\font\wr=jwr9

\def \al{\alpha}
\def \C{{\bold C}}
\def \de{\delta}
\def \f{\frac}
\def \ga{\gamma}
\def \GA{\Gamma}
\def \k{k}
\def \la{\lambda}
\def \N{{\Cal N}}
\def \om{\omega}
\def \ov{\overline}
\def \R{{\bold R}}
\def \RE{\,\hbox{Re}\,}
\def \rn{\R^n}
\def \sg{\sigma}
\def \suml{\sum\limits}
\def \t{\tau}
\def \us{\underset}
\def \ve{\varepsilon}
\def \vp{\varphi}
\def \vrs{\vp'(\sg)}
\def \vs{\vp(\sg)}
\def \Z{{\bold Z}}

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

\bigskip

\medskip


\centerline{\bf FREQUENCY DOMAIN CRITERION}
\centerline{\bf OF EXISTENCE OF STRANGE}
\centerline{\bf ATTRACTORS FOR DISCRETE}
\centerline{\bf SYSTEMS}

\bigskip

\bigskip

At the present time there are many ivestigations of chaotic
behaviour of nonlinear discrete systems. Main results of
these investigations may be found in books
[33,35,\linebreak
39,43,44,48,49]. The case of one-di\-men\-si\-on dynamics is
studied best of all in this case the phase space is either a segment of
the straight line or a circle, or  another set in  $\R^1$ [6,33,39,43,44].

In the present chapter an approach connected with lower frequency
estimates of highest Lyapunov exponents [21,22] is suggested.
This  approach is developed in the frame of the nonlinear
control theory: main results are formulated in terms of
transfer functions and frequency responses of linear parts
of the systems. On the
characteristics of nonlinear blocks $\vs$ we impose conditions
which are close
to the traditional ``inequalities of sector type" [17,24,25],
typical  for the theory of absolute stability
of nonlinear control systems.

%But for existence of a strange attractor it seems to be necessary for such sector conditions
%`to be turned inside out'.
Howeover it can be found that for the existence of a strange
attractor it is necessary ``to turn inside  out" such sector
condition.
Instead of customary and traditional conditions
$\vs/\sg\in[\k_1,\k_2]$ or $\vrs\in[\k_1,\k_2]$ for all $\sg\in\R^1$
here  we have to require that for almost all $\sg$ the condition
$\vp'\overline\in[k_1,k_2]$ is satisfied. In addition on certain
sets of nonzero measure $\vrs<\k_1$ and $\vrs>\mh>\k_2$ both are
satisfied.  For continuous systems such conditions and corresponding
frequency inequalities for trans\-fer fun\-c\-ti\-ons of the linear parts
one select empty sets in the space of
pa\-ra\-me\-ters. But for discrete  systems such con\-si\-de\-ra\-ti\-on permits
us to obtain frequency domain criterions of existence of strange
attractors. Let us consider examples of concrete discrete dynamic
systems of the first and second order, for which
frequency criterion, obtained here, defines conditions of existance
of strange attractors.

For discrete systems we  generally have no such ob\-vi\-ous\-ness of
representations in the phase spaces, which is characteristic for
the continuous systems with smooth trajectories filling this
space. However, this deficiency can partly be corrected. For this
purpose at the initial moment it is necessary to consider
a segment of a straight line, connecting the initial data and its further
iterations, i.e., the sequence of continuous curves
The decrease or increase of the lengths of these
curves gives the information about stability or instability or
other properties of solutions discrete systems.
The consideration of lengths of these continuous curves can also
be used as a basis for the definition of stability or
instability. This is especially important for noninjective
mapping, defining the cor\-res\-pon\-ding discrete systems (at
present they are widely known as the generators of
chaos).
The matter is that in specific cases two trajectories may stick
together: for the finite number of iterations. But the segment to
be iterated, which connects the initial data, can have more and
more increasing length even if this sticking is taken into
account. Another interesting effect in studies of an exponential
in\-sta\-bi\-li\-ty of strange attractors is the fact that in using the
lower estimates of the lengths of curves to be iterated it is
possible to do only with the definition of the notion of
lower length of a curve (by analogy with the direction, which
deals with the upper estimates of the Hausdorff dimension of
attractors, where it is sufficient to introduce only the outer
measure). This enables to study continuous curves without taking
care about the extent of their smoothness.
It turns out that in a number of cases the mechanism of
stretching is strong enough and it can be revealed and evaluated
by means of direct Lyapunov method, which can be connected with
an efficient algebraic apparatus: the Kalman-Szego lemma and the
Shepeljavyi lemma. The latter makes it possible to find the
estimates of stretching of the lower lengths of curves, which are
iterated, in terms of frequency domain inequalities, operating
with transfer functions of linear blocks.

The effect of ``sticking" is less strong than a `stretching'
in small neighbourhoods of the trajectories. This
effect is strengthened with increasing the size of
nei\-g\-h\-bo\-ur\-ho\-od. Such an increase of the size of
neighbourhood inevitably occurs when the effect of stretching is
prevailing . At certain moment the neighbourhood becomes so much
large that a `sticking' in turn becomes strong that it
completely compensates the stretching.

This situation makes it possible the coexistence of local exponential
instability on the attractor and  attracting pro\-per\-ti\-es (or at
least of compactness) of the attractor itself. This situation makes it
also possible to apply to the analysis of strange attractors of
discrete systems the rich arsenal of methods, approaches and
results of nonlinear analysis of the discrete control systems.

Consider a system
$$
x(\k+1)=f(x(\k),\k),\qquad x\in\R^n,\ \ \k\in\Z,
\eqno(1)
$$
where $f(x,\k)$ is a vector function: $\R^n\times\Z\to\R^n$, $\ \R^n$
is the Euclidean space of dimension $n$, $\ \Z$ is the set of
integer numbers. We shall denote by $x(\k,x_0^{})$ a solution of
system (1) with the initial data $x(0,x_0^{})=x_0^{}$.
Denote by $S(r,x_0^{})$ a sphere of radius $r$ with a centre at the
point $x_0^{}\in\rn$.

We shall need further a notion of lower length $l(\ga)$
 of a curve $\ga(s)$, $\ s\in[0,1]$, where $\ga(s)$
is a continuous function $[0,1]\to\rn$.

Firstly, let us consider the case when $\ga(s)$ is  an injective
mapping of the segment $[0,1]$ or a half-interval $(0,1]$ in $\R^n$.
In other words, here $\ga(s_1)\neq\ga(s_2)$ if $s_1\neq s_2$.
In this case, we define the lower length $l(s)$ of a curve $\ga$
in the following way.

Consider various finite and countable
partitions of the segment $Q=
[0,1]$ into sets of the form $Q_j=(\al_j^{},\beta_j)$, where

\vskip-2mm
$$
\eqalign{
& Q_i\cap Q_j=\emptyset,\qquad\forall\,i\neq j,\cr
& Q=\us{j}\to{\bigcup}\ov Q_j^{}.
} \eqno(2)
$$
Here $\emptyset$ is an empty set. In this case we call the lower
lenght $l(\ga)$ of curve $\ga(s)$ the following number (or
$+\infty$)

\vskip-2mm
$$
l(\ga)=\sup\suml_j\big|\ga(\beta_j^{})-\ga(\al_j^{})\big|,
$$
where a supremum is taken over all possible partitions (2) of the
segment $[0,1]$.

Suppose now that a set  $[0,1]$ may be divided into a
finite or a countable number of sets  $L_j\ $ ($j=1,\ldots,m$)
 of the form $(\al_j^{},\beta_j^{})$, where

\vskip-2mm
$$
\eqalign{
& L_i\cap L_j=\emptyset,\qquad\forall\,i\neq j,\cr
& Q=\us{j}\to{\bigcup}\ov L_j
}
$$

\vskip-2mm
\noindent
and a mapping $\ga(\cdot)$ is injective on each set $L_j$.

\newpage

In other words, here we consider the curves, being divided into
continuous injective pieces $\ga_j^{}(s): L_j\to\R^n$.
We shall assume that the partition $\{L_j\}$ considered
has one more property: for a subset $\{L_j\}^1$ of this partition
the following condition holds:
$$
\ga(L_j)\cap\ga(L_i)=\emptyset
$$
for any $L_j\in\{L_j\}^1$ and for any $i\neq j$.

For subset  $\{L_j\}^2$ the pieces $\ga(L_j)$  stick with one
another twice, i.e., for any $L_j\in\{L_j\}^2$
there exists the only $L_i\in\mh\in
\{L_j\}^2$, $i\neq j$ such that
$$
\ga(L_j)=\ga(L_i)
$$
and $\ga(L_j)\cap\ga(L_k)=\emptyset$, $\ \forall\,k\neq i,\ k\neq j$.

For a subset $\{L_j\}^3$ the corresponding pieces $\ga(L_j)$ ``stick with
one another three times", i.e., for any  $L_j\in\{L_j\}^3$
there exist precisely two different $L_i\in\{L_j\}^3$,
$\ L_k\in\{L_j\}^3$, $\ i\neq j$, $k\neq j$, $i\neq k$ such that
$$
\ga(L_j)=\ga(L_i)=\ga(L_k)
$$
and $\ga(L_j)\cap\ga(L_g)=\emptyset$, $\ \forall\,g\neq j,\ g\neq
i,\ g\neq k$.

Continuing this process in the same way, define further a subset
$\{L_j\}^4$, $\ldots$, $\{L_j\}^m$. Here $m$ is some natural number for
which
$$
\{L_j\}=\overset{m}\to{\underset{1}\to{\bigcup}}
\{L_j\}^k.
$$
Assume that such $m$ exists. It is clear that some of the subsets
$\{L_j\}^k$ may be empty.

\medskip

{\bf Definition 1.}
{\it
The following number (or $+\infty$)
$$
l(\ga)=\suml_{k=1}^m \frac1k\suml_{\{L_j\}^k} l(\ga(L_j))
$$
is called a lower length $l(\ga)$ of the curve $\ga(s)$.
}

\medskip

The expression
$$
\suml_{\{L_j\}^k} l(\ga(L_j))
$$
is the summation over all $L_j$, which belong to a subset
$\{L_j\}^k$.

It is clear that for the case that $\ga(\cdot)$ is
a diffeomorphism, the lower length coincides with the usual length
of a curve. The factors $k^{-1}$ make it possible to take into
account the case when some part of a curve ``passes" with an
increase of $s$" in two, three or more times. The necessity to
exclude ``the case of multiplicity" from consideration is typical
for the study of chaos.

Note that there exists a simple estimate
$$
l(\ga)\geq\frac1m\,l_0(\ga),\quad l_0(\ga)=\suml_{\{L_j\}}l(\ga(\ov
L_j)),
$$
that forms the basis of the proof of the criterion of
instability, which will be formulated. Here $l_0(\ga)$
may be interpreted as a lower length of a curve $\ga$
if we take into account ``the multiplicity of sticking"
as $l_0(\ga)$ may be defined by formula (2) for any continuous
curve $\ga(s)$. In the case when it is difficult or it is impossible
to take into consideration the value $l(\ga)$  for noninjective
$\ga(s)$ we can introduce the value $l_m(\ga)=l_0(\ga)/m$
where  $m$ is an upper estimate of maximum number of pre-images
of the points of a curve $\ga(s)$.   Further, let $l_m(\ga)$ be called
$m$-estimate of the lower length of the curve $\ga(s)$.

Introduce now the following notation for a segment of the straight
line in $\rn$:

\vskip-2mm
$$
\GA(s,x_0^{},y_0^{})=s(y_0^{}-x_0^{})+x_0,\qquad s\in[0,1].
$$

\smallskip

{\bf Definition 2.}
{\it
A solution $x(\k,x_0^{})$ is called locally ex\-po\-nen\-ti\-al\-ly
unstable if there exist positive numbers $C$, $\rho>1$, $\ve$
such that for any $\de\in(0,\ve)$ a point $y_0^{}\in
S(\de,x_0^{})$ can be found, which satisfies the following
conditions:

1) there exist positive numbers $\k\in\Z$ and $s\in[0,1]$ such
that
$$
\big|x(\k,x_0^{})-x(\k,\GA(s,x_0^{},y_0^{}))\big|>\ve,
\eqno(3)
$$

2) if for all $\k\in[0,\N]$, $\k\in\Z$, and $s\in[0,1]$
the inequality
$$
\big|x(\k,x_0^{})-x(\k,\GA(s,x_0^{},y_0^{}))\big|\leq\ve
\eqno(4)
$$
is satisfied, then for these values of $\k$ the estimate
$$
\l\big(x(\k,\GA(s,x_0^{},y_0^{}))\big)\geq ‘\rho^k\de.
\eqno(5)
$$
or
$$
l_m(x(\k,\GA(s,x_0,y_0)))\geq C\rho^k\de
\eqno(6)
$$
holds.
}

\medskip

Note that Definition 2 for noninjective mappings $f(x,\k)$
is some generalization of the usual definition of local
ex\-po\-nen\-ti\-al Lyapunov instability with Lyapunov exponent $\ln\rho$
(this definition is widely used for diffeomorphism). Note that
in our case inequalities (5), (6) does not necessarily imply
the property 1). For noninjective mappings, generally speaking,
the following case is possible: $\de\ll\ve$, for all $\k>\mh>0$, the
curves $x(\k,\GA(s,x_0^{},y_0^{}))$ remain in
$\ve$-neigh\-bo\-ur\-ho\-od
of solution $x(\k,x_0^{})$, and at the same time in this
neighbourhood  an exponential extension in
directions, de\-pen\-ding non\-smo\-oth\-ly on $\k$ takes place.

Consider system (1) with
$$
f(x,\k)=Ax+b\vp(c^*x),
$$
where $A$ is a constant $n\times n$-matrix, $b$ and $c$ are
constant  $n$-vectors, $\vs$ is a continuous piecewise
differentiable function having ordinary descontinuities of
derivative at the points $\sg=\sg_j^{}$. We suppose that all the
these points are isolated ones. In other words, we may
find a number $\t>0$ such that
$$
|\sg_j^{}-\sg_i^{}|\geq\t,\quad\forall\,i\neq j.
$$
We assume  that the inequalities $\det(A+\vrs bc^*)\neq0$,
$\ \forall\,\sg\neq\sg_j^{}$, and
$$
(\vrs-\mu_2^{})(\mu_1^{}-\vrs)\leq0,\quad\forall\sg\neq\sg_j^{}
\eqno(7)
$$
hold. Here  $\mu_1^{}<0$, $\mu_2^{}>0$ are some numbers.

Introduce a transfer function of system (1)
$$
W(p)=c^*(A-pI)^{-1}b\qquad p\in\C^1
$$

\medskip

{\bf Theorem  1.}
{\it
Suppose that for a number $\la>2$ the following conditions hold:

1) a matrix $\la^{-1}(A+\mu_1^{}bc^*)$ has one eigenvalue located
outside of the circle of radius $1$ with the centre at the zero
point and $n-1$ eigenvalues lying inside of this circle,

2) for all values $p\in\C^1$ such that $|p|=1$, the inequality
$$
\RE\left[(1+\mu_1^{}\ov{W(\la p)})(1+\mu_2^{}W(\la p))\right]<0
\eqno(8)
$$
is valid.

Then any solution of system {\rm(1)} is locally exponentially
unstable.
}

\medskip

{\bf Proof.}
By the Kalman-Szego lemma [37] from the con\-di\-ti\-on 2) of the theorem
it follows that there exists a symmetric real matrix  $H$ such
that
$$
\eqalign{
& \f1{\la^2}[Az+b\xi]^*H[Az+b\xi]-z^*Hz-\cr
& -(\xi-\mu_2^{}c^*z)(\mu_1^{}c^*z-\xi)<0,\cr
& \forall\ z\in\rn,\quad \forall\ \xi\in\R^1,\ \ |z|+|\xi|\neq0.
} \eqno(9)
$$
Assuming in (9) $\xi=\mu_1^{}c^*z$, we obtain the
inequality
$$
\eqalign{
& \left(\f1{\la}\,(A+\mu_1^{}bc^*)z\right)^*H\left(\f1{\la}\,
(A+\mu_1^{}bc^*)z\right)-z^*Hz<0,\cr
& \forall\ z\in\rn,\quad z\neq0.
}
$$
Hence and from the condition 1) of the theorem by the lemma of
Shepeljavyi [41] we obtain the matrix $H$ which has one negative
and $n-1$ positive eigenvalues.

Without loss of generality, suppose further that the vector $z$
and the matrix $H$ take the form
$$
H=\left(\matrix -1&0\cr 0&I\endmatrix\right),\qquad\left(\matrix
z_1\cr z_2\endmatrix\right),
$$
where $I$ is a unit $(n-1)\times(n-1)$-matrix, $z_1\in\R^1$,
$\ z_2\in\mh\in
\R^{n-1}$. In this case a quadratic form  $V(z)=z^*Hz$
takes the  form $V(z)=z^*Hz=
|z_2|^2-z_1^2$.

We now take  a number $\ve<\t/2$. From nondegeneracy of matrix
$A+\vrs bc^*$ in the bands $\{\sg_j^{}\leq\sg<\sg_{j+1}^{}\}$
it follows that the pre-image of any point of a mapping $Ax+b\vp(c^*x)$,
which is located in any $\ve$-neighbourhood of any point of $\rn$,
is a set consisting either of one or two points.

Hence it follows that the estimation of number $m$ in the
definition of the lower lengths of image of the segment which can
be found in such $\ve$-neighbourhoods, followed by
$\k$-iterations is as follows: $m\leq2^k$.

Fix now a vector $x_0^{}$ and choose for it  $y_0^{}$ such that
$z_1(0)=\mh=
\de$, $\ z_2(0)=0$, where $z(0)=x_0^{}-y_0^{}$,
$\ z_1(0)$ is the  first component of the vector $z(0)$, $\ z_2(0)$
is $(n-1)$-vector of other components of the vector  $z(0)$. Here $\de$
is some fixed number such that $\de<\ve$. Thus, we have
$$
\GA(s,x_0^{},y_0^{})=\left(\matrix s\,z_1(0)\cr 0\endmatrix\right)+x_0,
\qquad s\in[0,1].
$$

Let us prove estimate (6), assuming that for $\k\in[0,\N]$, $k{\in}\Z$,
inequality (4) is true.


For this purpose we consider a function
$\psi_k(s){=}\mh=
c^*x(\k,\GA(s,x_0,y_0))-\sg_{i(k)}^{}$. In this case
$\sg_{i(k)}^{}$ is a dis\-con\-ti\-nu\-i\-ty of the derivative of
$\vp(\sg)$ such that for some $s{\in}[0,1]$ the inequality
$\psi_k(s){\geq}0$ is satisfied and for some other values $s$
the inequality with the opposite sign is satisfied too.

Consider the following partition of the segment  $[0,1]$ into mutual
disjoint intervals of the form $(\al_{0j},\beta_{0j})$. Here
$\al_{0j}$ is a point for which  $\psi_0(\al_{0j})=0$ and
$\psi_0(s)>0$ (or $\psi_0(s)<0$) in some sufficiently small
right-sided neighbourhood $\al_{0j}$ (i.e, $s>\al_{0j}$).
For the point $\beta_{0j}$ the relations  $\psi_0(\beta_{0j})=0$
and $\psi_0(s)<0$ (or $\psi_0(s)>0$) are satisfied in some
sufficiently small right-sided neighbourhood $\beta_{0j}$
(i.e., $s>\mh>
\beta_{0j}$). In other words,  $\al_{0j}$ and $\beta_{0j}$
are the exit points of graphic $\psi_0(s)$ into the upper and
the lower half-planes when increasing $s$. Besides, if it is
necessary, we add the intervals $(0,\beta_{0j})$ and $(\al_{0j},1)$.

It is easy to see that the constructed system of intervals
$M_{0j}=
(\al_{0j},\beta_{0j})$ has the following properties
$$
M_{0j}\cap M_{0i}=\emptyset,\quad\underset{j}\to{\bigcup}
\ov M_{0j}=Q.
$$
In turn each a set $M_{0j}$ can be partitioned in the same way for
$\psi_1(s)$ into intervals $M_{0j,1j}=(\al_{0j,1j},\beta_{0j,1j})$.
Continuing this process further for $k{=}2,{\ldots},N$ and
reindexing   $M_{0j,1j,\ldots}$, as a result, we obtain a
partition of the segment $Q=[0,1]$ into the intervals $M_j$:
$$
M_{j}\cap M_{i}=\emptyset,\quad\underset{j}\to{\bigcup}
\ov M_{j}=Q.
$$
It is important for us that on any $\ov M_j$ there exists a
derivative$\vp'(\sg)$, where
$\sg=c^*x(\k,\GA(s,x_0,y_0))$, $\ s\in\ov M_j$, $\ k=0,\ldots,N$.

It is clear that under these conditions inequality
(7) is also true. Therefore, from this fact and from inequality
(9) it follows that for a difference
$x(\k,\GA(\beta_j^{},x_0^{},y_0^{}))-\mh-
x(\k,\GA(\al_j^{},x_0^{},y_0^{}))$,
$\ k=0,\ldots,\N-1$, we can write an estimate
$$
\eqalign{
& V\big(x(\k+1,\GA(\beta_j^{},x_0^{},y_0^{}))-x(\k+1,\GA
(\al_j^{},x_0^{},y_0^{}))\big)\leq\cr
& \leq\la^2V\big(x(\k,\GA(\beta_j^{},x_0^{},y_0^{}))-
x(\k,\GA(\al_j^{},x_0^{},y_0^{}))\big).     }
$$
Hence and from the special form of the matrix $H$ it follows
that
$$
\eqalign{
& \left|x(\k,\GA(\beta_j^{},x_0^{},y_0^{}))-x(\k,\GA
(\al_j^{},x_0^{},y_0^{}))\right|\geq\la^k(\beta_j^{}-\al_j^{}),\cr
& \forall\ j,\qquad\forall\ k=0,\ldots,\N.
}
$$
Summing by  $j$ the left and right sides of this inequality and
taking into account that it is possible to estimate
$m\leq2^k$, we obtain a relation
$$
l_m\big(x(\k,\GA(s,x_0^{},y_0^{}))\big)\geq\f1{2^k}\,\la^k\de,
\quad\forall k=0,\ldots,\N.
\eqno(10)
$$
Since $\la>2$, estimate (6) is proved.

Inequality (3) may be proved in the same way, taking into account
a special form of the matrix $H$. All the curves, considered
here, may be projected on the line $\{z_2=0\}$. Since for the lower
length of a projection an estimate being the same as (10) is
satisfied, we can conclude that for some $\k$ and $s$, inequality
(3) will be satisfied.

Let system (1) be represented in the form
$$
\eqalign{
& z(\k+1)=Pz(\k)+q\vp(\sg(\k)),\cr
& \sg(\k+1)-\sg(\k)=r^*z(\k)+\rho\vp(\sg(\k)),
} \eqno(11)
$$
where $P$ is a constant $(n-1)\times(n-1)$-matrix, $r$ and $q$
are constant $n$-vectors, $\vs$ is a continuous function. It is
clear that in this case the transfer function $W(p)$ for system
(11) can be written in the form
$$
W(p)=\f1{p-1}\,\big[r^*(P-pI)^{-1}q-\rho\big].
$$

It is well known that if $W(p)$ is a nondegenerated function
(i.e., its denominator is a polynomial of degree $n$ and does not
have common zeros with its numerator [17], then both systems
considered above are equivalent if $\det A=0$.

The following results is well known and almost obvious.

\medskip

{\bf Lemma.}
{\it
If all eigenvalues of the matrix $P$ are located inside of a unit
circle with centre in the origin and $\vs$ is a $2\pi$-periodic
function, then system {\rm(11)} is Levinson dissipative in a
cylindrical phase space
$\{z{\in}\rn,\sg\!\!\mod2\pi\}$.
}

\medskip

Recall that the Levinson dissipativity in this case means that
there exists a number $R$ such that for any solution
$z(\k,z_0,\sg_0)$,\,$\sg(\k,z_0,\sg_0)$ of system (11) the
inequality
$$
\varlimsup\limits_{\k\to+\infty}\big|z(\k,z_0,\sg_0)
\big|\leq R
\eqno(12)
$$
holds.

It is clear that in dissipativity region (12) there exists some
global attractor (i.e., a compact invariant and globally
attracting set). In case of a pointwise attraction it is, for
example, a set of all $\om$-limit points of system (1).

\medskip

{\bf Definition 3.}
{\it
A global attractor $K$ will be called a strange attractor, if for
any $x_0^{}\in K$ a solution $x(\k,x_0^{})$ is locally
exponential unstable one.
}

\medskip

From the theorem and lemma the next statement follows.

\medskip

{\bf Corollary.}
{\it
Let all conditions of the theorem and the lemma be satisfied.
Then there exists a strange attractor of system {\rm(1)}.
}

\medskip

Note that various conditions of Levinson
dis\-si\-pa\-ti\-vi\-ty  or the conditions of
boundedness of solutions for discrete sys\-tems are given in [27]. Note also
that various generalizations or weakenings of the conditions of
the theorem are possible if of more precise
estimates of sets $L_j$ are used.

\smallskip

{\it Example 1.}
Consider system (1) with $n=1$, $A=1$, $b=c=1$, $\vs$
being a  $2\pi$-periodic piecewise differentiable
function satisfying conditions (7).

Here $W(p)=(1-p)^{-1}$.

In this case the conditions 1) of the theorem take the form
$$
|1+\mu_1^{}|>\la>2,
\eqno(13)
$$
and the conditions 2) are the following
$$
\eqalign{
& (\la\cos\om)^2+(1-\la\sin\om)^2+\mu_1^{}\mu_2^{}+\cr
& +(\mu_1^{}+\mu_2^{})(1-\la\sin\om)<0,\quad\forall\om\in\R^1.
} \eqno(14)
$$

This inequality is equivalent to a condition
$$
\la^2+1+\mu_1^{}\mu_2^{}+\mu_1^{}+\mu_2^{}+\la|2+\mu_1^{}+\mu_2^{}|<0.
\eqno(15)
$$
If we suppose that $\mu_2^{}=-\mu_1^{}=\mu$,  then conditions
(13) and (15) take a very simple form: $\mu>\la+1>3$.

Thus, if $\mu>3$, then there exists  $\la>2$ such that all
conditions of the theorem are satisfied.

Thus, if $\mu>3$, then in the phase space, which in this case is a
circle, there is a strange attractor.

\midinsert
\moveright 2cm\vbox{
\special{em: graph 1.pcx}}
\vskip4cm
\centerline {Fig. 1.}
\endinsert

In Fig.1 one of the possible graphs of the
function $\vp(\sg)$ is shown, for which the conditions of the theorem are
satisfied. Here $\mu>3$.

\newpage


\smallskip

{\it Example 2.}
Consider system (1) with $n=1$, $A=0$, $b=c=
1$, $\vs$ which is a function piecewise differentiable satisfying
conditions (7) with $\mu_2^{}=-\mu_1^{}=\mu$, and such that
$\vs\in\mh\in
[\al,\beta]$, $\ \forall\sg\in[\al,\beta]$.

Here we may consider a compact phase space $[\al,\beta]$ and
apply the theorem. It is easily seen that the condition 1) of the
theorem holds if $\mu>\la>2$. This condition is also sufficient
for validity of theorem's condition 2).

Thus, for any function with graph, shown in Fig.1, where
$\mu_2^{}=-\mu_1^{}=\mu>2$ and which maps the segment
$[\al,\beta]$ into itself, all conditions of the theorem are satisfied.
Therefore, a local exponential instability takes place.

Note that a global stability of a fixed point for
the system takes place for $\mu<1$.
Emphasize once more that some weakening of the conditions of the
theorem may be achieved at the expense of more
detailed information about sets $L_j$ (see definition 1). For the
case under consideration, it means that if
iterations $\k{+}1,{\ldots},\k{+}\N$, $\ \forall\k\in\Z$ are known,
for which all
images of sufficiently small nei\-gh\-bo\-ur\-ho\-ods of discontinuity
points of the derivative $\sg_i$ do not intersect with a union
of these nei\-gh\-bo\-ur\-ho\-ods, then at the expense of this information
we may improve the estimate $\mu>2$. It is clear that the right
side of such inequality will depend
on $\N$.

\smallskip
{\it Example 3.}
Consider  system (11) with $n=2$ and a transfer function
$$
W(p)=\f{\beta p}{(p-1)(p-\al)},\qquad p\in\C^1,
$$
where $\al$ and $\beta$ are numbers, $\al\neq0$, $|\al|<1$,
$\beta>\mh>
0$, $\ \vs$ is a $2\pi$-periodic piecewise differentiable
function satisfying condition (7) with $-\mu_1^{}=\mu_2^{}=\mu$.

The systems of this type describe some  classes of rotators,
being excited by the periodic shocks [44], and phase locked
loop  [27].


It is clear that here all conditions of the lemma are satisfied.
Let us verify the conditions of the theorem.

Note that in this case the eigenvalues of a matrix
$\la^{-1}(A+\mathbreak+
\mu_1^{}bc^*)$ are roots of a polynomial
$$
(\la p-1)(\la p-\al)-\mu\beta\la p.
$$
It is easy to see that now the condition 1) of the theorem takes the
form
$$
\mu\beta+\al+1+\sqrt{(\mu\beta+\al+1)^2-4\al}>2\la,
$$
and the condition 2) will be satisfied if
$$
\mu\beta>\f{(1+\la)(\al+\la)}{\la}.
$$

Thus, the  system under consideration has a strange attractor in
the cylindrical phase space if the following inequalities
$$
\eqalign{
& (\mu\beta+\al+1)+\sqrt{(\mu\beta+\al+1)^2-4\al}>4,\cr
& \mu\beta>\f32\,(\al+2)
}
$$
are satisfied.

Note that the frequency criterion formulated here can be
extended in the natural way over the discrete systems with many
nonlinearities.




