\def\vk{\kappa}

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

\bigskip

\centerline{\bf ON STABILITY BY THE FIRST}
\centerline{\bf APPROXIMATION}

\bigskip

\bigskip

What  can be found in the neighbourhood of a solution of a
nonlinear differential equation if the Lya\-pu\-nov exponents of its
linearization are known? Most of the classical results in this
case are concerned with the investigation of the Lya\-pu\-nov
stability. A.M.Lya\-pu\-nov has proved that if a linear system of the
first approximation is a regular one and all of its Lya\-pu\-nov
exponents are negative, then the solution considered is
asymptotically stable [29]. O.Perron has shown [36] that the
requirement of regularity is essential. He gave an example of the
Lya\-pu\-nov unstable solution, a linearization along which was
irregular and had negative Lya\-pu\-nov exponents. N.G.Chetaev has
obtained the same theorem for instability [9,10]: if a linear
system of the first approximation is regular and at least one of
its Lya\-pu\-nov exponents is positive, then the solution is
Lya\-pu\-nov unstable.

When investigating the attractors of dynamical systems it is often
necessary to study not individual solutions but certain
ensambles of solutions [11,33,38,39,44]. By now many
computational methods have been created and many com\-pu\-ter experiments
have been carried out, where the Lya\-pu\-nov exponents of such
ensambles have been computed [3,33,44]. As far as such analysis
of attractors is concerned it is seems quite natural to replace
the laborious requirement that the first approximation is regular
by the requirement that the sign of Lyapunov exponents does not
change at "small perturbations of initial data $x_0$".
It turns out that such replacement is possible with the
help of Theorems 1,2 which will be proved below.


Note that we have to use other, "not Lyapunov" notion of
stability for the analysis of periodic solutions of autonomous
systems.
First of all, it is the notion of orbital
stability introduced by A.Poincare (further, it will also be
called the Poincare stability). In this case the well-known
An\-d\-ro\-nov-Vitt theorem and its generalization given by
B.P.De\-mi\-do\-vich for the case of aperiodic trajectoriies [13]
are concerned. B.P.Demidovich has shown that if a  linear
approximation along a certain bounded trajectory is regu\-lar, if it has one
zero Lya\-pu\-nov exponent, and all its other Lya\-pu\-nov exponents
are negative, then the trajectory considered is orbitally stable.
Theorem 3,  which will be proved below, permits us here also to take
off the property of regularity, replacing  it by the requirement
that Lyapunov exponents remain negative for some naturally ap\-pea\-ring
linearization at small perturbations of the initial states $x_0$.

When studying instability of trajectories on attractors
we have to use another concept of instability ---
Zhu\-kovs\-ky instability [24,26]. In order to illustrate the
difficulties, appearing here, let us recall the main definitions of
stability for a system
$$
\frac{dx}{dt}=f(x),\qquad x\in\br^n,\quad f\in\bc^2.
\eqno(1)
$$

\medskip

{\bf Definition 1.}
{\it
A solution $x(t,x_0^{})$ of system {\rm(1)} with initial data
$x(0,x_0^{})=x_0^{}$ is called Lya\-pu\-nov stable if for any number
$\ve>0$ there exists a number $\de(\ve)>0$ such that for any
vector $y_0^{}$, satisfying the inequality
$|x_0^{}-y_0^{}|\leq\de(\ve)$,
and for any $t\geq0$ the following inequality is satisfied:
$\big|x(t,x_0^{})-\mh-
x(t,y_0^{})\big|\leq\ve$.
If furthermore, for some number $\de_0$  and all $y_0^{}$ from
the ball $\{y|\ |y-x_0^{}|\leq\de_0\}$ the relation
$$
\liml_{t\to+\infty}\big|x(t,x_0^{})-x(t,y_0^{})\big|=0
$$
is satisfied, then the solution $x(t,x_0^{})$ is called Lya\-pu\-nov
asym\-p\-to\-ti\-cal\-ly  stable.
}

\medskip

Here $|\,\cdot\,|$ is an Euclidean norm in $\br^n$.

Introduce now the following notation: $L^+(x_0^{}){=}
\{x(t,x_0^{})|\mh
0\leq t<+\infty\}$. Thus, the set $L^+(x_0^{})$
is a positive semitrajectory of system (1).

\medskip

{\bf Definition 2.}
{\it
A solution $\!x({t}{,}x_0{)}$ of system (1) is called Poincare
stable (or orbital stable), if for any number $\ve>0$
there exists  a number $\de(\ve)>0$ such that for any vector
$y_0^{}$ satisfying the inequality
$|x_0^{}-y_0^{}|\leq
\de(\ve)$ and for any $t\geq0$ the inequality
$$
\rho\big(x(t,y_0^{}),\,L^+(x_0^{})\big)\leq\ve
\eqno(2)
$$
is satisfied. If furthermore, for some number $\de_0$ and all
$y_0^{}$ from the ball $\{y|\ |y-x_0^{}|\leq\de_0\}$ the relation
$$
\liml_{t\to+\infty}\rho\big(x(t,y_0^{}),\,L^+(x_0^{})\big)=0,
$$
is satisfied, then the solution $x(t,x_0^{})$ is called Poincare
asymptotically stable (or asymptotically orbitally
stable).
}

\medskip

Here $\rho(z,L)$ is the distance between a point  $z$ and a set $L$:
$$
\rho(z,L)=\inf\limits_{y\in L}|z-y|.
$$

In order to define Zhu\-kovs\-ky stability, we shall need the following
set of homeomorphisms
$$
\hom=\big\{\tau(\,\cdot\,)|\,\tau{:}[0,+\infty)\to[0,+\infty),
\ \tau(0)=0\big\}.
$$
Functions $\tau(t)$ from the set $\hom$ will play a role of time
re\-pa\-ra\-met\-ri\-za\-ti\-ons for the trajectories of system (1).

\medskip

{\bf Definition 3.}
{\it
A solution $x(t,x_0^{})$ of system (1) is called Zhu\-kovs\-ky stable
if for any number $\ve>0$ there exists a number $\de(\ve)>0$
such that for any vector $y_0^{}$, satisfying inequality
$|x_0^{}-y_0^{}|\leq\de(\ve)$, a function $\tau(\,\cdot\,)\in\hom$
can be found such that the inequality
$\big|x(t,x_0^{})-
x(\tau(t),y_0^{})\big|\leq\ve$, $\forall\,t\geq0$ is satisfied.

If furthermore, for some number $\de_0>0$ and any $y_0^{}$
from the ball $\{y|\ |x_0^{}-y_0^{}|\leq\de_0\}$ a function
$\tau(\,\cdot\,)\in\hom$ can be found such that the relation
$$
\liml_{t\to+\infty}\big|x(t,x_0^{})-x(\tau(t),y_0^{})\big|=0
$$
is satisfied, then the solution $x(t,x_0^{})$ is called Zhu\-kovs\-ky
asymptotically  stable.
}

In other words, Zhu\-kovs\-ky stability is Lya\-pu\-nov stability under an
adequate reparametrization each of the perturbed trajectories.

Recall that Lya\-pu\-nov (or Poincare, Zhu\-kovs\-ky) insta\-bi\-li\-ty
means the negation of corresponding type of stability.

Obviously, Lya\-pu\-nov  stability implies Zhu\-kovs\-ky stabili\-ty and
Zhu\-kovs\-ky stability implies Poincare stability.

Note that in all these definitions we suppose as usual that
all solutions considered are defined for all $t\in[0,+\infty)$.

Recall that all tree definitions are equivalent for equi\-lib\-ri\-um
points. For periodic solutions it is easy to prove the
equivalence of Poincare  stability and Zhu\-kovs\-ky stability
[24,26]. For nonlinear
systems the situation often appears when a periodic solution is
Zhu\-kovs\-ky asymptotically  stable (and is consequently Poincare
stable), but is Lya\-pu\-nov unstable [24,26]. In this case we
introduce a concept of asymptotic phase $c(y_0^{})$, which
corresponds for Zhukovsky asymptotic stability to
re\-pa\-ra\-met\-ri\-za\-ti\-on $\tau(t)=t+c(y_0^{})$.
Hence it follows that Lya\-pu\-nov instability can not
cha\-rac\-te\-ri\-ze a property of  ``repulsion of trajectories
from each other" on such internally unstable objects as  strange
attractors.

Poincare instability also can not characterize the ``re\-pul\-si\-on
of trajectories" on a strange attractor.
Though the reason of it quite different.
In numerical experiments often at least one
se\-mi\-tra\-jec\-to\-ry $x(t,x_0^{})$ 
fills the attractor everywhere densely. (This is the situation
with Lorenz attractor [11].)
But then inequality (2) is satisfied for any
$\ve>0$ and any point $y_0^{}$ of the attractor. Consequently,
all such trajectories  $x(t,x_0^{})$ are Poincare stable.

But in experiments on a  strange attractor in a course of time
``repulsion of the trajectories from each other" is observed. Such
``repulsion" corresponds to Zhu\-kovs\-ky instability. That is
way Zhu\-kovs\-ky  instability describs the behaviour of
trajectories on strange attractors most adequately.

Various examples, demonstrating Zhu\-kovs\-ky unstable\linebreak
flows on two-dimensional compact manifolds, are given in [24,26].

A very important fact in connexion with Zhukovsky stability and
instability is that certain linearizations, dif\-fe\-rent from
classical ones, appear here in a natural way. To describe
these linearization we suppose that all
considered semitrajectories $x(t,x_0^{})$, $\,t\geq0$, of system
(1) are disposed in a compact set $G$ and $f(x)\neq0$,
$\forall\,x\in G$. In this case the following result on the existence
of a special reparametrization of perturbed trajectories is
true.

\smallskip

{\bf Lemma 1} [24,26].
{\it
For any number $T>0$ there exists a number $\de(T)>0$ such that
for  any vector $y_0^{}$ from the
set $\{y|\ |x_0^{}-y|\leq
\de(T), \ (y-x_0^{})^*f(x_0^{})=0\}$
a differentiable function $\tau(\,\cdot\,)\in\hom$  can be found,
which satisfies the relation
$$
\big(x(t,x_0^{})-x(\tau(t),y_0^{})\big)^*f\big(x(t,x_0^{})\big)=0,
\quad \forall\ \ t\in[0,T].
\eqno(3)
$$
In addition, we have

\vskip-2mm
$$
\eqalign{
& \frac{d\tau(t)}{dt}=1{-}\frac{f(x(t,x_0^{}))^*}{|f(x(t,x_0^{}))|^2}
\,\bigg(\frac{\pr f}{\pr x}\,\big(x(t,x_0^{})\big)+\cr
& +\left(\frac{\pr f}{\pr x}\,\big(x(t,x_0^{})\big)\right)^*\bigg)\times
\big(x(t,x_0^{})-x(\tau(t),y_0^{})\big)+\cr
& +O\big(|x(t,x_0^{})-x(\tau(t),y_0^{})|^2\big).
} \eqno(4)
$$
}

\smallskip

Here $\frac{\pr f}{\pr x}\,(x(t,x_0^{}))$ is   the Jacobi
matrix  for the vector function $f$ at the point $x(t,x_0^{})$.
By $O(v)$ we denote a value, such that for sufficiently small $v$
the estimate $|O(v)|\leq cv$ is satisfied, where $c$ is a number.
Here symbol $*$ denotes the conjugation.

From relation (4) it follows that for the re\-pa\-ra\-met\-ri\-za\-ti\-on
$\tau(t)$, indicated in the lemma, a linear system of the first
approximation along the trajectory $x(t,x_0^{})$ has the form
$$
\eqalign{
& \frac{dz}{dt}=\bigg[\frac{\pr f}{\pr x}\,\big(x(t,x_0^{})\big)-
\frac{f(x(t,x_0^{}))\,f(x(t,x_0^{}))^*}{|f(x(t,x_0^{}))|^2}\cdot\cr
& \cdot\left(\frac{\pr f}{\pr x}\,\big(x(t,x_0^{})\big)+
\left(\frac{\pr f}{\pr x}\,\big(x(t,x_0^{})\big)\right)^*
\right)\bigg]\,z,
} \eqno(5a)
$$

\vskip-3mm
$$
f\big(x(t,x_0^{})\big)^*z=0.
\eqno(5b)
$$

Consider also a classic linearization
$$
\frac{dy}{dt}=\frac{\pr f}{\pr x}\,\big(x(t,x_0^{})\big)\,y.
\eqno(6)
$$

\medskip

{\bf Lemma 2.}
{\it
If $y(t)$ is a solution of system {\rm(6)}, then
$$
z(t)=\left(I-
\frac{f(x(t,x_0^{}))\,f(x(t,x_0^{}))^*}{|f(x(t,x_0^{}))|^2}\right)y(t)
$$
is a solution of system {\rm(5a), (5b)}.
}

\medskip

This statement can be verified by the substitution of the
vector function $z(t)$ in the right and left sides
of system (5).

Recall now that the definition of the Lya\-pu\-nov exponents for a linear
system is as follows
$$
\frac{dx}{dt}=A(t)\,x,\quad x\in\br^n,
\eqno(7)
$$
where $A(t)$ is a continuous $n\times n$-matrix.
For this purpose denote by $X(t)$ a fundamental matrix of system
(7) with initial data $X(0)=I$, where $I$ is a unit matrix.
Consider the square roots $\rho_i(t)$ of
eigenvalues of a matrix $X(t)^*X(t)$, which are called singular
numbers of a matrix $X(t)$. Futher, we shall suppose that
$\rho_1^{}(t)\geq\rho_2^{}(t)\geq\cdots\geq\rho_n^{}(t)$.

A very important and nearly obvious fact is that singular
numbers  of matrix $X(t)$ have a very simple
geo\-met\-ri\-cal interpretation.
The operator $\X(t)$ transformes a ball of unit radius in the
space $\br^n$ into an ellipsoid whose principal semiaxes coincide
with singular numbers $\rho_i(t)$. Thus, sin\-gu\-lar numbers
characterize the compressing and stretching pro\-per\-ti\-es of the
operator $\X(t)$: $\ \br^n\to\br^n$. In particular, the estimate
holds: $\rho_n(t)\,|x|\leq|X(t)x|\leq\mh\leq
\rho_1^{}(t)\,|x|$, $\ \forall\,x\in\br^n$.

The following simple statement resuls from the above.

\medskip

{\bf Lemma 3.}
{\it
For singular numbers $\al_i(t)$ of a matrix $X(t)S$ an estimate
$\vk_1^{}\rho_i^{}(t)\leq\al_i(t)\leq\vk_2^{}\rho_i^{}(t)$ is
true,\linebreak

\pagebreak
\noindent
where $\vk_1^{}$ and $\vk_2^{}$ are the minimal and the maximal
singular numbers of the $n\times n$-matrix $S$ respectively.
}

\medskip

Recall that a number
$$
\nu_j^{}=\ln\varlimsup\limits_{t\to+\infty}\rho_j^{}(t)^{1/t}.
\eqno(8)
$$
is called a Lya\-pu\-nov exponent $\nu_j^{}$ of system (7)

From Lemma 3 it follows that in defining the Lya\-pu\-nov exponents
the condition $X(0)=I$ can be rejected since for the matrices
$X(t)S$ and $X(t)$ under the condition $\det S\neq0$ the value $\nu_j^{}$
defined from formula (8) remains the same.

We begin now to investigate the Lya\-pu\-nov stability
and Zhu\-kovs\-ky stability for solutions $x(t,x_0^{})$ of system (1),
assuming that $x_0^{}\in\OM$, where $\OM$ is a bounded open
set in $\br^n$.

When investigating the Lya\-pu\-nov stability let us consider here a
more general case of a system
$$
\frac{dx}{dt}=f(t,x),\quad x\in\br^n,
\eqno(9)
$$
where $f(t,x)$ is a twice continuously differentiable vector
function. Notice that without loss of generality, we can consider the
solution $x(t,t_0,x_0)$ with the initial data\linebreak
$x(t_0,t_0,x_0)=x_0$ for $t_0=0$. Thus, we
consider a solution $x(t,x_0^{})$ of system (9) with
initial data $x(0,x_0^{})=x_0^{}$. We shall assume here that for
solutions of system (9) we have $x(t,x_0^{})$, $\ x_0^{}\in\OM$,
for singular numbers $\al_1^{}(t,x_0^{})\geq\mh\geq
\al_2^{}(t,x_0^{}) \geq\ldots
\geq\al_n(t,x_0^{})$ of the fundamental matrix of the
system
$$
\frac{dy}{dt}=\frac{\pr f(t,x)}{\pr x}\,\bigg|_{x=x(t,x_0^{})}y
\eqno(10)
$$
and for a continuous function $\al(t)$ a relation

\vskip-2mm
$$
\al_1^{}(t,x_0^{})\leq\al(t),\quad\forall\,t\geq0,\ \forall
x_0^{}\in\OM
\eqno(11)
$$
is satisfied.

\medskip

{\bf Theorem 1.}
{\it
Let the function $\al({t})$ be bounded on the interval $(0{,}{+}\infty)$.
Then the solution $x(t,x_0^{})$, $\ x_0^{}\in\OM$ is Lyapunov
stable. If furthermore, $\liml_{t\to+\infty}\al(t)=0$, then the
solution $x(t,x_0^{})$, $\ x_0^{}\in\OM$, is Lya\-pu\-nov asymptotically
stable.
}

\smallskip

{\bf Proof.}
Introduce here the following notations:

\vskip-2mm
$$
F_ty=x(t,y),\quad x(0,y)=y,
$$
i.e., $F_ty$ is a shift operator along solutions of system (9),

\vskip-2mm
$$
(F_ty)'=X(t,y),\quad X(0,y)=I,
$$
i.e., $(F_ty)'$ is the derivative with respect to $y$ of the
operator $F_ty$, which coincides with the fundamental matrix of
the linear system (10).

It is well known [50] that according to the suppositions, made for
any vectors $y$, $z$ and a number $t\geq0$, there exists a vector
$w$, for which the following relations

\vskip-3mm
$$
|w-y|\leq|y-z|,
\eqno(12)
$$

$$
|F_ty-F_tz|\leq\big|(F_tw)'\big|\,|y-z|
\eqno(13)
$$
are satisfied.

From here and from (11) it follows that for any $y_0^{}$, for
which $\{w|\ |w-
x_0^{}|\leq|x_0^{}-y_0^{}|\}\subset\OM$, the estimate
$$
\eqalign{
& |F_tx_0^{}-F_ty_0^{}|\leq|x_0^{}-y_0^{}|\,\sup\al_1^{}(t,w)\leq
\al(t)|x_0^{}-y_0^{}|,\cr
& \forall\,t\geq0
} \eqno(14)
$$
is satisfied. Here the supremum is taken over all $w$ from the
ball $\{w|\ |w-x_0^{}|\leq|x_0^{}-y_0^{}|\}$.

The statement of Theorem 1 directly follows from
es\-ti\-ma\-te (14).

Suppose now that instead of inequality (11) for the fundamental
matrix $X(t,x_0^{})$ of system (10) with the initial condition
$X(0,x_0^{})=I$ and for some vector function $\xi(t)$ the
following relations
$$
\eqalign{
& \big|\xi(t)\big|=1,\qquad \forall\,t\geq0,\cr
& \max\limits_i\inf\limits_{\OM}|X_i(t,x_0^{})\,\xi(t)\big|
\geq\al(t),\quad\forall\,t\geq0
} \eqno(15)
$$
are satisfied. Here $X_i(t,x_0)$ is the  $i$-th
row of the matrix $X(t,x_0)$.

\medskip

{\bf Theorem 2.}
{\it
Let for a function $\al(t)$ the relation
$$
\varliminf\limits_{t\to+\infty}\al(t)=+\infty
\eqno(16)
$$
be satisfied. Then the solution $x(t,x_0^{})$,  $\ x_0^{}\in\OM$
is Lya\-pu\-nov unstable.
}

\medskip

{\bf Proof.}
Having fixed some pair $x_0^{}\in\OM$ and $t>0$, choose in any
$\de$-neighbourhood of a point $x_0^{}$ a vector $y_0^{}$ such that
$$
x_0^{}-y_0^{}=\de\,\xi(t).
$$
Assume that the number $\de$ is so small that
$\{w|\,|w{-}x_0^{}|{\leq}\de\}{\subset}\mh\subset
\OM$.

Introduce also the following notation:
$\ F_{it}z=x_i(t,z)$ is the $i$-th component of vector function
$x(t,z)$.

It is well known [50] that for any fixed numbers $t,i$ and
vectors $x_0,y_0$ there exists a vector $w_i{\in}\br^n$, for
which we have
$$
\eqalign{
& |x_0-w_i|\leq|x_0-y_0|,\cr
& F_{it}x_0-F_{it}y_0=X_i(t,w_i)(x_0-y_0).
} \eqno(17)
$$
Using the last relation, we obtain the estimate
$$
\eqalign{
& |F_tx_0-F_ty_0|=\sqrt{\sum\limits_i\big|X_i(t,w_i)(x_0-y_0)\big|^2}\geq\cr
& \geq\de\max\left\{\big|X_1(t,w_1)\xi(t)\big|,\ldots,
\big|X_n(t,w_n)\xi(t)\big|\right\}\geq\cr
& \geq\de\max\limits_i\inf|X_i(t,w)\,\xi(t)\big|\geq\de\al(t).
}
$$
Here the infimum is taken over the set
$w\in\big\{w|\ |x_0-w|\leq\mh\leq|x_0-y_0|\big\}$.

From this estimate and from condition (16) it follows that for
any positive numbers $\ve$ and $\de$ there exists a vector
$y_0^{}$ and a number $t$, such that

\vskip-2mm
$$
|x_0^{}-y_0^{}|\leq\de,\quad|F_tx_0^{}-F_ty_0^{}|\geq\ve.
$$
The last inequalities show that the solution $x(t,x_0^{})$
is Lya\-pu\-nov unstable.

Let us discuss now the results obtained. Theorem 1 states
the Lya\-pu\-nov asymptotic stability for a flow of solutions with
the initial data from $\OM$ if their linearization have negative
Lya\-pu\-nov exponents. Here there are no effects discovered by
O.Perron for irregular linearization of individual solutions
[36]. Roughly speaking, the requirement of uniform
ex\-po\-nen\-ti\-al decrease ``with respect to $t_0$" for singular numbers of
fun\-da\-men\-tal matrix of the linearization (and this is one of the
properties of regular stable linear systems) in classical
theorems of Lya\-pu\-nov and his successors [29,30] is replaced by the
requirement of uniformity ``with respect to $x_0^{}$". Thus, the
effects of Perron are possible only on the boundaries of the flow which is
stable by the first approximation.

The requirements (15), (16) of Theorem 2 are in essence
re\-qu\-i\-re\-ments of positiveness for at least one of the Lya\-pu\-nov
exponents of linearizations of the flow of solutions with the initial
data from $\OM$ with the following addition: ``unstable directions
$\xi(t)$" (or unstable manifolds) of these solutions
co\-or\-di\-na\-te\-wi\-se continuously depend on the initial data
$x_0^{}$. Re\-al\-ly, if this property take place, then considering, if
it is necessary, the region  $\OM$ as a union of regions $\OM_i$
of an arbitrary small diameter, where conditions (15) and (16)
are satisfied, we obtain the Lya\-pu\-nov instability for the
flow of solutions under the initial data from $\OM$.

Consider now system (1) and linear approximation (5a), (5b) for
solutions $x(t,x_0)$, $\ x_0\in\OM$ assuming that
$x(t,x_0)\in\mh\in
\OM$,
$\forall\,t\geq0$, and $f(x)\neq0$, $\forall\,x\in\ov\OM$.
It is well known that $f(x(t,x_0))$ is a solution of system (6).
Therefore it is possible to consider a linear independent system
of solutions  $f(x(t,x_0))$, $\ y_2(t),\ldots,y_n(t)$ of system
(6) and matrices

\vskip-3mm
$$
\eqalign{
& Y(t,x_0)=\big(f(x(t,x_0)),\wt Y(t)\big),\cr
& \wt Y(t,x_0)=\big(y_2(t),\ldots,y_n(t)\big),\cr
& Z(t,x_0)=\left(I-\frac{f(x(t,x_0))\,f(x(t,x_0))^*}
{\big|f(x(t,x_0))\big|^2}\right)Y(t).
}
$$

\vskip-2mm
\noindent
Suppose that for singular numbers $\beta_1(t,x_0){\geq}{\ldots}
{\geq}\beta_{n}^{}(t,x_0)$ of the fundamental matrix $Z(t,x_0)$ of
system (5a), (5b) and for a continuous function $\beta(t)$
the estimate

\vskip-2mm
$$
\beta_1(t,x_0)\leq\beta(t),\qquad\forall\,t\geq0,
\quad\forall\,x_0\in\OM
\eqno(18)
$$

\vskip-2mm
\noindent
is valid.

\medskip

{\bf Theorem 3.}
{\it
If
$$
\liml_{t\to+\infty} \beta(t)=0,
$$
then the solution $x(t,x_0)$, $x_0\in\OM$ is Zhu\-kovs\-ky
asym\-p\-to\-ti\-cal\-ly
stable.
}

\medskip

{\bf Proof.}
Fix a number $T$ such that $\beta(t)\leq1/2$,
$\forall\,t\geq T$. By the number $T$ by virtue of Lemma 1 we can
choose a number $\de(T)>0$ such that for $x_0\in\OM$ and any
$y_0$ satisfying the relations
$$
|y_0-x_0|\leq\de(T),\quad (y_0-x_0)^*\,f(x_0)=0,
$$
a differentiable function $\tau(t)$ can be found such that it
satisfies relations (3), (4).

A number $\de(T)$ can also be chosen such that the inclusion
$$
\left\{y|\,|x_0-y|\leq\de(T)\right\}\subset\OM
$$
will be satisfied. Let us perform now the
substitution of an independent variable $x(t)\to x\big(\tau(t)\big)$
in systems (1):
$$
\frac{dx}{dt}=\frac{dx}{d\tau}\,\frac{d\tau}{dt}=
f\big(x(\tau(t))\big)\,\frac{d\tau(t)}{dt}
\eqno(19)
$$
It is clear that this system has some analog of
the first integral (3), a linearization of system (19) along the
solution $x(t,x_0)$ by virtue of (4) has form (5a), (5b).
Therefore, repeating the reasoning used in proving Theorem 1, we
obtain by an obvious equality $x(\tau(t),x_0)=x(t,x_0)$
the following
$$
\left|x(T,x_0)-x\big(\tau(T),y_0\big)\right|\leq
\beta(T)\,|x_0-y_0|\leq\frac12\,|x_0-y_0|.
\eqno(20)
$$
Considering further vector $q_0=x(T,x_0)$ as new initial data
and using again Lemma 1, we repeat all arguments given above on
the interval $[0,T]$ for the solution $x(t,q_0)$.
In this case, taking into account (20), we get:
$$
\left|x(2T,x_0)-x\big(\tau(2T),y_0\big)\right|\leq\frac14
|x_0-y_0|.
$$
Further, we shall continue the same reasoning and we obtain
$$
\eqalign{
& \left|x(t,x_0)-x\big(\tau(t),y_0\big)\right|\leq
\sup\limits_{t\geq0}\beta(t)|x_0-y_0|,\cr
& \liml_{t\to+\infty}\left|x(t,x_0)-x\big(\tau(t),y_0\big)
\right|=0.
}
$$
These relations show that the solution $x(t,x_0)$ is
Zhu\-kovs\-ky asym\-p\-to\-ti\-cal\-ly stable.

Consider now a point $x_0{\in}\OM$ and a set $\Phi(x_0)=
\big\{u|\,u\in\mh\in
\OM,\ u^*f(x_0)=0,\ |u-x_0|\leq\ga\big\}$. Here
$\ga$ is a number.

Assume now that instead of inequality (18) for matrix $Z(t,u)$
previously introduced for some vector function $\xi(t)$, the
relations
$$
\eqalign{
& \xi(t)^*f(x_0)=0,\quad \big|\xi(t)\big|=1,\quad\forall
t\geq0,\cr
& \max\limits_i\inf\limits_{\Phi(x_0)}
\big|Z_i(t,u)\,\xi(t)\big|\geq\al(t),\quad\forall\,t\geq0
} \eqno(21)
$$
are valid.

\medskip

{\bf Theorem 4.}
{\it
Let for the function $\al(t)$ relation {\rm(16)} be satisfied. Then
the solution $x(t,x_0)$ is Zhu\-kovs\-ky unstable.
}

\medskip

{\bf Proof.}
Suppose the opposite, i.e., let the solution $x(t,x_0)$
be Zhu\-kovs\-ky stable. In this case there exists a number $\de>0$
such that for any $y_0$, satisfying inequality $|x_0-y_0|\leq\de$,
and $(x_0-y_0)^*f(x_0)=0$,
there exists a reparametrization $\tau(t)$, for which
relations (3) and (4) are satisfied for all $t\geq0$ [24,26].
Thus, we can here on pass to system (19) for all $t\geq0$.
Repeating the reasoning of the proof of Theorem 2 and
taking into account that system (5a), (5b) is a linearization of
system (19) along the solution $x(t,x_0)$, we obtain that for any
$t>0$ there exists a vector $y_0$ such that $|x_0-y_0|\leq\de$,
$$
\eqalign{
& (x_0-y_0)^*f(x_0)=0,\cr
& \left|x(t,x_0)-x\big(\tau(t),y_0\big)\right|\geq
\al(t)\,\de.
}
$$
From here,  equality (3), and from condition (16) Zhu\-kovs\-ky
instability of $x(t,x_0)$ the solution follows.

Note that Theorem 3 and 4 essentially differ from the results of
[13,26] because they do not contain the re\-qu\-i\-re\-ment of regularity
of the linearization. It is assumed instead that
the stability (or instability) by the first approximation is preserved
at small perturbations of the initial data $x_0$.

