\input amstex
\documentstyle{amsppt}

\magnification=\magstep1
\TagsOnRight
\NoRunningHeads

\def\R{\Bbb R}
\def\CC{\bold C}
\def\C{\Bbb C}
\def\a{\alpha}
\def\b{\beta}
\def\g{\gamma}
\def\d{\delta}
\def\t{\tau}
\def\l{\lambda}
\def\la{\langle}
\def\ra{\rangle}
\def\L{\Lambda}   
\def\p{\psi}
\def\ep{\varepsilon}
\def\D{\Delta}
\def\f{\varphi}
\def\m{\mu}
\def\e{\eta}
\def\h{\chi}
\def\o{\omega}
\def\r{\rho}
\def\x{\xi}
\def\n{\nu}
\def\z{\zeta}
\def\dd{\varDelta}
\def\F{\Phi}
\def\P{\Psi}
\def\O{\Omega}
\def\LL{\eufm L}
\def\M{\eufm M}
\def\N{\eufm N}
\def\0{\bold 0}
\def\th{\theta}
\def\G{\Gamma}
\def\fa{\forall}
\def\lra{\Leftrightarrow}
\def\ex{\exists}
\def\tr{\triangle}
\def\pa{\partial}
\def\ti{\tilde}
\def\Ti{\Tilde}
\def\wt{\widetilde}
\def\na{\nabla}
%\def\ci{\circ}
\def\ol{\overline}
\def\sm{\setminus}
\def\te{\text}
\def\A{\Cal A}
\def\B{\Cal B}
\def\D{\Cal D}
\def\RR{\Cal R}
\def\PP{\Cal P}
\def\H{\Cal H}
\def\s{\sigma}
\def\Si{\Sigma}


\topmatter

\author
V.E.Chernyshev
\endauthor

\title
Heteroclinic contours that generate stable chaos
\endtitle

\abstract
This is an English version of the Introduction to the author's thesis.
\endabstract

\date
June 1998
\enddate

\endtopmatter


\head{ Introduction}\endhead

\vskip .3cm
Statements of the dissertation are numerated as
follows. The first numeral (Roman) is the number of
chapter, the second numeral is the number of section,
and the third numeral is the number of the statement
in its section.

\vskip .2cm

Consider a system of differential equations of the form
  $$ \dot x = X(x), \quad x \in \R^3, \quad X \in \C^r(\R^3), \ r
                \ge 3. \tag "\bf (1)"$$
Assume that our system generates a flow, i.e., that any its
trajectory is defined on the whole axis. Denote this flow
by $g^t$.

{\bf  Definition.}
{\it An invariant set $\G$ is called a heteroclinic contour
if $\G$ is connected, compact, and consists of a finite
family of trajectories
$\g_i, \ i \in 1:m,$  
and of their $\a$-limit sets and  $\o$-limit sets,
$$ \G = \{ \cup_{i\in 1:m}\g_i\} \cup \{\cup_{i \in 1:m}
\a(\g_i)\} \cup \{ \cup_{i \in 1:m} \o(\g_i)\}.$$
It is assumed that any limit set belonging to the 
contour $\G$ is either a hyperbolic rest point 
or a hyperbolic closed trajectory.}

{\bf Definition.}
{\it We say that a heteroclinic contour} $\G$ 
{\it is a heteroclinic cycle if it is possible to
numerate its trajectories so that}
$\o(\g_i) = \a(\g_{i+1}), \ i \in 1:m-1, \quad \o(\g_m) =
\a(\g_1),\quad $  {\it and } $\a(\g_i) \cap \a(\g_j) = \emptyset$ 
{\it for }  $ i
\ne j (\te {mod}\, m).$

{\bf Definition.} {\it We say that a heteroclinic cycle of
a three-dimensional autonomous system belongs to the Lorenz type
if any its limit set is either a saddle-node rest point or
a closed trajectory with orientable stable and unstable
manifolds.}

{\bf Definition.} {\it We say that a heteroclinic contour is
equidimensional if the dimensions of stable manifolds of
all its limit sets are the same.}

Below we consider equidimensional heteroclinic cycles of the
Lorenz type such that among their limit sets there are both
rest points and closed trajectories. Our goal is to give
conditions under which contours of this type generate 
persistent chaos (in the sense of the following two definitions).

{\bf Definition.}{\it We say that an invariant set} $J$
{\it is chaotic if

(1) the set} $J$ {\it contains a dense trajectory of the
flow} $g^t$;

(2) {\it the union of closed trajectories is dense in}
$J$;

(3) {\it the set} $J$ {\it exhibits sensitive dependence
with respect to initial data, i.e., there exists a number}
$\ep > 0 $ {\it such that for any point}  
$x\in J$ {\it and for any number}  $\d > 0 $ 
{\it there is a point}  $y$ {\it
and a number} $\ t > 0\  $ {\it such that}
 $\ \r(x,y) < \d\ $ {\it and}  $\ \r(g^tx,g^ty) >
\ep,\ $ {\it where} $\r(x,y)$ {\it denotes the
distance between} $x$ {\it and} $y$.

{\bf Definition.}{\it We say that a heteroclinic
cycle $\G$ generates persistent chaos if for any
neighborhood $V(\G)$ of the set $\G$ there exists
a number $\delta>0$ such that any system whose
$C^1$ distance from system (1) is less than
$\delta$ has a chaotic invariant set belonging
to} $V(\G)$.

Now we formulate our conditions I--III.
Consider an equidimensional heteroclinic cycle of the
Lorenz type consisting of trajectories
$\g_i, \ i \in 1:m, \ m \ge 2,$ and of their limit
sets $\a(\g_i), \ \o(\g_i), \ i \in 1:m.$  The 
numeration of the trajectories is chosen so that
$\o(\g_i) = \a(\g_{i+1}), \ i \in 1:m-1,\ $,
$\ \o(\g_m) = \a(\g_1), \ $  and  $\a(\g_i) \cap \a(\g_j) =
\emptyset $ if   $i \ne j$. 
Any limit set $\a(\g_i),
\ i \in 1:m,$ is either a saddle rest point
$O_i$ or a saddle closed trajectory $P_i$  
such that its stable and unstable manifolds are
orientable.
We assume that the limit set $\a(\g_1)$ 
is a closed trajectory $P_1$.

Since one of the limit sets of our equidimensional cycle
is a closed trajectory, for any limit set, the dimension
of its stable manifold equals two.
Denote by $\ \m_i > 0 > \l_i > \n_i\ $ the eigenvalues of
the Jacobi matrix $\ D\, X(O_i)$ at a saddle rest
point $O_i \in \G$.

Our first condition has the following form:

$ \quad \te {\bf I.} \quad \l_i > -\m_i, \quad \l_i - \m_i > \n_i.$

Our second condition is related to the character of
approach of trajectories $\g_i$ of our heteroclinic cycle
to their limit sets. This is a general position condition.

$ \quad \te {\bf II.}$ {\it There exists a continuous bundle} $P$
{\it of the planes}  $P(x), \ x \in \G,$ {\it invariant with
respect to the differential}  $D\, g^t,
\ t \in \R,$ {\it of the shift}  $g^t$ {\it for time}  $t$ {\it along
trajectories of system (1). The plane} $P(O_i)$ {\it coincides
with the plane} $ \la v_i^s,v_i^u \ra$ {\it spanned by the 
eigenvetors} $ \
v_i^s, \ v_i^u$ {\it corresponding to the eigenvalues}  $\m_i, \
\l_i.$ {\it For a point} $x \in P_j$, {\it the plane}  $P(x)$ {\it is
the tangent plane at} $x$ {\it of the unstable manifold}
$\ W_j^u\ $ {\it of the closed trajectory}  $P_j$.


Let  $P_{l_1}, \ P_{l_2}, \dots, P_{l_k}$ be all closed
trajectories belonging to the cycle  $\G, \ l_1 = 1 < l_2 < \dots < l_k.$
We show in Theorem I.1.1 that condition II implies, in
particular, the following property: the unstable manifold
$W_{l_i}^u$ of the closed trajectory $P_{l_i}$ intersects
transversally the stable manifold  $\
W_{l_i +1}^s\ $ of the next limit set in the cycle $\G$
along a trajectory  $\g_{l_i}: \ W_{l_i}^u \cap W_{l_i+1}^s =
\g_{l_i}$. In addition, trajectories of the cycle  $\G$  
approach rest points along leading directions (the latter
term means directions corresponding to the eigenvalues
$\l_i$).

The third condition (similarly to Condition I) is not
a general position condition.

$ \quad \te {\bf III.} $ {\it It is possible to fix
an orientation on the bundle } $P$ {\it so that this
orientation is continuous on the set } $\G \setminus \{
\cup \, \g_{l_i}, \ i \in 1:k\}$ {\it and has the
following properties}.
 {\it In the planes} $P(O_i)$,
{\it the orientation is determined by the frame} $(v_i^s,v_i^u)$, 
{\it where} \ $v_i^s \quad
(v_i^u)$ {\it is the limit position of the vector} $X(x)$ {\it as}  $x \to
O_i, \quad x \in \g_{i-1} \quad (x \in \g_i).$  {\it In the plane}
$P(x)$, {\it where a point} $x$ {\it belongs to a closed trajectory}
$P_{l_i}, \ i \in 1:k,$ {\it the orientation is determined by the frame} $\
(X(x),v^u(x)),\ $  {\it where } $\ v^u(x)\ $ {\it is the tangent vector
to the unstable manifold} $W^u(x)$ {\it of the point} $x \in P_{l_i},$
{\it directed toward the intersection of} $W^u(x)$ {\it with the trajectory}
$\g_{l_i}$.

%\vskip .3cm
\newpage

Chapter I: ``{\bf Heteroclinic contours of the Lorenz type}".
\smallskip
In this chapter, we investigate general properties of
heteroclinic contours.

Section 1 of Chapter 1 is devoted to equidimensional 
contours of the Lorenz type. By analogy with heteroclinic
cycles of the Lorenz type, we say that a heteroclinic
contour is of the Lorenz type if any its limit set is
either a saddle rest point or a saddle closed trajectory.
It is assumed that our contour contains both rest
points and closed trajectories.

Condition II formulated above for heteroclinic cycles
can be stated in the same way for heteroclinic contours.
Sometimes other general position conditions are applied
in the investigation of dynamics near heteroclinic
contours. They are formulated in Section 1. In
Theorem I.1.1, we prove that these conditions are
equivalent to Condition I in the case of an
equidimensional contour of the Lorenz type.

{\bf Definition.} {\it We say that a heteroclinic
contour is ramification free if it is possible to
numerate its trajectories so that}
$\o(\g_i) = \a(\g_{i+1}), \ i \in
1:m-1,$  {\it and}  $ \ \a(\g_i) \cap \a(\g_j) = \emptyset, \ i,j \in
1:m, \ i \ne j$.

In Section 2, we study ramification free heteroclinic contours
of the Lorenz type that satisfy Conditions I and II and the
following additional assumption: the limit sets
$\a(\g_i), \ i \in 2:m,$ are saddle rest points, while
the limit set $\a(\g_1)$ is a saddle closed trajectory
with orientable stable and unstable manifolds. Such a
contour is called a simple contour of the Lorenz type.

{\bf  Theorem I.2.2.}
{\it A simple contour} $\G${\it of the Lorenz type has a neighborhood}
$V(\G)$ {\it with the following property: system (1) has in} $V(\G)$
{\it a smooth invariant surface} $Q$ {\it containing the contour} $\G$.
{\it In addition, the bundle} $P(x), \ x \in \G,$ {\it from Condition II
coincides with the bundle of tangent planes} $T_xQ, \ x \in\G,$ {\it of
the surface} $Q$.

Further, we describe in Section 2 the behavior of trajectories
of system (1) on the surface $Q$. The description is given
in terms of the Poincar\'e mapping $h$ sending points of a
transversal $s_1\subset Q$ of the trajectory $\g_1$  
along trajectories of system (1) to a transversal 
$s_{m+1} \subset Q$ of the trajectory $\g_m$.
The contour $\G\subset Q$ divides the curve $s_1$
into two parts $s_1^+$ and $s_1^-$.


It is shown in Theorem I.2.3 that under Condition III,
the mapping $h$ is defined on one of the curves $s_1^+$ or
$s_1^-$ and can be written (in proper coordinates) in the
following form:
  $$\aligned
  & h(\e_1) = C\e_1^E + q(\e_1^E), \\
  & h'(\e_1) = CE\e_1^{E-1} + r(\e_1^{E-1}),
      \endaligned $$
where $E < 1$, and the functions $q, r$ are infinitesimally small
as $\e_1 \to 0.$

Since $E < 1$, the mapping $h$ is expanding for small
$\e_1$. Consider the part of the
surface $Q$ covered by trajectories intersecting the
curve on which the mapping $h$ is defined. It follows
from our reasons that it is natural to call this part
of $Q$ the unstable surface of the contour $\G$.

Below, in Chapter II, we show that simple cycles of
the Lorenz type satisfying Codition III generate 
persistent chaos. By definition, a saddle-focus
rest point cannot be a limit set in a simple cycle
of the Lorenz type. We extend the class of heteroclinic
cycles generating persistent chaos. In Section 3,
we consider a class of heteroclinic contours
having saddle-focus rest points as their limit sets.
This class includes simple heteroclinic contours
of the Lorenz type satisfying Conditions I -- III
considered above. It is shown in Theorem I.3.1 that
any neighborhood of a contour of this class
contains a simple heteroclinic contour
of the Lorenz type satisfying Conditions I -- III.
If the unperturbed contour is a cycle, then the
generated simple contour is also a cycle. It
follows that cycles of the introduced class
generate chaos.

\vskip .3cm


Chapter II. ``{\bf Persistent chaotic invariant sets generated by
cycles of the Lorenz type}".

\smallskip
In this chapter, we prove the main results of the
dissertation. Everywhere in this chapter, we assume that the
unperturbed system (1) has a heteroclinic cycle $\G$ of
the Lorenz type satisfying Conditions I, II, and III.
We also assume that all systems of differential equations
(both unperturbed and perturbed) are of class $C^4$.

Denote by $g^t_\ep$ the flow of the perturbed system
  $$\dot x = X(x) + Y(x), \quad Y \in \C^r(\R^3), \quad r \ge 4,
       \quad ||Y||_{\C^1} < \ep, \tag "\bf (2)"$$
let $D\, g_\ep^t$ be its derivative.

The first step in the proof of the main result of Chapter II
is to establish the following statement (having an independent
interest). In this statement, we construct an invariant
strongly stable fiber bundle over a neighborhood of
a cycle $\G$.

{\bf ’heorem II.2.1.}
{\it Assume that an equidimensional cycle} $\G$ {\it of the Lorenz type
satisfies Conditions I, II, and III. Then there exist numbers} 
$\ep_0 > 0, \ C >
0 , \ \l^i < 0, \ \L^i < 0, \ \l^i < \L^i, \ i \in 1:m,$ \
{\it a neighborhood $V(\G)$ of the heteroclinic cycle} $\G$, \
{\it and neighborhoods}  $V_i \subset V(\G)$  {\it of the limit sets}
  $\a(\g_i), \ i \in 1:m,$ {\it such that for}  $\ep < \ep_0$
{\it there exists a continuous} $D\, g_\ep^t$ {\it - invariant}
{\it decomposition}
  $$ \R^3 = E_\ep^{ss}(x) \oplus E_\ep^v(x), \quad x \in V(\G),
   \tag "\bf (3)"$$
{\it where}  $E_\ep^{ss}(x)$ {\it is a line and}  $E_\ep^v(x)$
{\it is a plane for which the following estimates hold:}
  $$||D\, g_\ep^t(x)v|| \le C\exp (\l^i t)||v||, \quad t \ge 0,
 \quad  v \in E_\ep^{ss}(x), \ [x,\ g_\ep^t(x)] \in V_i ,
      \tag "\bf (4)"$$
 $$ ||D\, g_\ep^t(x)v|| \le C\exp (\L^i t)||v||, \quad t \le 0,
  \quad v \in E_\ep^v(x), \ [x,\ g_\ep^t(x)] \in V_i.
  \tag "\bf (5)"$$
{\it Here} $[x,g_\ep^t(x)]$ {\it denotes the arc of the trajectory
of a point} $x$   {\it with ends}  $x$ {\it and} $ g_\ep^t(x)$.
{\it In addition, the bundle of lines}  $E_\ep^{ss}(x), \ x \in V(\G),$
{\it is locally Lipschitz continuous.}

The heteroclinic cycle $\G$ does not have a neighborhood
invariant with respect to the mapping $g_\ep^t$, hence
we have to give a special definition.

{\bf  Definition.}
 {\it The bundles} $E_\ep^{ss}(x), \
E_\ep^v(x),$
$ \ x \in V(\G),$ {\it over a neighborhood} $V(\G)$ {\it
of the cycle} $\G$ {\it are invariant with respect to} $D\, g_\ep^t$ {\it
if}
  $$D\, g_\ep^t(x)E_\ep^{ss}(x) = E_\ep^{ss}(g_\ep^t(x)),
         \quad D\, g_\ep^tE_\ep^v(x) = E_\ep^v(g_\ep^t(x)),$$
{\it provided} $g_\ep^\t(x) \in V(\G)$ {\it for} $\t \in [0,t].$

We prove Theorem II.1.1 in two steps. In Section 1, we
construct decomposition (3) over the heteroclinic cycle $\G$
for the unperturbed system (1). In this proof, we show that
both terms in (3) are locally Lipschitz continuous and
satisfy estimates (4) and (5).


In Section 2, we construct the needed decomposition (3)
over a neighborhood $V(\G)$ of the heteroclinic cycle $\G$.
First we extend the decomposition constructed in Section 1
to a neighborhood $W(\G)$ of $\G$ preserving its local
Lipschitz continuity. The extension
  $$ \R^3 = \wt E_1^{ss}(x) \oplus \wt E_1^v(x), \quad x \in
                            W(\G),$$
is not necessarily invariant with respect to $D\, g_\ep^t$.
We obtain invariant terms in (3) by analogous reasons, let
us describe them for the first term.

Consider a bundle $\LL$ over a neighborhood $W(\G)$
of the heteroclinic cycle $\G$ such that its fiber $\LL(x)$ 
over $x \in W(\G)$ is the set of all lines transverse
to the plane $\wt E_1^v(x)$. Any such line is the graph
of a linear mapping $\ P(x): \ \wt
E_1^{ss}(x) \to \wt E_1^v(x),$\ we identify the mapping
and the line.

Take a neighborhood $V(\G)$ of $\G$ such that it belongs to
$W(\G)$ together with its closure.
Consider the space $L$ of continuous sections of the bundle $\LL$,
i.e., the family of continuous mappings $P: \ V(\G) \to \LL$\
such that any $P(x)$ is a linear mapping from $\wt E_1^{ss}(x)$
to $\wt E_1^v(x)$.  

We take the neighborhood $V(\G)$ to be closed, hence
$L$ is a Banach space in the $\C$ norm. 
For $\ep$ small enough, we define a mapping $H_\ep$  
taking a closed convex subset of $L$ to itself and
contracting on this subset.
This mapping is a graph transformation of linear
mappings $P(x), \ x
\in V(\G)$, under $D\, g_\ep^{-l\t}$ for some fixed  $l \in N, \
\t \in \R$.
It follows that the bundle $E_\ep^{ss}(x), \ x \in V(\G),$
corresponding to the fixed point of $H_\ep$ is invariant
with respect to $D\, g_\ep^{-l\t}.$ 
Now the uniqueness of a fixed point of the contraction
$H_\ep$ implies the invariance of $E_\ep^{ss}(x), \ x \in V(\G)$,
under $D\, g_\ep^t$ for all $t \in \R$.
Further, we prove that the bundle $E_\ep^{ss}(x), \ x \in
V(\G)$, is locally Lipschitz in $V(\G)$.

Similarly one constructs the second term $E_\ep^v(x),$
$ \ x \in V(\G)$, in decomposition (3). Note that the family
$E_\ep^v(x), \ x \in V(\G),$ may be not Lipschitz
in a neighborghood of $\G$.

We apply the bundle $\ E_\ep^{ss}(x), \ x \in V(\G),\ $  in
Section 3 (Theorem II.3.1) to construct a strongly stable
one-dimensional lamination $\ W_\ep^{ss}(x), \ x \in V(\G),\ $ in
a neighborhood of the cycle $\G$. Let us give a definition.

{\bf  Definition.}
{\it  We define a one-dimensional lamination on a
neighborhood} $V(\G)$
{\it as a decomposition of this neighborhood into a union
of smooth one-dimensional submanifolds called laminae such that
for any point} $x \in V(\G)$ {\it there is a neighborhood} $U$ {\it of
this point, a subset} $W \subset I^{n-1}$, {\it and a homeomorphism}
$\f: \ I \times I^{n-1} \to U$ {\it mapping}
$I\times w, \ w \in W,$ {\it diffeomorphically onto a connected
component of the intersetion of some lamina with} $U$. {\it
In addition, in some local coordinates} $\  v, \  w \ $, {\it
the derivative} $\ \frac{\pa \f}{\pa v}(v,w)\ $ {\it is continuous in}
$\ v, w.\ $

In the definition above, $I$ denotes $(-1,1)$, and we use
coordinates $\ v, \ w, \ v \in I, \ w \in I^{n-1}$ in the
$n$-dimensional cube $I^n = I \times I^{n-1}$.

The tangent bundle $W_\ep^{ss}(x), \ x \in V(\G),$ of our 
lamination is the bundle of lines 
$$
E_\ep^{ss}(x), \ x \in V(\G).
$$  
Since this last bundle is Lipschitz continuous, it is easy to
show that the homeomorphism $\f$ from the definition of a
lamination satisfies the Lipschitz condition with respect
to $w$ locally in our case. In addition, it is shown in
Sect. 3 that the family of curves
$W_\ep^{ss}(x), \ x \in V(\G)$, is invariant under the
mapping $g_\ep^\t$, i.e.,
$ \ g_\ep^\t W_\ep^{ss}(x) \cap V(\G) \subset
W_\ep^{ss}(g_\ep^\t(x)), \ x \in V(\G), $\ $ \t \in \R,\ $
and the mapping $g_\ep^\t$ contracts along laminae.
The last statement has the following meaning: there 
exist numbers $\l > 0,\  K > 0, \ \s > 0,$  such that
for
$z,y \in W_\ep^{ss}(x) \cap B_\s(x), \ x \in V(\G),$  
the inequality
$$ d(g_\ep^t(z),g_\ep^t(y)) \le K\exp (-\l t)\, d(z,y),$$
holds if the points $g_\ep^\t(y), \ g_\ep^\t(z) \in V(\G)$
for $\t \in [0,t]$. (Here $d(z,y)$ denotes the distance
between the points $z, y$, and $B_\s(x)$ is the ball of
radius $\s$ centered at $x$.)


For a segment $\wt\g$ of a trajectory of system (2), different
from a rest point and such that $\wt\g \subset V(\G)$,
we define its strongly stable set
$$ W_\ep^{ss}(\wt\g,\d) = \bigcup_{x \in \wt\g}
W_\ep^{ss}(x,\d),$$
where $W_\ep^{ss}(x,\d) = W_\ep^{ss}(x) \cap B_\d(x).$ 
For $\d$ small enough, this set is a two-dimensional smooth
manifold.

{\it The set} $\ W_\ep^{ss}(\wt\g,\d)$\ {\it is called the
strongly stable manifold of the segment} $\wt\g$ {\it of a
trajectory of system} (2) {\it of size} $\d$.


Let $S$ be an arbitrary transversal to trajectories of
system (2) belonging to the neighborhood  $V(\G)$.  
For a point $x \in S$, the connected component of the
intersection $\ W_\ep^{ss}(\wt\g,\d) \cap S,\ $
containing the point $x$ lying on a small enough segment
$\wt\g$ of its trajectory does not depend on the
segment $\wt\g$ if the number $\d$ is small enough.
If $S$ is a smooth transversal, then the component
$w_\ep^{ss}(x,\d)$ is a smooth curve.

{\it  We say that} $w_\ep^{ss}(x,\d)$  {\it is the
strongly stable manifold (or a strongly stable
curve) of the point}
$x$ {\it of size} $\d$ {\it on the transversal} $S$.

It is shown at the end of Sect. 3 that if a Poincare
mapping is defined for two transversals, then this
mapping contracts along strongly stable curves.

In Sect. 4, we construct a chaotic invariant set $J$
for the unperturbed system (1). The following statement
is proved.

{\bf Theorem II.4.2.}
{\it For any neighborhood} \ $\wt V$ {\it of the heteroclinic
cycle} $\G$, {\it satisfying conditions} I, II, {\it and} III, {\it
there exists a chaotic invariant set} $J \subset \wt
V.$

The invariant set $J$ is defined as the set of all
trajectories of system (1) through points of an invariant
set $I$ of the Poincare mapping $F$ of a transversal
$S(1)$ \ $ \subset  \wt V. $



We construct the invariant set $I$ in Lemmas
II.4.1 -- II.4.8. This set is constucted as the image
of the space $\O$ whose elements are two-sided
infinite sequences $\o = \{\o_s\}, \ s \in Z,$  
under a one-to-one mapping $\P$. Elements of the
sequences above belong to an infinite set $E$
of integer $k$-dimensional vectors.
If the space $\O$ is endowed with a standard
metric, then the mapping $\P$ is a homeomorphism
$ \ \P : \ \O \to \P\O = I.$  This statement is
proved in Lemma II.4.9. Consider the shift
homeomorhism $\s$ on the space $\O$. It follows
from the construction of the mapping $\P$ that
this mapping conjugates the mapping
$\s$ on the space $\O$ with the mapping $F$ on $I$.
Thus, a symbolic dynamics for the mapping
$\ F|_I\ $ is constructed. This construction is
applied in Theorem II.4.1 to show that the invariant
set $I$ is chaotic. This method for establishing
chaotic structure of an invariant set is well known
[22, 2]. Note that in our case, the alphabet $E$
used in construction of symbolic sequences is
infinite. As a result, we see that both the space
$\O$ and the set $I$ are not compact, and it is impossible
to extend the mapping $F$ to the closure of the 
set $I$ preserving its continuity.

Coding of points of the set $I$ by sequences
$\{\o_s\}, \ s \in Z,$ has a simple geometric
interpretation. If a vector $\o_s$ has coordinates
$(e_1^s,\dots,e_k^s), \ e_i^s \in N,$ then the 
trajectory of the point $x$ makes, after its
$s$th crossing of the transversal $S(1)$, 
$e_1^s$ turns around the closed trajectory
$P_{l_1}$ not leaving a small neighborhood
of this closed trajectory, then it makes $e_2^s$
turns around the closed trajectory $P_{l_2}$ and
so on, then it makes $e_k^s$ turns around
$P_{l_k} \subset \G$, and then it crosses the  
transversal $S(1)$ at $(s+1)$th time.

It is easy to show that the shift $\s$ has the same
basic properties as the shift on the space of
sequences with finite number of symbols. Thus,
we can prove that the mapping $\s$ is chaotic.

In Theorem II.4.1, we apply the topological
conjugacy of the mappings $F|_I$ and $\s$ to show
that the set $I$ is chaotic.

By this theorem, the invariant set $J$ defined above
has a dense half-trajectory, and closed trajectories
are dense in $J$.

The third property from the definition of a chaotic set
(the sensitive dependence on the initial point)
is established in Theorem II.4.2. Finally, we see 
that the invariant set $J$ of system (1) is chaotic.
The invariant set $\ol J$ is also chaotic. Since the
set $I$ is not compact, $\ol J\ne J$.

Theorem II.4.3 and its corollaries describe the
structure of invariant sets $\ol J \cap W_{l_i}^u, \ i \in 1:k.$ 
In particular, it is shown that the heteroclinic cycle
$\G$ belongs to the closure of the invariant set $J$.


In the last Sect. 5, we study the perturbed system (2)
such that the perturbation $Y(x)$ is $\C^1$-small.  
Let us formulate the main result.

{\bf Theorem II.5.1.}
{\it  Assume that system (1) has a heteroclinic cycle} $\G$ {\it
of the Lorenz type satisfying conditions} I, II, III. {\it
For any neighborhood} $V(\G)$ {\it of the cycle} $\G$ {\it
there exists a number} $\ep_0$ {\it such that if}
 $\ep < \ep_0$, {\it then the perturbed system (2) has a
chaotic set lying in} $ V(\G)$.

The proof of this theorem is contained in Lemmas
II.5.1 --II.5.7.  This proof is to some extent parallel
to the proof of Theorem II.4.2. Indeed, the chaotic
set $J(\ep)$ is constructed as the union of trajectories
of system (2) through points of a chaotic invariant set
$I(\ep)$ for the mapping $F(\ep)$ defined by first return
to the transversal $S(1)$ of trajectories of system (2)
belonging to a neighborhood $W(\G)$ of the heteroclinic 
cycle $\G$. Here and below, dependence on $\ep$  means
that we consider the perturbed system (2) such that
$\ ||Y||_{\C^1} < \ep$.

The mapping $F(\ep)|_{I(\ep)}$ is topologically conjugate
to the shift $\s$ on an invariant subset of the space $\O(\ep)$;
this space consists of infinite sequences $ \ \o =
\{\o_s\}_{s = -\infty}^{+\infty}, \ \o_s \in E(\ep), \ s \in Z.$
The subset invariant under $\s$ on which the shift $\s$  
is chaotic is determined by a condition on regularity
of intersection of some sets. This last condition is
quite complicated and similar to condition from [2].

The main difference with the case of the nonperturbed
system is that the alphabet $E(\ep)$ applied for coding
of points of the set $I(\ep)$ may be finite for some
perturbations. In this case, the invariant set $I(\ep)$  
is compact and locally maximal (see Lemma II.5.8.)
In the next lemma, we show that if the alphabet
$E(\ep)$ is finite, then the invariant set $J(\ep)$
constructed in Theorem II.5.1 is compact and locally 
maximal. By definition, the invariant set $J(\ep)$  
does not contain rest points of the perturbed system
belonging to the neighborhood $W(\G)$; it also does
not contain trajectories double-asymptotic to rest
points. Since the set $J(\ep)$ is compact, in the case
of finite alphabet $E(\ep)$, the chaotic set is
separated from rest points of the perturbed system.

It is shown in Theorem II.5.2 that, under some
additional condition that guarantees that the
mapping $F(\ep)|_{I(\ep)}$ is topologically
conjugate to the shift $\s$ on the whole space
$\O(\ep)$, the set $J(\ep)$ is a maximal compact
chaotic invariant set belonging to the neighborhood
$W(\G)$ and containing neither rest points of
system (2) nor their double-asymptotic 
trajectories.

In Theorem II.5.3, we describe the structure of the
invariant set $\ol {J(\ep)} \setminus J(\ep)$ in the 
case where the first condition of Theorem II.5.2
(the condition of finiteness of the set $E(\ep)$)
is violated. The proof of this theorem is contained
in Lemmas II.5.10 -- II.5.14.

Conditions of Theorem II.5.2 providing the maximality
of the invariant set $J(\ep)$ in the neighborhood $W(\G)$
are formulated in terms of the perturbation $Y(x)$. At
the end of the last section, we construct a perturbation
$Y(x)$
satisfying the conditions of Theorem II.5.2 and having 
arbitrarily small $\C^1$-norm.

This construction is contained in Theorem II.5.5. This
theorem shows that it is possible to separate a chaotic
invariant set $\ol {J(\ep)}$ from rest points by an
arbitrarily $\C^1$-small perturbation of system (1).

\medskip
Translated into English by S.~Yu.~Pilyugin.

\end
