la-2d)\tau}
\sqrt{\big(r-Z(\tau)\big)^2+Y(\tau)^2}\,d\tau\leq\cr
& \leq M, \qquad \forall\ t\geq0.
} \eqno(14)
$$
Then, choosing
$$
\kappa^2=\frac{d(\la-2d)}{(\la-2)M},
$$
we get the following estimate for $\la$:
$$
\frac {4\,dM}{(\la-2)(\la-2d)}<1. \eqno(15)
$$

It is easy to see that in ine\-qu\-a\-li\-ty (15) we can consider only
such $\la$ for which the left side of the ine\-qu\-a\-li\-ty is close to
$1$. In this case the ine\-qu\-a\-li\-ty $d/\kappa\geq2(b-1)$
is satisfied if $\la>4b-2$.

From the above reasoning, the results of [24,25,46],
and estimate (15) we have the following theorem.

\medskip

{\bf Theorem 3.}
{\it
If $d>b-2$ and ine\-qu\-a\-li\-ty {\rm(15)} holds for
$\la=2(d+b+1)/(1-S)$,
then for the Hausdorff dimension $\dim K$ of the
attractor $K$ we have the estimate $\dim K\leq\mh\leq
2+S$. If
this ine\-qu\-a\-li\-ty is valid for $\la=2(d+b+1)$, then any solution of
system {\rm(13)} tends to a certain  equilibrium as $t\to+\infty$.
}

\medskip

It is well known [24,25] that on attractor of system (13)
one can chosen the number $M$ such that it satisfies the
ine\-qu\-a\-li\-ty $M\leq\ga$,  where
$$
\ga=\cases
\ds r &\hbox{for}\ \ b\leq2,\\
\ds  \frac{br}{2\sqrt{b-1}} &\hbox{for}\ \ b\geq2.
\endcases
$$
Note that if $M=\ga$ and $1<b<d$, then
estimate  (15)  coincides with the well-known
Eden-Foias-Temam estimate [15].

Sometimes estimate (9) can be made more precise, taking into
account the peculiarities of the structure of the linear system.
For example, let us consider a system
$$
\eqalign{ &
\ds  \dot x_{i1}=a_{i1}x_{i1}+b_{i1}(t)\sg,\cr    &
\ds  \dot x_{i2}=a_{i2}x_{i2}+b_{i2}(t)x_{i1},\cr    &
\ds  \dot x_{i3}=a_{i3}x_{i3}+b_{i3}(t)x_{i2},\cr    &
\ds  \hbox to 6cm{. \dotfill .}\cr    &
\ds  \dot x_{im(i)}=a_{im(i)}x_{im(i)}+b_{im(i)}(t)x_{im(i)-1},
} \eqno(16)
$$

$$
\sg=\sum_{i=1}^rx_{im(i)},
$$
where $a_{ij}$ are  complex numbers; $b_{ij}(t)$  are complex
con\-ti\-nuo\-us func\-ti\-ons. We now  introduce
$$
\al_{ij}={Re}\,a_{ij},\quad
\beta_{ij}=\sup\limits_{t\geq0}\big|b_{ij}(t)\big|.
$$

\medskip

{\bf Theorem 4.}
{\it
Let for some number $\la$ the inequalities
$\la>\max\limits_{ij}\al_{ij}$,
$$
Q:=\sum_{i=1}^r
\prod_{j=1}^{m(i)}\frac{\beta_{ij}}{\la-\al_{ij}}<1.
\eqno(17)
$$
be valid.

Then
$$
\big|\sg(t)\big|\leq\frac C{1-Q}e^{\la t},
\qquad\forall\,t\geq0, \eqno(18)
$$
where $C$ is some constant depending on the initial data
$x_{ij}(0)$ and the values $\beta_{ij}$, $\ \al_{ij}$.
}

\medskip

{\bf Proof.}
For system (16) we have the following simple estimate
$$
\big|\sg(t)\big|\leq Ce^{\la t}+\sum_{i=1}^r\big|
\wt x_{im(i)}(t)\big|,\qquad\forall\,t\geq0, \eqno(19)
$$
where $\wt x_{ij}(t)$ are solutions of the corresponding scalar
dif\-fe\-ren\-ti\-al equ\-a\-ti\-ons of system (16) with initial data $\wt
x_{ij}(0)=0$, and $C$ is a certain sufficiently large number depending
on  $x_{ij}(0)$, $\al_{ij}$, $\beta_{ij}$.

On the other hand, it is well known (for example, from the
Wazewski ine\-qu\-a\-li\-ty) that there exist the numbers\linebreak
$R>0$ and $\ga>\la$ for which the estimate
$$
\big|\sg(t)\big|\leq R e^{\ga t},\qquad\forall\, t\geq0
\eqno(20)
$$
is valid.

Taking into consideration the estimate from the equ\-a\-ti\-ons of
system (16), we shall sequentially get
$$
\eqalign{ &
\ds \big|\wt x_{i1}(t)\big|\leq
e^{\al_{i1}t}\intl_0^tRe^{(\ga-\al_{i1})\tau}
\beta_{i1}\,d\tau\leq\frac{R\beta_{i1}}
{\la-\al_{i1}}e^{\ga t},\cr    &
\ds  \big|\wt x_{i2}(t)\big|\leq e^{\al_{i2}t}\intl_0^t
\frac{R\beta_{i1}}{\la-\al_{i1}}
e^{(\ga-\al_{i2})\tau}\beta_{i2}\,d\tau\leq\cr    &
\ds \leq\frac{R\beta_{i2}}
{(\la-\al_{i2})}\frac{\beta_{i1}}{(\la-\al_{i1})}
e^{\ga t},\cr    &
\ds \hbox to 6cm{.\dotfill.}\cr    &
\ds  \big|\wt x_{im(i)}(t)\big|\leq R
e^{\ga t}\prod_{j=1}^{m(i)}
\frac{\beta_{ij}}{\la-\al_{ij}}.
} \eqno(21)
$$
Thus, from ine\-qu\-a\-li\-ty (19) and above estimates one can derive
$$
\big|\sg(t)\big|\leq Ce^{\la t}+QRe^{\ga t},
\qquad\forall\,t\geq0. \eqno(22)
$$
Let us now make of use estimate (22) instead of  estimate (20) when
considering equalities (16) and estimate (19). Then we obtain the
following ine\-qu\-a\-li\-ty
$$
\big|\sg(t)\big|\leq Ce^{\la t}(1+Q)+Q^2Re^{\ga t},\qquad
\forall\,t\geq0.
$$
Repeating this procedure $k$ times, we get the following
estimate:
$$
\big|\sg(t)\big|\leq Ce^{\la t}(1+Q+\ldots+Q^k)+Q^kRe^{\ga t},
\qquad\forall\,t\geq0.
$$
Passing in this ine\-qu\-a\-li\-ty to the limit as $k\to+\infty$ we
obtain estimate (18).

\smallskip

Note also that the estimate of the exponential growth of
solutions (18) is invariant with respect to linear
tran\-s\-for\-ma\-ti\-ons of the phase space.

Let us remark that in the case $\la<0$ Theorems 1 and 4
are criteria of the stability of linear systems.
Sometimes Theorem 4  gives better result than a well-known
frequency domain circle criterion of stability [25]. It can be shown,
for example, in the case:
$r{=}1$, $\ \big|b_{1j}(t)\big|{\leq}1$,
$\ \forall\,t{\geq}0$, $\ \forall\,j$, $\ m(1)>1$.
Here estimate (18) gives the following condition of stability:
$\al_{1j}{<}0$, $\ \forall\,j$, $\ |\al_{11}\al_{12}{\ldots}\al_{1m(1)}|>1$.
The circle criterion leads to this result in the case
$b_{12}(t)\equiv\mh\equiv b_{13}(t){\equiv{\ldots}\equiv} b_{1m(1)}(t)\equiv1$
only.

Now consider the estimates of Lyapunov  exponents
$\nu_j^{}$  in the case $j=1,\ldots,n$.
For this purpose we continue our study of the following system
$$
\frac{dx}{dt}=A(t)x,\qquad x\in\br^n,
\eqno(23)
$$
where $A(t)$ is a real continuous $n\times n$-matrix.

Denote by $u_1(t){,}{\ldots}{,}u_n(t)$ an orthonormal system of
eigen\-vec\-tors of the matrix $X(t)^*X(t)$ such that
$X(t)^*X(t)u_j(t){=}\mh=
\rho_j(t)^2u_j(t)$. Recall [16] that the
linear operator $X(t)$ transforms the or\-tho\-nor\-mal system
$u_1(t),\ldots,u_n(t)$ into the or\-tho\-go\-nal one and
$\big|X(t)u_j(t)\big|=\rho_j(t)$.

We now consider a certain linear $k$-dimensional subspace
$L_k\subset\br^n$ and nonnegative function $\al(t)$.

\medskip

{\bf Lemma 3.}
{\it
Suppose that for any vector $v\in L_k$, the estimate
$|X(t)v|\geq\al(t)|v|$ is valid; then $\rho_k(t)\geq\al(t)$.
}

\medskip

{\bf Proof.}
Let us fix a value of the parameter $\ t\ $ and choose a vector $v{\in}L_k$
such  that $v^*u_j(t)=0$,  $\ \forall\,j=1,\ldots,k-1$.
It is obvious that there exists nonzero vector $v$ satisfying
these hypotheses. Assuming
$$
v=\sum_{j=k}^n\al_ju_j(t),\qquad \al_j\in\br^1,
$$
we find
$$
\al(t)^2|v|^2\leq\big|X(t)v\big|^2=\sum_{j=k}^n\al_j^2\rho_j(t)^2
\leq\rho_k(t)^2|v|^2.
$$

Lemma 3 is proved.

\smallskip

We now consider a symmetrical matrix $H$, which has no less
than $k$ negative eigenvalues, a quadratic form $V(x)=\mh=
x^*Hx$, and the set $\Omega=\big\{x|\,x^*Hx<0\big\}$.

\medskip

{\bf Lemma 4.}
{\it
Suppose that for some function $\la(t)$ and any vector $z\in\Omega$
the following ine\-qu\-a\-li\-ty
$$
z^*HA(t)z+\la(t)z^*Hz\leq0,\quad\forall\ t\geq0
\eqno(24)
$$
is valid.

Then there exists a positive number $\beta$ such that
$$
\rho_k(t)\geq \beta\exp\left(-\int_0^t\la(\tau)\,d\tau\right),
\quad\forall\ t\geq0. \eqno(25)
$$
}

\medskip

{\bf Proof.}
Let us rewrite estimate (24) as follows:
$$
\left(V\big(x(t)\big)\exp2\int_0^t\la(\tau)\,d\tau
\right)^{\bullet}\leq0.
$$
Hence we find that
$V\big(x(t)\big)\leq V\big(x(0)\big)\exp\left(-2\int_0^t\la(\tau)
\,d\tau\right)$. From here and the condition on the spectrum of the
matrix $H$ we obtain that there exist $k$-dimensional hyperplane
$L_k$ and the number $\beta$ such that for any $x(0){\in}L_k$
we have the estimate $|x(t)|{\geq}\beta|x(0)|\exp
\big({-}\intl_0^t\la(\tau)d\tau\big)$. This estimate and Lemma
3 prove Lemma 4. The following assertion is proved in the same
way.

\medskip

{\bf Lemma 5.}
{\it
If for any vector $v\in L_k$ the following estimate
$\big|X(t)v\big|\leq
\al(t)|v|$is true, then $\rho_{n-k+1}^{}(t)\leq\al(t)$.
}

\medskip

Consider now the symmetrical matrix $M$ having no less than $k$
positive eigenvalues, the quadratic form $W(x)=\mh=
x^*Mx$, and the set $\Phi=\big\{x|\,x^*Mx>0\big\}$.

\medskip

{\bf Lemma 6.}
{\it
Suppose that for some function $\psi(t)$ and any vector $z\in\Phi$
we have
$$
z^*MA(t)z+\psi(t)z^*Mz\leq0,\quad\forall\ t\geq0.
\eqno(26)
$$
Then there exists a positive number $C$ such that
$$
\rho_{n-k+1}^{}(t)\leq C\exp\left(-\int_0^t\psi(\tau)
\,d\tau\right),\quad\forall\ t\geq0. \eqno(27)
$$
}

\medskip

{\bf Proof.}
Consider the system
$$
\dot y=-A(t)^*y
$$
and its fundamental matrix $Y(t)$ with initial data
$Y(0)=I$. It is well known [12] that
$$
Y(t)^*Y(t)=\big(X(t)^*X(t)\big)^{-1}.
$$
Therefore $\ga_j^{}(t)=\rho_{n-j+1}(t)^{-1}$, where $\ga_j^{}(t)$
are singular numbers of the matrix $Y(t)$ such that
$\ga_1(t)\geq\ldots\geq\ga_n(t)$.

We now introduce the following notations: $H{=}-M$, $\la(t){=}-\psi(t)$,
$\,A_0(t)=-A(t)^*$, $\,C=\beta^{-1}$. Then condition (26) takes
the same form as (24) with $A(t)=A_0(t)$. It follows from Lemma 4
that
$$
\ga_k(t)\geq\beta\exp\intl_0^t\psi(\tau)\,d\tau,
\quad\forall\ t\geq0.
$$
Thus,
$$
\rho_{n-k+1}^{}(t)=\ga_k(t)^{-1}\leq
C\exp\left(-\int_0^t\psi(\tau)
\,d\tau\right),\quad\forall\ t\geq0.
$$

Consider system (23) with $A(t)=A+BU(t)$, where $A$
is a constant $n\times n$-matrix, $B$ is a constant
$n\times m$-matrix, $U(t)$ is a continuous $m\times n$-matrix.

Let us introduce the Hermitian forms $F_k(z,\xi)$ of the complex vector
variables $z\in\bc^n$, $\,\xi\in\bc^m$, $\ k=1,\ldots,n$.

Assume that there exist the $n\times m$-matrices $Q_k$
and the positive number $\ve$ such that the following inequalities
$$
F_k(x,Q_k^*x)\geq\ve|Q_k^*x|^2,\qquad\forall\ x\in\br^n
\eqno(28)
$$
are satisfied. We also suppose that
$$
F_k\big(x,U(t)x\big)\geq0,\qquad\forall\ x\in\br^n,\quad
\forall\ t\geq0. \eqno(29)
$$

\medskip

{\bf Theorem 5.}
{\it
Let the pair $(A,B)$ be controllable, the pairs $(A,Q_k)$
be observable and for some set of the numbers
$\la_1\leq\ldots\leq\la_n$ the following hypotheses be satisfied:

1) the matrix $A+BQ_k^*+\la_kI$ has $k$ or more eigenvalues with
positive real parts;

2) for all $k=1,\ldots,n$ and all $\om\in\br^1$ we have
$$
F_k\big(\big[(i\om-\la_k)I-A\big]^{-1}B\xi,\xi\big)\leq0,
\qquad\forall\ \xi\in\bc^m. \eqno(30)
$$

Then for the singular numbers $\rho_k(t)$ of the
fundamental matrix of system {\rm(23)} there exist numbers
$\beta_k>0$ such that
$$
\rho_k(t)\geq\beta_ke^{-\la_kt},\qquad\forall\ t\geq0,\ \,
k=1,\ldots,n. \eqno(31)
$$
}

\medskip

{\bf Proof.}
From the hypothesis 2) of the theorem by the Yakubovich-Kalman lemma
[24,25] it can be found the symmetrical matrix $H_k$ for which
the following ine\-qu\-a\-li\-ty
$$
\eqalign{
& 2z^*H_k\big[(A+\la_kI)z+B\xi\big]+F_k(z,\xi)\leq0,\cr
& \forall\ z\in\br^n,\ \forall\ \xi\in\br^m.
} \eqno(32)
$$
is true. Setting in (32) $\xi=Q^*x$ and taking into account (28),
we obtain
$$
2z^*H_k(A+\la_kI+BQ_k^*)z\leq-\ve|Q_k^*z|^2,\qquad\forall\ z\in\br.
\eqno(33)
$$
From ine\-qu\-a\-li\-ty (33) and the hypothesis 1) of the theorem it follows
that the matrix $H_k$ has no less than $k$ negative eigenvalues
[24,25].

From relations (29) and (32) we concluded that for the
function $V_k(x)=x^*H_kx$ and any solution $x(t)$ of system (23)
we have
$$
\eqalign{
& \dot V_k\big(x(t)\big)+2\la_kV_k\big(x(t)\big)=2x(t)^*H_k
\big[(A+\la_kI)x(t)+\cr
& +BU(t)x(t)\big]+F_k\big(x(t),U(t)x(t)\big)-\cr
&  -F_k\big(x(t),U(t)x(t)\big)\leq0, \qquad\forall\ t\geq0.
}
$$
This ine\-qu\-a\-li\-ty and Lemma 4 result in the assertion of Theorem 5.

\smallskip

By the same proof and Lemma 6 we can obtain the following result.

\medskip

{\bf Theorem 6.}
{\it
Let the pair $(A,B)$ be controllable, the pairs $(A,Q_k)$
be  observable and for some set of numbers $\la_1\geq\ldots\geq \la_n$
the following hypotheses be satisfied:

1) the matrix $A+BQ_k+\la_kI$ has $k$ or more eigenvalues with
negative real parts;

2) for all $k=1,\ldots,n$ and all $\om\in\br^1$ we have
$$
F_k\big([(i\om-\la_k)I-A]^{-1}B\xi,\,\xi\big)\leq0,\quad
\forall\ \xi\in\bc^m. \eqno(34)
$$

Then for the singular numbers $\rho_k(t)$ of the fundamental
matrix of system {\rm(23)} there exist positive numbers
$\beta_k$ such that
$$
\rho_{n-k+1}^{}(t)\leq\beta_ke^{-\la_kt},\quad\forall
\ t\geq0. \eqno(35)
$$
}

Let us now consider system (23) with $m=1$ and $U(t)=\mh=
u(t)C^*$,
where $C$ is a constant $n$-vector. Introduce the transfer
function of system (23) $W(p)=C^*(A-pI)^{-1}B$, $\ p\in\bc^1$.
Further, we assume that the function $W(p)$ is nondegenerate
[24,25] and the function $u(t)$ satisfies the inequalities
$0\leq u(t)\leq\mu$, $\ \forall\,t\geq0$.
Let us recall that $W(p)$ is nondegenerate if it has the
denumeretor of degree $n$ and the numeretor  has no common
zeroes with its denominator. The nondegeneracy is equivalent to
controllability and observability.

\medskip

{\bf Theorem 7.}
{\it
Let for some set of numbers $\la_1\leq\ldots\leq\la_n$ the
following hypotheses be satisfied:

1) the matrix $A+\la_kI$ has $k$ or more eigenvalues with
positive real parts;

2) for all $k=1,\ldots,n$ and all $\om{\geq}0$ the ine\-qu\-a\-li\-ty
$\mu^{-1}+\mh+
\hbox{Re}\,W(i\om-\la_k){\geq}0$ is valid.

Then the assertion of Theorem 5 is true.
}

\medskip

To prove of Theorem 7 it is sufficient to make use of Theorem 5
with the following Hermitian form
$$
F_k(z,\xi)=\xi^*(C^*z-\mu^{-1}\xi)+(C^*z-\mu^{-1}\xi)^*\xi
\eqno(36)
$$
and take $Q_k=\de C$, where $\de$ is a sufficiently small
positive number. The controllability of $(A,B)$ and observability
of $(A,Q_k)$ result from non\-de\-ge\-ne\-ra\-cy of $W(p)$.

\medskip

{\it Remark 1.}
Hypothesis 2) of Theorem 2 can be easily written in more generalized
form of circle criterion [24,25].

\medskip

{\it Remark 2.}
Let us recall the definition of Lyapunov exponents $\nu_j^{}$
$$
\nu_j^{}=\ln\varlimsup\limits_{t\to+\infty}
\rho_j^{}(t)^{1/t}.
$$

By this notation estimate (31) takes a very simple form:
$\nu_j^{}\geq-\la_j$.

\medskip

{\it Remark 3.}
The hypotheses of Theorem 7 are always satisfied for sufficiently
large $\la_n$. Therefore from Theorem 7 we can always find estimate
(31) for $\rho_n(t)$. On the other hand, for sufficiently small
$\mu$ the hypotheses of Theorem 7 are satisfied for $\la_k$, which are
close to $-\hbox{Re}\,\k_k^{}$. Here $\k_k^{}$ are eigenvalues
of the matrix $A$ such that
$\hbox{Re}\,\k_1^{}\geq\ldots\geq\hbox{Re}\,\k_n$.

The same considerations can be made by Theorem 6 for the upper
estimates $\nu_j^{}$.

Let us consider a system
$$
\frac{dx}{dt}=Ax+Bf(x),\qquad x\in\br^n,
\eqno(37)
$$
where $A$ and $B$ are constant $n\times n$ and $n\times m$-matrices,
$f(x)$ is a continuously differentiable vector function.

Denote  $\ga(x)=\hbox{Tr}\left(A+B\frac{\pr f}{\pr x}
\right)$. Here $\frac{\pr f}{\pr x}$ is a Jacobi matrix of
the vector-function $f(x)$ at the point $x$.

We now assume that the compact set  $K$ is invariant with
respect to the shift operator along the trajectories of system
(37), i.e., $x(t,K)=
K$, $\forall\,t\in\br^1$. By
$\dim K$ we denote the Hausdorff dimension $K$.

Let us suppose that condition (28) is satisfied and instead of
ine\-qu\-a\-li\-ty (29) we have the following relation:
$$
F_k\bigg(z,\frac{\pr f}{\pr x}z\bigg)\geq0,\qquad\forall\ z\in\br^n,
\quad\forall\ x\in K. \eqno(38)
$$

\medskip

{\bf Theorem 8.}
{\it
Let the pair $(A,B)$ be controllable, the pairs $(A,Q_k)$
be observable and for a set of the numbers
$\la_1\leq\ldots\leq\la_n$ the hypotheses 1) and 2) of Theorem 5 be
satisfied.

Let for some natural number $m$ and for
$s\in[0,1]$ the following ine\-qu\-a\-li\-ty
$$
\ga(x)+(1-s)\la_{m+1}^{}+\sum_{j=m+2}^n\la_j<0,
\qquad\forall\ x\in K
\eqno(39)
$$
be also valid. Then $\dim K<m+s$.
}

\medskip

{\bf Theorem 9.}
{\it
Let the pair $(A,B)$ be controllable, the pairs $(A,Q_k)$ be
observable and for some set of numbers $\la_1\geq\ldots\geq\la_n$
the hypotheses 1) and 2) of Theorem 6 be satisfied.

Let for some natural number $m$ and for
$s\in[0,1]$ the following ine\-qu\-a\-li\-ty
$$
\sum_{j=n-m+1}^n\la_j+s\la_{n-m}>0
\eqno(40)
$$
be true.\newline
Then $\dim K<m+s$.
}

\medskip

Suppose that system (37) is the Levinson dissipative one [12].
Suppose also that its domain of dissipativity $\Om$ is generic
in $\bc^1$-topology and it is a bounded simply connected open set in $\br^n$.
In other words, there exists a positive number $\ve$ such that
for any continuously dif\-fe\-ren\-ti\-ab\-le vector-function $g(x)$,
satisfying the following condition
$$
\big|f(x)-g(x)\big|+\left|\frac{\pr f}{\pr x}-\frac{\pr g}{\pr x}
\right|\leq\ve,\qquad\forall\ x\in\br^n,
$$
and for any $x_0\in\br^n$ a number $\ T(x_0,g)$ can be found,
for which $\ x(t,x_0)\in\Om$,
$\ \forall\,t\geq T(x_0,g)$.
Here $x(t,x_0)$ is a solution of the system $\dot x=Ax+Bg(x)$
with initial data $x(0,x_0)=x_0$.

\medskip

{\bf Theorem 10.}
{\it
Let all hypotheses of Theorem 5 and condition {\rm(38)} for any
$z\in\br^n$ and for any $x\in\Om$, and condition {\rm(39)}
for any $x\in\Om$ and for $m=2$, $\,s=0$ be satisfied.

Then any solution of system {\rm(37)} tends to a stationary set
as
$t\to+\infty$.
}

\medskip

{\bf Theorem 11.}
{\it
Let all hypotheses of Theorem 6, con\-di\-ti\-on {\rm(38)} for any
$z\in\br^n$ and for any $x\in\Om$, and condition {\rm(40)}
for $m=2$, $\,s=0$ be satisfied.

Then any solution of system {\rm(37)} tends to a stationary
set as $t\to+\infty$.
}

\medskip

The following estimate
$$
\eqalign{ &
\ds \rho_1^{}(t)\cdots\rho_d^{}(t)\rho_{d+1}^s(t)\leq\beta
\exp\bigg\{\int_0^t\bigg[\ga\big(x(\tau)\big)+\cr    &
\ds +(1-s)\la_{d+1}^{}+\sum_{j=d+2}^n\la_j
\bigg]\,d\tau\bigg\},
}
$$
which results from Theorem 5 is the basis for the proof of
Theorems 8 and 10. Here $\beta$ is some constant,
$\rho_j^{}(t)$ are singular numbers of the fundamental matrices
of linear systems, ob\-tai\-ned as a result of the linearization of
system (37) along its solutions $x(t)$.

Further, the proofs of Theorems 8 and 10 are the same as those
given in [23--25].

Theorems 9 and 11 are proved in a similar way.

Let in system (37) be $m=1$, $\ f(x)=\vp(C^*x)$, where $C$
is a constant $n$-vector, and the following ine\-qu\-a\-li\-ty
$$
0\leq\frac{d\vp}{d\sg}\leq\mu,\qquad\forall\ \sg\in
\{\sg=r^*x,\ x\in K\} \eqno(41)
$$
be valid.

\medskip

{\bf Theorem 12.}
{\it
Let all hypotheses of Theorem 7 be satisfied.

If inequalities {\rm(39)} and  {\rm(41)} are valid, then
$\dim K<\mh<
m+s$.

If the simply-connected domain of dissipativity $\Om$ is ge\-ne\-ric
in $\bc^1$-topology and inequalities {\rm(39)},  {\rm(41)} hold
for all $x\in\Om$ and for $m=2$, $\,s=0$,  then any solution of
system {(37)} tends to the stationary set as $t\to+\infty$.
}

\medskip

This theorem as well as Theorem 7 is deduced from Theorems 8 and
10 by the Her\-mi\-ti\-an form (36).

Theorem 12 generalizes the results of R.Smith [45] and gives us
more precise estimates. Note that recently years for
analysing the global stability the method of using the compound
matrices has been developed [28,32].

As an example of application of the theorems obtained, let us
consider the Lorenz system
$$
\dot X{=}-d(X-Y),\quad\dot Y{=}(r-Z)X-Y,\quad\dot Z{=}-bZ+XY
\eqno(42)
$$
and some of its generalizations. Here $d$, $\,r$, $\,b$
are positive numbers.

We now write system (42) in the form of (37), where
$$
A=\left(\matrix
-d&d &0\cr
 0&-1&0\cr
 0&0&-b
\endmatrix \right),\quad
B=\left(\matrix 0&0\cr     1&0\cr     0&1\endmatrix\right),
$$

$$
f(x)=\left(\matrix(r-Z)X\cr     XY
\endmatrix\right).
$$
Let us construct the Hermitian form $F_k(z,\xi)$ as follows:
$F_k(z,\xi)=\bar\xi_1z_2+\bar\xi_2z_3+\xi_1\bar z_2+\xi_2\bar
z_3+ M\big(\k|z_1|^2+\k^{-1}|z_2|^2+\mh+
\k^{-1}|z_3|^2\big)$. Here
$\xi_j^{}$ and $z_j^{}$ are components of the vectors $\xi$ and $z$,
the positive numbers $M$ and $\k$ will be chosen later on.

In the case under consideration ine\-qu\-a\-li\-ty (38) takes the form:
$2\big[(r-Z)z_2+Yz_3\big]z_1+M\big(\k|z_1|^2+\k^{-1}|z_2|^2+\mh+
\k^{-1}|z_3|^2\big)\geq0$. It can be easily seen that this
condition is satisfied for all $z\in\br^3$ and for the following
ine\-qu\-a\-li\-ty:
$$
(r-Z)^2+Y^2\leq M^2. \eqno(43)
$$
It is well known [24,25] that a certain domain of dis\-si\-pa\-ti\-vi\-ty
$\Om$ and, therefore, any invariant compact set $K$ of the Lorenz
system are situated in the set, defined by ine\-qu\-a\-li\-ty (43) with
$$
M=\cases
r& \hbox{for}\ \  b\leq 2,\\
\ds \frac{br}{2\sqrt{b-1}}& \hbox{for}\ \ b\geq2.
\endcases \eqno(44)
$$
Thus, if $M$ is defined by equality (44), then condition (38) is
satisfied.

In our case we have
$$
\eqalign{
& [pI-A]^{-1}B\xi=\cr
& =\left(\frac{d}{(p+d)(p+1)}\xi_1,
\frac1{(p+1)}\xi_1,\frac1{(p+b)}\xi_2\right)^T,\quad
p\in\bc^1.
}
$$
Here the notation $(\ \,)^T$ is an operation of transposition.
Therefore condition (30) is as follows:
$$
2(\la_k-1)\geq\k^{-1}M+\frac{\k d^2M}{(\la_k-d)^2}\,,
\qquad 2(\la_k-b)\geq\k^{-1}M.
$$
If we take here $k=3$, $\ \k=d^{-1}(\la_3-d)$ and write
ine\-qu\-a\-li\-ty(39) in the form: $\la_3(1-s)<d+b+1$, then we
finally obtain the following assertion.

\medskip

{\bf Corollary 2.}
{\it
If $d\geq b-2$ and for some $s\in[0,1]$
$$
\left(\frac{d+b+1}{1-s}-1\right)\left(\frac{d+b+1}{1-s}-
d\right)>Md, \eqno(45)
$$
then $\dim K<2+s$. If {\rm(45)} holds for $s=0$,
then any solution of system {\rm(42)} tends to a certain
equilibrium as $t\to+\infty$.
}

\medskip

It can be remarked that from Corollary 2 we can  find a well-known
estimate of Eden-Foias-Temam [11]. Note that earlier by using the other
methods we have obtained a similar result (see Theorem 3). In our opinion, the
advantage of the ``frequency" approach is that it is
``almost automatically"
transferred to more complicated equations than system (42).
Let us consider, for example, the following system [44]:
$$
\eqalign{ &
\ds \dot X_1=\sg_1(Y_1-X_1),\quad \dot Y_1=-Y_1+X_1(r_1-Z),\cr    &
\ds \dot Z=-bZ+X_1Y_1+X_2Y_2,\cr    &
\ds \dot X_2=\sg_2(Y_2-X_2),\quad \dot Y_2=-Y_2+X_2(r_2-Z),
} \eqno(46)
$$
where $r_1$, $\,r_2$, $\,b$, $\,\sg_1$, $\,\sg_2$ are positive
numbers. Without loss of generality, we shall assume that
$r_2\geq r_1$ and denote: $\al_0^{}=\min(1,\sg_1,b)$,
$\ L=r_2$ for $b\leq2\al_0^{}$, $\ \,L=r_2b\big(4\al_0^{}
(b-\mh-
\al_0^{})\big)^{-1/2}$ for $b\geq2\al_0^{}$,
$$
V(x)=Y_1^2+Y_2^2+\frac{r_2-r_1}{\sg_1}X_1^2+
(r_2-Z)^2.
$$

\medskip        
{\bf Lemma 7.}
{\it
There exists a domain of dissipativity $\OM$ for system (46) such
that $\OM\subset\big\{V(x)\leq L^2\big\}$.
}

\medskip

The proof of the lemma is based on the consideration of the
ine\-qu\-a\-li\-ty $\dot V\big(x(t)\big)+2\al V\big(x(t)\big)\leq
\frac12\,r_2^2b^2(b-\al)^{-1}$ for $\al\in[0,\al_0^{}]$.

Write system (46) in the form:
$$
A=\left(\matrix -\sg_1&\sg_1&0&0&0\cr
0&-1&0&0&0\cr
0&0&-b&0&0\cr
0&0&0&-\sg_2&\sg_2\cr
0&0&0&0&-1\endmatrix\right),\
B=\left(\matrix 0&0&0\cr     1&0&0\cr     0&1&0\cr
0&0&0\cr     0&0&1 \endmatrix\right),
$$

$$
f(x)=\left(\matrix X_1(r_1-Z)\cr
X_1Y_1+X_2Y_2\cr
X_2(r_2-Z)
\endmatrix\right).
$$
Let us construct the Hermitian form $F_k(z,\xi)$ in the fol\-lo\-wing
way: $F_k(z,\xi)=\bar\xi_1z_2+\bar\xi_2z_3+\bar\xi_3z_5+\xi_1\bar z_2+
\xi_2\bar z_3+\xi_3\bar z_5+\mh{+}
L_1\big(\k_1^{}|z_1|^2{+}\k_1^{-1}|z_2|^2+
(\k_1^{-1}+\k_2^{-1})|z_3|^2{+}\k_2^{}|z_4|^2{+}\k_2^{-1}
|z_5|^2\big)$. Here $L_1=L+r_2-r_1$. The parameters  $\k_1^{}$
and $\k_2^{}$ will be defined later on.

In our case ine\-qu\-a\-li\-ty (38) will be
satisfied, if
$$
(r_1-Z)^2+Y_1^2\leq L_1^2,\qquad (r_2-Z)^2+Y_2^2\leq L_1^2.
\eqno(47)
$$
It follows from Lemma 7 that these inequalities are valid in the
set $\Om\subset\big\{V(x)\leq L^2\big\}$. Therefore, condition (38)
is satisfied.

Here condition (30) is equivalent to the inequalities
$$
-2\hbox{Re}\frac1{i\om+1-\la_k}{\geq}
\frac{L_1}{|i\om+1-\la_k|^2}
\left(\frac1{\k_1^{}}{+}\frac{\k_1^{}\sg_1^2}
{|i\om{+}\sg_1-\la_k|^2}\right),
$$

$$
-2\hbox{Re}\frac1{i\om+1-\la_k}{\geq}
\frac{L_1}{|i\om+1-\la_k|^2}
\left(\frac1{\k_2^{}}{+}\frac{\k_2^{}\sg_2^2}
{|i\om{+}\sg_2{-}\la_k|^2}\right),
$$

$$
-2\hbox{Re}\frac1{i\om+b-\la_k}\geq L_1\left(\k_1^{-1}+\k_2^{-1}\right)
\frac1{|i\om+b-\la_k|^2}.
$$
Setting $\k_i=\sg_i^{-1}|\la_k-\sg_i|$, we shall write these
conditions as follows:
$$
\la_k-1\geq\frac{L_1\sg_i}{|\la_k-\sg_i|},
$$

$$
2(\la_k-b)\geq L_1\left(\frac{\sg_1}{|\la_k-\sg_1|}{+}
\frac{\sg_2}{|\la_k-\sg_2|}\right).
\eqno(48)
$$

\medskip

{\bf Corollary 3.}
{\it
Let $\min(\sg_1,\sg_2)>\max(1,b)$ and there exist numbers
$\la_4\in(\sg_1,\sg_2)$ and $\la_5>\max(\sg_1,\sg_2)$ such that
ine\-qua\-li\-ti\-es {\rm(48)} and $\sg_1+\sg_2+b+2>\la_5+(1-s)\la_4$,
$\ s\in[-1,1]$ are valid.

Then for any invariant compact set $K$ of system {\rm(46)}
we have $\dim K<3+s$.
}

\medskip

We shall formulate a conjecture which is a
certain natural ge\-ne\-ra\-li\-za\-ti\-on of the results obtained.

Let us consider a system
$$
\dot x=G(x),\qquad x\in\br^n
\eqno(49)
$$
and denote by $\k_j^{}(x)$ the eigenvalues of the mat\-rix
$\frac{\pr G}{\pr x}$, assuming that $\hbox{Re}\,\k_1^{}(x)
{\geq}{\ldots}{\geq}\hbox{Re}\,\k_n(x)$.

\medskip

{\bf Conjecture}.
If for some $s\in[0,1]$
$$
\hbox{Re}\,\k_1^{}(x)+\ldots+\hbox{Re}\,\k_d^{}(x)+
s\,\hbox{Re}\,\k_{d+1}^{}(x)<0,\qquad\forall\ x\in K,
$$
then $\dim K<d+s$,
where $K$ is a certain invariant compact set of system (49).




