Appearance of discrete Lorenz attractors in the transitions from saddle to
saddle-focus

I.I. Ovsyannikov

Constructor University, Germany;
Lobachevsky State University of Nizhny Novgorod, Russia;

E-mail<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Abstract. Strange attractors of the Lorenz type are among the most important
objects of this theory, because these chaotic attractors preserve their
chaotic properties under small perturbations. This feature is very important
for applications, experiments, where certain tolerance is unavoidably present,
but it does not destroy chaotic attractors. The discrete Lorenz attractor is a
discrete-time analogue of the classical Lorenz attractor, but its dynamics is
richer - for example, it allows homoclinic tangencies, hence discrete Lorenz
attractors are wild attractors. Another important property of these attractors
is that they can be born in bifurcations near triply degenerate fixed or
periodic points. This makes possible analytic proof of their existence in
systems that allow the existence of such points. Triply degenerate fixed
points appear in global bifurcations - homoclinic and heteroclinic tangencies.
In order to get Lorenz-like attractors, the dynamics of the first return map
along the homoclinic or heteroclinic cycle should be effectively at least
three-dimensional, i.e. there should not exist lower-dimensional invariant
manifolds. This can be achieved by adding some special conditions, global or
local. Global degeneracies are related to the existence of non- simple
homoclinic tangencies or non-simple heteroclinic orbit in a cycle, these cases
were studied before. Local conditions either require the cycle to contain at
least one saddle-focus, or add certain relations on the multipliers of the
fixed point such that the leading stable direction of the saddle either
disappears or alternates. All these cases were already studied before except
one, related to the transition from saddle to saddle-focus. In the present
paper this case is investigated, and the existence of a cascade of parameter
domains containing systems with discrete Lorenz attractors is proved. In
particular, it includes the Belyakov bifurcation, when the saddle becomes a
saddle-focus through a collision of the eigenvalues, and the 3DL bifurcation,
when the dimension of the leading stable subset alternates between 1 and 2.
This paper completes the list of the simplest bifurcations of homoclinic and
heteroclinic tangencies by studying the last possible case.

Keywords: Homoclinic tangency, rescaling, 3D Hénon map, Belyakov bifurcation,
Lorenz attractor.

Mathematics Subject Classification: 37C29, 37G25, 37G35

## Introduction

Dynamical chaos is the research area that is being rapidly developed in the
last decades. Chaotic phenomena are often represented by strange attractors.
They can be of quite different types, and hyperbolic attractors and Lorenz
attractors are the mostly known and well investigated. Such attractors are
remarkable by the fact that they preserve their ‘‘strangeness’’ under small
smooth perturbations, and, besides, hyperbolic attractors are structurally
stable111We note that many well-known types of chaotic dynamics, such as
Hénon-like attractors, spiral attractors (R$\ddot{o}$ssler attractors,
attractors observed in the Chua circuits) etc, are, in fact, the so-called
quasiattractors (in the terms introduced by Afraimovich and Shilnikov [10])
because stable periodic orbits can appear in them under arbitrary small smooth
perturbations.. Until the nearest time, only these attractors were known to
compose the class of the so-called genuine strange attractors. The situation
was drastically changed after the paper [1] by Turaev and Shilnikov in which
the theory of the wild hyperbolic attractors was founded and an example of a
wild spiral attractor was presented.

Wild hyperbolic attractors allow homoclinic tangencies, unlike the hyperbolic
and Lorenz attractors, and therefore they belong to Newhouse domains [2, 3]
but nevertheless stable periodic orbits as well as other stable invariant
subsets are not born under perturbations. Thus, wild hyperbolic attractors
should also be referred to as genuine strange attractors222Moreover, by
definition (see [1]), such attractors are chain transitive and stable closed
invariant sets. The chain transitivity means that any point of the attractor
$\Lambda$ is admissible by $\varepsilon$-orbits from any other point of
$\Lambda$; the stability means the existence of an open adsorbing domain
containing the attractor such that any orbit entering into the domain tends to
$\Lambda$ (exponentially fast). Thus, this definition is, in fact, the
definition of attractor by Ruelle and Conley.. An important subset of wild
hyperbolic attractors is composed by discrete Lorenz attractors which appear,
in particular, in Poincare maps for periodically perturbed flows with the
Lorenz attractors [4]. The definition of such an attractor is given in papers
[5, 6]. It is well-known that Lorenz attractors do not allow homoclinic
tangencies [7, 8]. However the latter can appear under non-autonomous periodic
perturbations. Nevertheless, bifurcation of these tangencies do not lead to
the appearance of stable periodic orbits. The reason of it is that Lorenz-like
attractors possess the pseudo-hyperbolic structure333This, very briefly, means
that the differential $Df$of the corresponding map $f$ in the restriction onto
the absorbing domain ${\cal D}$ of attractor $\Lambda$ admits an invariant
splitting of form $E^{ss}_{x}\oplus E^{uc}_{x}$, for any point $x\in{\cal D}$,
such that $Df$ is strongly contracting along directions $E^{ss}$ and expands
volumes in transversal to $E^{ss}$ sections $E^{uc}$ (see [1, 4] for details).
However, for a long time, specific examples of systems of differential
equations with such attractors were not known. And quite recently, such an
example was proposed in [9], this is a four-dimensional extension of the well-
known Lorenz model.

One of important peculiarities of the discrete Lorenz attractors is that such
attractors can be born at local bifurcations of periodic orbits having three
or more multipliers lying on the unit circle. This means that the
corresponding attractors can be found in particular models which have a
sufficient number of parameters to provide the mentioned degeneracy. The
following 3D Hénon map

$\bar{x}=y,\;\;\bar{y}=z,\;\;\bar{z}=M_{1}+Bx+M_{2}y-z^{2}$ (0.1)

controlled by three independent parameters $M_{1}$, $M_{2}$ and $B$ is an
example of such a model. In the papers [5, 11, 12] it was shown that the map
(0.1) possesses a discrete Lorenz-like attractor in some open parameter domain
close to the point $(M_{1}=1/4,B=1,M_{2}=1)$ when the map has a fixed point
with the triplet $(-1,-1,+1)$ of multipliers. Recently, discrete Lorenz
attractors were also found in the orientation-reversing case $B<0$, near point
$(M_{1},M_{2},B)=(7/4,-1,-1)$, see [6].

###### Remark 1

In paper [13] two universal scenarios of the development of chaos from a
stable fixed point to a strange attractor were represented for 3D maps. One of
these scenarios leads to the appearance of a discrete Lorenz-like attractor.
This means that the discrete Lorenz attractors can be freely met in various
dynamical models. In particular, such an attractor was found very recently in
a nonholonomic model of a rattleback [14] and a model of two-component fluid
convection [15].

The fact that strange attractors of other nature than Hénon-like ones (e.g.
Lorenz-like attractors) can appear under bifurcations of homoclinic tangencies
in multidimensional case (the dimension for maps is $\geq 4$) was announced
first in the paper [16]. Concerning the discrete Lorenz-like attractors, in
the paper [12] it was shown that they appear in three parameter general
unfoldings in the case of three-dimensional diffeomorphisms having a
homoclinic tangency to a fixed point of the saddle-focus type with a Jacobian
equal to one. Analogous results were obtained in papers [17, 18, 19] while
studying three parameter general unfoldings of three-dimensional
diffeomorphisms with a nontransversal contracting-expanding heteroclinic cycle
containing two fixed points which are saddle and saddle-focus [17] or are both
saddle-foci [18, 19].

Note that in all these cases it was assumed that at least one of the fixed
points has the saddle-focus type. Formally, together with the conditions on
the Jacobians of the fixed points, it meant that the so-called effective
dimension $d_{e}$ of the problem (see [20]) equals to three i.e. is maximal
for three-dimensional diffeomorphisms. When all fixed points are saddles, i.e.
when all three multipliers of the fixed point are real, in the general case
there exists a global two-dimensional invariant manifold, such that the
transverse to it dynamics is trivial. Then, in order to keep the effective
dimension at least three, additional degeneracies should be introduced that
prevent from existence of such invariant manifolds. The full list of such
conditions was identified in [21].

These degeneracies can be divided into global and local. Global degeneracies
mean that the homoclinic and heteroclinic cycles contain non-simple orbits.
Birth of discrete Lorenz attractors in such bifurcations were studied in paper
[22] in the homoclinic case and paper [23] in the heteroclinic case. Local
degeneracies occur when the eigenvalues at saddle fixed point satisfy
resonance conditions. In paper [24] the birth of discrete Lorenz attractors
was studied when the stable eigenvalues are resonant, namely they satisfy
condition $\lambda_{1}=-\lambda_{2}$. Another possibility is to have multiple
eigenvalues, such that in small perturbations the fixed point can become
either a hyperbolic saddle or a saddle-focus. The continuous-time analog of
this bifurcation was studied by L. Belyakov in paper [25]. So we call this
bifurcation in discrete-time systems a discrete Belyakov bifurcation.

In the present paper we study cases of homoclinic and heteroclinic tangencies
to fixed points having the saddle type, such that one of the fixed points
undergoes the discrete Belyakov bifurcation.

The paper is organized as follows. In section 1 there is the statement of the
problem and the main result – Theorem 1 – is formulated. In section 2 the
first return map is constructed in both cases, and the non-degeneracy
conditions are obtained. At the end of the section the rescaling lemma 1 is
formulated. The proof of the lemma is in section 3.

##  1 Statement of the problem and main results

We consider two cases of $C^{r}$-diffeomorphisms, $r\geq 3$, satisfying the
following conditions:

Case I: A homoclinic Belyakov bifurcation

I.A) $f_{0}$ is three-dimensional;

I.B) $f_{0}$ has a fixed point $O$ with multipliers
$(\lambda_{1},\lambda_{2},\gamma)$ where $\lambda_{1}=\lambda_{2}=\lambda$ and
$|\lambda|<1<|\gamma|$;

I.C) The Jacobian $J(O)\equiv|\lambda^{2}\gamma|$ at point $O$ is equal to
one;

I.D) Unstable manifold $W^{u}(O)$ and stable manifold $W^{s}(O)$ have a
quadratic tangency at the points of a homoclinic orbit $\Gamma_{0}$.

Case II: A heteroclinic cycle with Belyakov bifurcation

II.A) $f_{0}$ is three-dimensional;

II.B) $f_{0}$ has two fixed points $O_{1}$ and $O_{2}$ with multipliers
$(\lambda_{1},\lambda_{2},\gamma_{1})$ and $(\nu_{1},\nu_{2},\gamma_{2})$
respectively, such that $\lambda_{1}=\lambda_{2}=\lambda$,
$|\lambda|<1<|\gamma_{1}|$, $|\nu_{2}|<|\nu_{1}|<1<|\gamma_{2}|$;

II.C One of the Jacobians $J_{1}=|\lambda_{1}\lambda_{2}\gamma_{1}|$ and
$J_{2}=|\nu_{1}\nu_{2}\gamma_{2}|$ is greater than $1$, and another one is
smaller than $1$;

II.D.a Unstable manifold $W^{u}(O_{1})$ and stable manifold $W^{s}(O_{2})$
intersect transversely at the points of heteroclinic orbit $\Gamma_{12}$,
unstable manifold $W^{u}(O_{2})$ and stable manifold $W^{s}(O_{1})$ have
quadratic tangencies at the points of heteroclinic orbit $\Gamma_{21}$;

II.D.b Unstable manifold $W^{u}(O_{1})$ and stable manifold $W^{s}(O_{2})$
have quadratic tangencies at the points of heteroclinic orbit $\Gamma_{12}$,
unstable manifold $W^{u}(O_{2})$ and stable manifold $W^{s}(O_{1})$ intersect
transversely at the points of heteroclinic orbit $\Gamma_{21}$.

We will study bifurcations of single-round periodic orbits in generic
parametric unfoldings $f_{\mu}$ such that
$\left.f_{\mu}\right|_{\mu=0}=f_{0}$. The main attention will be paid to the
birth of discrete Lorenz attractors in local bifurcations. First, we have to
identify the necessary set of parameters required to unfold the original
system.

Figure 1: The multipliers in the fixed point $O$ in Case I, or $O_{2}$ in Case
II a) for $\mu_{2}>0$; b) for $\mu_{2}<0$.

In Case I we take, as usual, the splitting distance of manifolds $W^{u}(O)$
and $W^{s}(O)$ with respect of some of the points of $\Gamma_{0}$ as the first
parameter $\mu_{1}$. At the bifurcation moment we have
$\lambda_{1}=\lambda_{2}$ (condition I.B), and we introduce parameter
$\mu_{2}$ to unfold this condition, such that for $\mu_{2}=0$ the multipliers
are equal $\lambda_{1}(\mu)=\lambda_{2}(\mu)$, for $\mu_{2}>0$ both
eigenvalues are real and for $\mu_{2}<0$ they are a complex-conjugate pair,
see fig. 1.

The third parameter will control the Jacobian at $O$ in the way that

$\mu_{3}=1-|\lambda_{1}(\mu)\lambda_{2}(\mu)\gamma(\mu)|.$ (1.1)

In Case II value $\mu_{1}$ will be also a splitting parameter of a quadratic
tangency, $\mu_{2}$ will also control the multipliers of the fixed point
$O_{1}$ such that they are real for $\mu_{2}>0$ and complex for $\mu_{2}<0$.
Also note that due to condition II.B, near one saddle point the phase volumes
are contracted and near another one the volumes are expanded. Thus, we are
dealing with the contracting-expanding (or mixed) case, which requires one
more parameter $\mu_{3}$, that will control the values of the Jacobians
$J_{1}$ and $J_{2}$. Similar to [17, 18, 19, 23], the following value will
play this role:

$\mu_{3}=S(f_{\mu})-S(f_{0}),$ (1.2)

where $S(f)$ is a functional defined as $\displaystyle S(f)=-\frac{\ln
J_{1}}{\ln J_{2}}$.

In any neighbourhood of initial diffeomorphism $f_{0}$ there exist
diffeomorphisms $f_{\mu}$ having saddle fixed points with non-trivial leading
and non-leading stable invariant subspaces. This means that homoclinic or
heteroclinic orbits to such fixed points can potentially have additional
degeneracies. In order to prevent from other bifurcations, we have to add
generality conditions. Namely, we require the homoclinic or heteroclinic
orbits to be simple in some neighbourhood of $f_{0}$.

The main result of the present paper is the following theorem.

######  Theorem 1 (Main Theorem)

Let $f_{\mu}$ be the three-parametric family under consideration, that is,
$f_{0}$ satisfies A)–D), and $f_{\mu}$ is a general unfolding under conditions
B), C) and D). Then,

1) In the homoclinic Case I in any neighbourhood of the origin $\mu=0$ in the
parameter space there exist infinitely many domains $\delta_{k}$, where
$\delta_{k}\to(0,0,0)$ as $k\to\infty$, such that the diffeomorphism $f_{\mu}$
has a discrete Lorenz attractor at $\mu\in\delta_{k}$.

2) In the heteroclinic Case II in any neighbourhood of the origin $\mu=0$ in
the parameter space there exist infinitely many domains $\delta_{kj}$, where
$\delta_{kj}\to(0,0,0)$ as $k,j\to\infty$, such that the diffeomorphism
$f_{\mu}$ has a discrete Lorenz attractor at $\mu\in\delta_{kj}$.

The proof of the Theorem is based on the Rescaling Lemma 1, and is given after
the lemma’s formulation.

##  2 Construction of the first return map

The first return maps near homoclinic or heteroclinic cycles is a composition
of local and global maps. The local maps are defined in some small
neighbourhoods of fixed points, and the global maps connect these
neighbourhoods along the global pieces of homoclinic or heteroclinic orbits.

We consider a sufficiently small neighbourhood $U$ of the homoclinic of the
heteroclinic cycle. It is composed as a union of a small neighborhood $U_{0}$
of fixed point $O$ with a finite number of small neighborhoods $U_{i}$ of
those points of $\Gamma$ which do not belong to $U_{0}$ in Case I. In the
heteroclinic Case II, it is composed of small neighbourhoods $U_{1}$ and
$U_{2}$ of fixed points $O_{1}$ and $O_{2}$ and a finite number of small
neighborhoods $U_{i}$ of those points of $\Gamma_{12}$ and $\Gamma_{21}$ which
do not belong to $U_{1,2}$. Each single-round periodic orbit lying entirely in
$U$ has exactly one intersection point with each of $U_{i}$.

The local maps $T_{0},T_{10},T_{20}$ near respective fixed points
$O,O_{1},O_{2}$ is defined as a restriction of $f_{\mu}$ onto $U_{0}$, $U_{1}$
and $U_{2}$.

We select two homoclinic points $M^{+}\in U_{0}\cap\Gamma$ and $M^{-}\in
U_{0}\cap\Gamma$, such that $M^{+}\in W^{s}_{loc}(O)$ and $M^{-}\in
W^{u}_{loc}(O)$ and their respective small neighbourhoods $\Pi^{+}\subset
U_{0}$ and $\Pi^{-}\subset U_{0}$. It is clear that there exists a natural
number $q$ such that $M^{+}=f_{0}M^{-}$. We define the global map as
$T_{1}(\mu)=f_{\mu}^{q}:\Pi^{-}\to\Pi^{+}$.

In the heteroclinic case, we select in $U_{1}$ heteroclinic points
$M_{1}^{+}\in W^{s}_{loc}(O_{1})$ and $M_{1}^{-}\in W^{u}_{loc}(O_{1})$ and in
$U_{2}$ heteroclinic points $M_{2}^{+}\in W^{s}_{loc}(O_{2})$ and
$M_{2}^{-}\in W^{u}_{loc}(O_{2})$, such that
$M_{1}^{-},\;M_{2}^{+}\in\Gamma_{12}$ and
$M_{2}^{-},\;M_{1}^{+}\in\Gamma_{21}$. There exist numbers $q_{1}$ and $q_{2}$
such that $M_{2}^{+}=f_{0}^{q_{1}}M_{1}^{-}$ and
$M_{1}^{+}=f_{0}^{q_{2}}M_{2}^{-}$. We define global maps along heteroclinic
orbits $\Gamma_{12}$ and $\Gamma_{21}$ as
$T_{12}=f_{\mu}^{q_{1}}:\Pi_{1}^{-}\to\Pi_{2}^{+}$ and
$T_{21}=f_{\mu}^{q_{2}}:\Pi_{2}^{-}\to\Pi_{1}^{+}$

Begin iterating $\Pi^{+}$ under the action of $T_{0}$. Starting from some
number $k_{0}$ images $T_{0}^{k}\Pi^{+}$ will have a nonempty intersection
with $\Pi^{-}$. The first return map $T_{k}\equiv T_{1}T_{0}^{k}$ is thus
defined for every $k\geq k_{0}$ on an infinite set of stripes
$\sigma_{0}^{k}\subset\Pi^{+}$ accumulating to $W^{s}(O)$ as $k\to\infty$.
Their images under the iterations of the local map are stripes
$f^{k}_{\mu}\sigma_{0}^{k}=\sigma_{1}^{k}\subset\Pi^{-}$ which accumulate to
$W^{u}(O)$. Single-round periodic orbits lie in intersections
$\sigma_{0}^{k}\cap T_{1}\sigma_{1}^{k}$.

Analogously to the homoclinic case, in the heteroclinic case there exists a
number $k_{0}$ such that images $T_{10}^{k}\Pi_{1}^{+}$ have a non-empty
intersection with $\Pi_{1}^{-}$ for $k>k_{0}$ and respectively $j_{0}$ such
that $T_{20}^{j}\Pi_{2}^{+}$ have a non-empty intersection with $\Pi_{2}^{-}$
for $j>j_{0}$. The first return maps $T_{kj}=T_{21}T_{20}^{j}T_{12}T_{10}^{k}$
is therefore defined on subsets $V_{kj}\subset\Pi_{1}^{+}$ that accumulate to
$W^{s}(O_{1})$ as $k,j\to\infty$. In order to write explicit formulas for the
first return map we need to write both local and global maps in the most
suitable form.

###  2.1 Properties of the local maps

Assume that a fixed point possesses $n$ leading stable eigenvalues
$\lambda_{1},\ldots,\lambda_{n}$ and $m$ non-leading stable eigenvalues
$\lambda_{n+1},\ldots,\lambda_{n+m}$ and an unstable eigenvalue $\gamma$. This
means that $|\lambda_{1}|=\ldots=|\lambda_{n}|=\lambda<1$ and
$|\lambda_{i}|<\lambda$ for $i>n$. The main normal form has then the following
representation in some $C^{r}$-smooth local coordinates (see [26, 27, 28,
29]):

$\begin{array}[]{l}\bar{x}_{1}\;=\;A_{1}(\mu)x_{1}+\tilde{H}_{2}(y,\mu)x_{2}+O(\|x\|^{2}\|y\|)\\\
\bar{x}_{2}\;=\;A_{2}(\mu)x_{2}+\tilde{R}_{2}(x,\mu)+\tilde{H}_{4}(y,\mu)x_{2}+O(\|x\|^{2}\|y\|)\\\
\bar{y}\;=\;\gamma(\mu)y+O(\|x\|\|y\|^{2}),\\\ \end{array}$ (2.1)

where $\operatorname{spec}A_{1}(0)=\\{\lambda_{1},\ldots,\lambda_{n}\\}$,
$\operatorname{spec}A_{2}(0)=\\{\lambda_{n+1},\ldots,\lambda_{n+m}\\}$,
$\tilde{H}_{2,4}(0,\mu)=0$, $\tilde{R}_{2}(x,\mu)=O(\|x\|^{2})$. In
coordinates (2.1) the invariant manifolds of saddle fixed point $O$ are
locally straightened: stable $W^{s}_{loc}(O):\\{y=0\\}$, unstable
$W^{u}_{loc}(O):\\{x_{1}=0,\;x_{2}=0\\}$ and strong stable
$W^{ss}_{loc}(O):\\{x_{1}=0,\;y=0\\}$.

If the fixed points is a saddle ($O_{2}$), then matrices $A_{1}(\mu)$ and
$A_{2}(\mu)$ are one-dimensional, $\operatorname{spec}A_{1}=\\{\nu_{1}\\}$ and
$\operatorname{spec}A_{2}=\\{\nu_{2}\\}$. Multiple iterations of the local map
$T_{20}^{j}:\;U_{2}\to U_{2}$ near such a point can be represented in the form
close to linear (see, for example, [28, 29]):

$\begin{array}[]{l}u_{1j}=\nu_{1}^{j}(\mu)u_{10}+\hat{\nu}^{j}\eta_{1j}(u_{0},v_{k},\mu)\\\
u_{2j}=\hat{\nu}^{j}\eta_{2j}(u_{0},v_{k},\mu)\\\
v_{0}=\gamma_{2}^{-j}v_{j}+\hat{\gamma}_{2}^{-j}\eta_{3j}(u_{0},v_{j},\mu).\end{array}$
(2.2)

Here $0<\hat{\nu}<\nu_{1}$, $\hat{\gamma}_{2}>\gamma_{2}$, functions
$\eta_{j}$ are uniformly bounded in $j$ together with their derivatives up to
order $(r-2)$, and their higher-order derivatives tend to zero.

According to [29, 30], an important role in the global dynamics is played by
local structures near the saddle fixed point. The first such object is an
extended unstable manifold  $W^{ue}(O)$, see figure 2. By definition, it is a
two-dimensional invariant manifold, tangent to the leading stable direction
(those, corresponding to $\lambda_{1}$) at the saddle point and containing the
unstable manifold $W^{u}(O)$. Unlike the previous ones, the extended unstable
manifold is not uniquely defined and $C^{1+\varepsilon}$–smooth. Locally,
$W^{ue}_{loc}(O)=W^{ue}(O)\cap U_{0}$, and the equation of $W^{ue}_{loc}(O)$
has the form $x_{2}=\psi(x_{1},y)$, where $\psi(0,y)\equiv 0$ and
$\psi^{\prime}_{x_{1}}(0,0)=0$. Note that despite the fact that $W^{ue}(O)$ is
non-unique, all of them have the same tangent plane at each point of
$W^{u}(O)$.

Another essential structure is the strong stable invariant foliation on
$W^{s}(O)$, see figure 2. In $W^{s}(O)$ there exists a one-dimensional strong
stable invariant submanifold $W^{ss}(O)$, which is $C^{r}$–smooth and touches
at $O$ the strong stable direction. Moreover, manifold $W^{s}(O)$ is foliated
near $O$ by the leaves of an one-dimensional invariant foliation $F^{ss}$
which is $C^{r}$-smooth, unique and contains $W^{ss}(O)$ as a leaf.

Figure 2: A part of the strong stable foliation $F^{ss}$ containing the strong
stable manifold $W^{ss}$; and a piece of one of the extended unstable
manifolds containing $W^{u}$ and being transversal to $W^{ss}$ at $O$.

The existence of these structures implies that locally in the neighborhood
$U_{2}$ the dynamics can be locally decomposed into a dynamics on the two-
dimensional invariant manifold $W^{ue}(O_{2})$ and strong transversal
contraction towards it along the leaves of the foliation $F^{ss}$. If such a
decomposition spreads globally along the whole homoclinic (heteroclinic)
cycle, it will imply the existence of a global two-dimensional invariant
manifold. In such a case, the existence of three-dimensional chaotic objects
(discrete Lorenz attractors) is not possible. Thus, in order to have these
attractors it is essential to prevent from existence of such global invariant
manifolds. These conditions will be discussed in details in the next
subsection.

If a fixed point experiences a transition from a saddle to a saddle-focus ($O$
or $O_{1}$), then for $\mu_{2}>0$, when it has real eigenvalues and becomes a
saddle, the matrices $A_{1}(\mu)$ and $A_{2}(\mu)$ are one-dimensional. But
for $\mu\leq 0$ their dimensions are different, namely $A_{1}(\mu)$ is two-
dimensional and $A_{2}(\mu)$ zero-dimensional. It means that normal forms
(2.1) for $\mu_{2}<0$ and $\mu_{2}>0$ can not be conjugated smoothly in a
parametric family. So in order to construct a smooth family of local maps, we
need to use another form of the linear part. A normal form proposed in [31]
for multiple eigenvalues plays well this role. Namely, the following formula
explicitly defines $\mu_{2}$ and provides a smooth conjugation of families of
matrices with real and complex-conjugated eigenvalues:

$A_{1}(\mu)=\begin{pmatrix}\lambda&1\\\ \mu_{2}&\lambda\end{pmatrix}.$ (2.3)

Indeed, when $\mu_{2}>0$ the eigenvalues of $A_{1}$ are real,
$\lambda_{1,2}=\lambda\pm\sqrt{\mu_{2}}$ and for $\mu_{2}<0$ they are complex-
conjugate $\lambda_{1,2}=\lambda\pm i\sqrt{-\mu_{2}}$. To use this matrix in
the first return map we need to find a suitable form for its multiple
iterations $A_{1}^{k}(\mu)$. Namely, if $\mu_{2}>0$:

$A_{1}^{k}(\mu)=\begin{pmatrix}\displaystyle\frac{(\lambda+\sqrt{\mu_{2}})^{k}+(\lambda-\sqrt{\mu_{2}})^{k}}{2}&\displaystyle\frac{(\lambda+\sqrt{\mu_{2}})^{k}-(\lambda-\sqrt{\mu_{2}})^{k}}{2\sqrt{\mu_{2}}}\\\
\displaystyle\sqrt{\mu_{2}}\frac{(\lambda+\sqrt{\mu_{2}})^{k}-(\lambda-\sqrt{\mu_{2}})^{k}}{2}&\displaystyle\frac{(\lambda+\sqrt{\mu_{2}})^{k}+(\lambda-\sqrt{\mu_{2}})^{k}}{2},\end{pmatrix}$
(2.4)

if $\mu_{2}<0$

$A_{1}^{k}(\mu)=\begin{pmatrix}\displaystyle\frac{(\lambda+i\sqrt{-\mu_{2}})^{k}+(\lambda-i\sqrt{-\mu_{2}})^{k}}{2}&\displaystyle\frac{(\lambda+i\sqrt{-\mu_{2}})^{k}-(\lambda-i\sqrt{-\mu_{2}})^{k}}{2i\sqrt{-\mu_{2}}}\\\
\displaystyle
i\sqrt{-\mu_{2}}\frac{(\lambda+i\sqrt{-\mu_{2}})^{k}-(\lambda-i\sqrt{-\mu_{2}})^{k}}{2}&\displaystyle\frac{(\lambda+i\sqrt{-\mu_{2}})^{k}+(\lambda-i\sqrt{-\mu_{2}})^{k}}{2}\end{pmatrix}$
(2.5)

and if $\mu_{2}=0$:

$A_{1}^{k}(\mu)=\begin{pmatrix}\lambda^{k}&k\lambda^{k-1}\\\
0&\lambda^{k}\end{pmatrix}.$ (2.6)

It is clear that $A_{1}^{k}(\mu)$ depends smoothly on $\mu_{2}$. Formulas
(2.4)–(2.6) can be written in a more compact form. Define angle $\phi$ as
follows:

$\phi=\left\\{\begin{array}[]{l}\displaystyle\operatorname{arctanh}\frac{\sqrt{\mu_{2}}}{\lambda},\;\;\mbox{if}\;\;\mu_{2}>0\\\
0,\;\;\mbox{if}\;\;\mu_{2}=0\\\
\displaystyle-\arctan\frac{\sqrt{-\mu_{2}}}{\lambda},\;\;\mbox{if}\;\;\mu_{2}<0.\\\
\end{array}\right.$ (2.7)

Then finally

$A^{k}(\mu)=(\lambda^{2}-\mu_{2})^{k/2}\begin{pmatrix}C_{k}(\phi)&S_{k}(\phi)\\\
\mu_{2}S_{k}(\phi)&C_{k}(\phi)\end{pmatrix},$ (2.8)

where

$C_{k}(\phi)=\left\\{\begin{array}[]{l}\cosh
k\phi,\;\;\mbox{if}\;\;\mu_{2}\geq 0\\\ \cos
k\phi,\;\;\mbox{if}\;\;\mu_{2}<0.\\\ \end{array}\right.$ (2.9)

and

$S_{k}(\phi)=\left\\{\begin{array}[]{l}\displaystyle\frac{\sinh
k\phi}{\sqrt{\mu_{2}}},\;\;\mbox{if}\;\;\mu_{2}>0\\\
\displaystyle\frac{k}{\lambda},\;\;\mbox{if}\;\;\mu_{2}=0\\\
\displaystyle-\frac{\sin
k\phi}{\sqrt{-\mu_{2}}},\;\;\mbox{if}\;\;\mu_{2}<0.\\\ \end{array}\right.$
(2.10)

are smooth functions of $\mu_{2}$. Now, multiple iterations near $O$,
$T_{0}^{k}:U_{0}\to U_{0}$ (or near $O_{1}$), for any $k$ can be also
calculated in a form close to linear. For all small $\mu$ iterations
$T_{0}^{k}:(x_{0},y_{0})\to(x_{k},y_{k})$ can be represented as:

$\begin{array}[]{l}(x_{1k},x_{2k})^{\top}=A_{1}^{k}(\mu)(x_{10},x_{20})^{\top}+\hat{\lambda}^{k}\xi_{1k}(x_{0},y_{k},\mu)\\\
y_{0}=\gamma^{-k}y_{k}+\hat{\gamma}^{-k}\xi_{2k}(x_{0},y_{k},\mu).\end{array}$
(2.11)

Here also, $0<\hat{\lambda}<\lambda$, $\hat{\gamma}>\gamma$, functions
$\xi_{k}$ are uniformly bounded together with their derivatives up to order
$(r-2)$ and their higher-order derivatives tend to zero.

###  2.2 Properties of the global maps

Consider the homoclinic Case I. Assume that for $\mu=0$, the coordinates of
the homoclinic points are $M^{-}(0,0,y^{-})\in W^{u}_{loc}(O)$,
$M^{+}(x_{1}^{+},x_{2}^{+},0)\in W^{s}_{loc}(O)$. Then the global map for all
small $\mu$ can be written as a Taylor expansion near point $M^{-}$:

$\begin{array}[]{l}\bar{x}_{10}-x_{1}^{+}=a_{11}x_{1k}+a_{12}x_{2k}+b_{1}(y_{k}-y^{-})+O(\|x_{k}\|^{2}+\|x_{k}\|\cdot|y_{k}-y^{-}|+(y_{k}-y^{-})^{2}),\\\
\bar{x}_{20}-x_{2}^{+}=a_{21}x_{1k}+a_{22}x_{2k}+b_{2}(y_{k}-y^{-})+O(\|x_{k}\|^{2}+\|x_{k}\|\cdot|y_{k}-y^{-}|+(y_{k}-y^{-})^{2}),\\\
\bar{y}_{0}=y^{+}+c_{1}x_{1k}+c_{2}x_{2k}+d(y_{k}-y^{-})^{2}+O(\|x_{k}\|^{2}+\|x_{k}\|\cdot|y_{k}-y^{-}|+(y_{k}-y^{-})^{3}),\end{array}$
(2.12)

where all the coefficients smoothly depend on parameters. We note also that
$d\neq 0$ since $W^{u}(O)$ and $W^{s}(O)$ have a quadratic tangency for
$\mu=0$, i.e. $y^{+}(0)=0$. With that we define the first control parameter to
be $\mu_{1}\equiv y^{+}(\mu)$. Map $T_{1}$ is a diffeomorphism, so that

$J_{T1}=\mbox{det}\;\begin{pmatrix}a_{11}&a_{12}&b_{1}\\\
a_{21}&a_{22}&b_{2}\\\ c_{1}&c_{2}&0\end{pmatrix}\;\neq 0$ (2.13)

In particular, this means that $b_{1}^{2}+b_{2}^{2}\neq 0$ and
$c_{1}^{2}+c_{2}^{2}\neq 0$ for $\mu=0$.

For small $\mu_{2}>0$ the fixed point $O$ becomes a saddle point. Therefore,
additional degeneracies may occur with the homoclinic orbit. Namely, the
homoclinic tangency can become non-simple. Next we give the definition of it
and formulate the non-degeneracy condition in coordinates.

Denote as $P^{ue}(M^{-})$ a tangent plane to to $W^{ue}(O)$ at the point
$M^{-}$. The homoclinic tangency of $W^{u}(O)$ and $W^{s}(O)$ is called simple
if image $T_{1}(P^{ue}(M^{-}))$ intersects transversely (i.e. generically) the
leaf $F^{ss}(M^{+})$ of invariant foliation $F^{ss}$, coming through point
$M^{+}$. If this condition is not fulfilled, such a quadratic tangency is
called non-simple. There are two possibilities of codimension-one non-simple
homoclinic orbits: 1) inclination flip, when the surface
$T_{1}(P^{ue}(M^{-}))$ is tangent to surface $W^{s}(O)$, but $W^{u}(O)$ is not
tangent to the leaf $F^{ss}(M^{+})$, and 2) orbit flip, when
$T_{1}(P^{ue}(M^{-}))$ is transverse to $W^{s}(O)$, but $W^{u}(O)$ is tangent
to the leaf $F^{ss}(M^{+})$. It is clear that in both cases further forward
iterations of $W^{ue}(O)$ will not tend to be tangent to $W^{ue}_{loc}(O)$,
hence a global two-dimensional invariant manifold will not exist.

The notion of a simple quadratic homoclinic tangency was introduced in [16],
and it is a variant of the so-called quasi-transversal intersection [2]. A
simple homoclinic tangency of arbitrary order was defined in [32].
Bifurcations of non-simple homoclinic tangencies were studied first in [33],
where it was called ‘‘generalized homoclinic tangency’’, and also [34]. Birth
of discrete Lorenz attractors in bifurcations of non-simple homoclinic and
heteroclinic orbits were studied in [22, 23].

Now consider small $\mu_{2}>0$ and a homoclinic tangency (i.e. $\mu_{1}=0$),
for which $O$ is a saddle with stable multipliers
$\lambda_{1,2}=\lambda\pm\sqrt{\mu_{2}}$ and unstable one $\gamma$. In the
notation of section 1, this means that $\dim W^{ss}(O)=\dim(W^{s}(O)\backslash
W^{ss}(O))=\dim W^{u}(O)=1$, $\dim W^{ue}(O)=2$ and invariant foliation
$F^{ss}$ consists of one-dimensional leaves. The stable eigenvectors at $O$
are: leading $l^{s}=(1,\sqrt{\mu_{2}},0)^{\top}$ and non-leading
$l^{ss}=(-1,\sqrt{\mu_{2}},0)^{\top}$. The leaves of strong stable invariant
foliation $F^{ss}$ are straight lines parallel to $l^{ss}$ and tangent plane
$P^{ue}$ to extended unstable manifold $W^{ue}(O)$ has equation
$x_{2}=x_{1}\sqrt{\mu_{2}}$. The homoclinic tangency will be simple if the
following two conditions hold: tangent vector $l^{u}$ to $W^{u}(O)$ is not
parallel to $l^{ss}$ and surface $T_{1}(P^{ue})$ intersect $W^{s}(O)$
transversely. The latter will be fulfilled if there is a tangent vector to
$P^{ue}$, whose image under $T_{1}$ has a non-zero $y$–component.

Tangent to $W^{u}(O)$ vector $l^{u}$ has coordinates $(0,0,1)^{\top}$. Its
image under $T_{1}$ is obtained by means of the differential of (2.12)
evaluated at $M^{-}$. It gives vector
$DT_{1}(M^{-})l^{u}=(b_{1},b_{2},0)^{\top}$. This vector will be parallel to
$l^{ss}$ (orbit flip) and the tangency will be non-simple if
$b_{1}\sqrt{\mu_{2}}-b_{2}=0$. Thus, condition $b_{2}\neq 0$ guarantees that
orbit flip does not happen for sufficiently small $\mu_{2}$. Now take any
tangent to $P^{ue}$ vector $(x_{1},x_{1}\sqrt{\mu_{2}},y)$ with non-zero
$x_{1}$ (recall that the image of vector $(0,0,y)$ lies in $W^{s}$ at $\mu=0$)
and apply $DT_{1}(M^{-})$ to it. The $y$–component of the resulting vector
will be $(c_{1}+c_{2}\sqrt{\mu_{2}})x_{1}$. If this expression is zero for
some parameter value, then surfaces $T_{1}P^{ue}$ is tangent to $W^{s}(O)$,
and the tangency is also non-simple (inclination flip). This additional
degeneracy will not happen for sufficiently small $\mu_{2}$ if $c_{1}\neq 0$.
Thus we have the following non-degeneracy conditions:

$b_{2}(0)\neq 0,\;\;c_{1}(0)\neq 0.$ (2.14)

Thus, the global map $T_{1}$ in Case I is given by equation (2.12) with non-
degeneracy conditions (2.13) and (2.14).

Now consider the heteroclinic Case II. Let for $\mu=0$ the previously selected
heteroclinic points have the following coordinates: $M_{1}^{-}(0,0,y^{-})\in
W^{u}_{loc}(O_{1})$, $M_{1}^{+}(x_{1}^{+},x_{2}^{+},0)\in W^{s}_{loc}(O_{1})$,
$M_{2}^{-}(0,0,v^{-})\in W^{u}_{loc}(O_{2})$,
$M_{2}^{+}(u_{1}^{+},u_{2}^{+},0)\in W^{s}_{loc}(O_{2})$. The heteroclinic
cycle is composed of a fixed point $O_{1}$ that changes from saddle to saddle-
focus when $\mu_{2}$ changes sign, and a saddle point $O_{2}$.

In the Case IIa heteroclinic orbit $\Gamma_{12}$ is transversal and
heteroclinic orbit $\Gamma_{21}$ consists of quadratic tangencies. Then, the
global map $T_{12}$ can be written for all small $\mu$ as a Taylor expansion
near point $M_{1}^{-}$:

$\begin{array}[]{l}u_{10}-u_{1}^{+}=a_{11}x_{1k}+a_{12}x_{2k}+b_{1}(y_{k}-y^{-})+O(\|x_{k}\|^{2}+\|x_{k}\|\cdot|y_{k}-y^{-}|+(y_{k}-y^{-})^{2}),\\\
u_{20}-u_{2}^{+}=a_{21}x_{1k}+a_{22}x_{2k}+b_{2}(y_{k}-y^{-})+O(\|x_{k}\|^{2}+\|x_{k}\|\cdot|y_{k}-y^{-}|+(y_{k}-y^{-})^{2}),\\\
v_{0}=c_{1}x_{1k}+c_{2}x_{2k}+d_{1}(y_{k}-y^{-})+O(\|x_{k}\|^{2}+\|x_{k}\|\cdot|y_{k}-y^{-}|+(y_{k}-y^{-})^{2}).\end{array}$
(2.15)

The global map $T_{21}$ is written as a Taylor expansion near point
$M_{2}^{-}$:

$\begin{array}[]{l}\bar{x}_{10}-x_{1}^{+}=a_{31}u_{1j}+a_{32}u_{2j}+b_{3}(v_{j}-v^{-})+O(\|u_{j}\|^{2}+\|u_{j}\|\cdot|v_{j}-v^{-}|+(v_{j}-v^{-})^{2}),\\\
\bar{x}_{20}-x_{2}^{+}=a_{41}u_{1j}+a_{42}u_{2j}+b_{4}(v_{j}-v^{-})+O(\|u_{j}\|^{2}+\|u_{j}\|\cdot|v_{j}-v^{-}|+(v_{j}-v^{-})^{2}),\\\
\bar{y}_{0}=\mu_{1}+c_{3}u_{1j}+c_{4}u_{2j}+d_{2}(v_{j}-v^{-})^{2}+O(\|u_{j}\|^{2}+\|u_{j}\|\cdot|v_{j}-v^{-}|+(v_{j}-v^{-})^{3}).\end{array}$
(2.16)

The non-degeneracy conditions on Jacobians have the form:

$J_{12}=\mbox{det}\;\begin{pmatrix}a_{11}&a_{12}&b_{1}\\\
a_{21}&a_{22}&b_{2}\\\ c_{1}&c_{2}&d_{1}\end{pmatrix}\;\neq 0,$ (2.17)
$J_{21}=\mbox{det}\;\begin{pmatrix}a_{31}&a_{32}&b_{3}\\\
a_{41}&a_{42}&b_{4}\\\ c_{3}&c_{4}&0\end{pmatrix}\;\neq 0$ (2.18)

and also

$d_{1}\neq 0,\;d_{2}\neq 0.$ (2.19)

Other possible degeneracies that should be prevented, are related to the
existence of non-simple heteroclinic orbits. For quadratic heteroclinic
tangencies the condition to be non-simple is defined in a similar way as for
homoclinic ones – we consider the extended unstable manifold $W^{ue}(O_{2})$
and its image under $T_{21}$. If this image is transverse to the leaf
$F^{ss}(M_{1}^{+})$, the tangency is called simple. Similarly, we obtain two
possibilities of non-simple tangencies – the orbit flip and the inclination
flip. For transversal heteroclinic orbits, it is also possible to have an
orbit flip, when image $T_{12}W^{ue}(O_{1})$ is tangent to the leaf
$F^{ss}(M_{2}^{+})$. Such bifurcation was studied first in [23].

For the orbit $\Gamma_{12}$, the extended unstable manifold $W^{ue}(O_{1})$ is
tangent to $x_{1}$ direction as $\mu_{2}\to 0$, and the leaf of invariant
foliation $F^{ss}$ at point $M^{+}_{2}$ is $\\{v=0,\;u_{1}=u_{1}^{+}\\}$, the
tangent vector to this leaf is $l_{ss}=(0,1,0)^{\top}$. The image
$T_{12}P^{ue}(M_{1}^{-})$ has two linearly independent tangent vectors
$l_{1}=(a_{11},a_{21},c_{1})^{\top}$ and $l_{2}=(b_{1},b_{2},d_{1})^{\top}$.
The heteroclinic intersection will be simple if vectors $l_{1}$, $l_{2}$ and
$l_{ss}$ do not lie in the same plane, that is:

$A_{11}(0)=\left.\left(a_{11}-\frac{b_{1}c_{1}}{d_{1}}\right)\right|_{\mu=0}\neq
0.$ (2.20)

For the orbit $\Gamma_{21}$ the extended unstable manifold $W^{ue}(O_{2})$ is
tangent to $u_{1}$ direction, and the leaves of invariant foliation $F^{ss}$
in $U_{1}$ tend to be parallel to $x_{1}$ as $\mu_{2}\to 0$, the tangent
vector to this leaf is $l_{ss}=(1,0,0)^{\top}$. The image
$T_{21}P^{ue}(M_{2}^{-})$ has two linearly independent tangent vectors
$l_{1}=(a_{31},a_{41},c_{3})^{\top}$ and $l_{2}=(b_{3},b_{4},0)^{\top}$. The
heteroclinic intersection will be simple if vectors $l_{1}$, $l_{2}$ and
$l_{ss}$ do not lie in the same plane, that is:

$b_{4}(0)c_{3}(0)\neq 0.$ (2.21)

To summarize, in the Case IIa the global maps are given by formulas (2.15) and
(2.16) with non-degeneracy conditions (2.17)–(2.21).

Analogously, we construct the global maps in the Case IIb, when orbit
$\Gamma_{12}$ consists of quadratic tangencies and $\Gamma_{21}$ is
transversal. The global maps $T_{12}$ and $T_{21}$ have the following form:

$\begin{array}[]{l}u_{10}-u_{1}^{+}=a_{11}x_{1k}+a_{12}x_{2k}+b_{1}(y_{k}-y^{-})+O(\|x_{k}\|^{2}+\|x_{k}\|\cdot|y_{k}-y^{-}|+(y_{k}-y^{-})^{2}),\\\
u_{20}-u_{2}^{+}=a_{21}x_{1k}+a_{22}x_{2k}+b_{2}(y_{k}-y^{-})+O(\|x_{k}\|^{2}+\|x_{k}\|\cdot|y_{k}-y^{-}|+(y_{k}-y^{-})^{2}),\\\
v_{0}=\mu_{1}+c_{1}x_{1k}+c_{2}x_{2k}+d_{1}(y_{k}-y^{-})^{2}+O(\|x_{k}\|^{2}+\|x_{k}\|\cdot|y_{k}-y^{-}|+(y_{k}-y^{-})^{3}).\end{array}$
(2.22)

and

$\begin{array}[]{l}\bar{x}_{10}-x_{1}^{+}=a_{31}u_{1j}+a_{32}u_{2j}+b_{3}(v_{j}-v^{-})+O(\|u_{j}\|^{2}+\|u_{j}\|\cdot|v_{j}-v^{-}|+(v_{j}-v^{-})^{2}),\\\
\bar{x}_{20}-x_{2}^{+}=a_{41}u_{1j}+a_{42}u_{2j}+b_{4}(v_{j}-v^{-})+O(\|u_{j}\|^{2}+\|u_{j}\|\cdot|v_{j}-v^{-}|+(v_{j}-v^{-})^{2}),\\\
\bar{y}_{0}=c_{3}u_{1j}+c_{4}u_{2j}+d_{2}(v_{j}-v^{-})+O(\|u_{j}\|^{2}+\|u_{j}\|\cdot|v_{j}-v^{-}|+(v_{j}-v^{-})^{2}),\end{array}$
(2.23)

and the non-degeneracy conditions are the non-zero Jacobians:

$J_{12}=\mbox{det}\;\begin{pmatrix}a_{11}&a_{12}&b_{1}\\\
a_{21}&a_{22}&b_{2}\\\ c_{1}&c_{2}&0\end{pmatrix}\;\neq 0,$ (2.24)

and

$J_{21}=\mbox{det}\;\begin{pmatrix}a_{31}&a_{32}&b_{3}\\\
a_{41}&a_{42}&b_{4}\\\ c_{3}&c_{4}&d_{2}\end{pmatrix}\;\neq 0$ (2.25)

the conditions of quadratic tangency and transverse orbits:

$d_{1}\neq 0,\;d_{2}\neq 0,$ (2.26)

and absence of non-simple orbits:

$b_{1}(0)c_{1}(0)\neq 0.$ (2.27)

and

$A_{41}(0)=\left.\left(a_{41}-\frac{b_{4}c_{3}}{d_{2}}\right)\right|_{\mu=0}\neq
0.$ (2.28)

######  Lemma 1 (Rescaling lemma)

Let $f_{\mu_{1},\mu_{2},\mu_{3}}$ be the family under consideration. Then, in
the parameter space $(\mu_{1},\mu_{2},\mu_{3})$ in Case I there exist
infinitely many regions $\Delta_{k}$ accumulating to the origin as
$k\to\infty$, and in Case II there exist infinitely many regions $\Delta_{kj}$
accumulating to the origin as $k,j\to\infty$, such that the first return map
in appropriate rescaled coordinates and parameters is asymptotically
$C^{r-1}$-close to the map (0.1). with parameters

1) In Case I:

$\begin{array}[]{l}M_{1}=-d^{2}\gamma^{2k}(\mu_{1}+\ldots)\\\
M_{2}=L(\mu)b_{2}\lambda^{k}\gamma^{k}\sin(k\phi+\psi)+\ldots\\\
B=J_{T1}(\lambda^{2}\gamma)^{k}+\ldots\end{array}$ (2.29)

2) In Case IIa:

$\begin{array}[]{l}M_{1}=-d_{1}^{2}d_{2}\gamma_{1}^{2k}\gamma_{2}^{2j}(\mu_{1}+\ldots)\\\
M_{2}=L(\mu)b_{4}d_{1}(\lambda\gamma_{1})^{k}(\nu_{1}\gamma_{2})^{j}\sin(k\phi+\psi)+\ldots\\\
B=J_{12}J_{21}(\lambda^{2}\gamma_{1})^{k}(\nu_{1}\nu_{2}\gamma_{2})^{j}+\ldots\end{array}$
(2.30)

3) In Case IIb:

$\begin{array}[]{l}M_{1}=-d_{1}^{2}d_{2}\gamma_{1}^{2k}\gamma_{2}^{2j}(\mu_{1}+\ldots)\\\
M_{2}=L(\mu)b_{1}d_{2}A_{41}(\lambda\gamma_{1})^{k}(\nu_{1}\gamma_{2})^{j}\sin(k\phi+\psi)+\ldots\\\
B=J_{12}J_{21}(\lambda^{2}\gamma_{1})^{k}(\nu_{1}\nu_{2}\gamma_{2})^{j}+\ldots\end{array}$
(2.31)

where $L(\mu)$ is some bounded from zero function, the parameters are varied
such that $\sin(k\phi+\psi)$ is kept small, and dots stand for terms that tend
to zero as $k,j\to\infty$.

The rescaling normal form for the first return map $T_{k}$ or $T_{kj}$ is the
3D Hénon map (0.1), in both Cases $\rm I$ and $\rm II$. The statement of the
main Theorem 1 immediately follows from this lemma: indeed, in the Hénon map
(0.1) there exist parameter domains $\Delta^{+}$ and $\Delta^{-}$ (for $B>0$
and $B<0$ respectively), at which the map possesses a discrete Lorenz
attractor. Then, in the Case I for every sufficiently large $k$ they
correspond to subsets $\delta_{k}$ that converge to the origin in the space of
original parameters $(\mu_{1},\mu_{2},\mu_{3})$ as $k\to\infty$. In the
heteroclinic Case II, for every sufficiently large $k,j$ such that the product
of Jacobians of local maps
$(\lambda^{2}\gamma_{1})^{k}(\nu_{1}\nu_{2}\gamma_{2})^{j}$ is kept finite,
domains $\Delta^{\pm}$ correspond to subsets $\delta_{kj}$ that also converge
to the origin as $k,j\to\infty$.

##  3 Proof of the rescaling Lemma.

###  3.1 Homoclinic Case I

Using formulas (2.11), (2.12), we obtain the following expression for the
first return map $T_{k}\equiv
T_{1}T_{0}^{k}:\sigma_{0}^{k}\subset\Pi^{+}\to\Pi^{+}$

$\begin{array}[]{l}\bar{x}_{1}-x_{1}^{+}=a_{11}\lambda^{k}(x_{1}C_{k}(\phi)+x_{2}S_{k}(\phi))+a_{11}\hat{\lambda}^{k}\xi_{1k}^{1}(x,y,\mu)+\\\
\qquad\qquad+a_{12}\lambda^{k}(x_{1}\mu_{2}S_{k}(\phi)+x_{2}C_{k}(\phi))+a_{12}\hat{\lambda}^{k}\xi_{1k}^{2}(x,y,\mu)+\\\
\qquad\qquad+b_{1}(y-y^{-})+O(\lambda^{2k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y-y^{-}|+(y-y^{-})^{2}),\\\
\bar{x}_{2}-x_{2}^{+}=a_{21}\lambda^{k}(x_{1}C_{k}(\phi)+x_{2}S_{k}(\phi))+a_{21}\hat{\lambda}^{k}\xi_{1k}^{1}(x,y,\mu)+\\\
\qquad\qquad+a_{22}\lambda^{k}(x_{1}\mu_{2}S_{k}(\phi)+x_{2}C_{k}(\phi))+a_{22}\hat{\lambda}^{k}\xi_{1k}^{2}(x,y,\mu)+\\\
\qquad\qquad+b_{2}(y-y^{-})+O(\lambda^{2k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y-y^{-}|+(y-y^{-})^{2}),\\\
\gamma^{-k}(\bar{y}+\left(\hat{\gamma}/\gamma\right)^{-k}\xi_{2k}(\bar{x},\bar{y},\mu))=\mu_{1}+c_{1}\lambda^{k}(x_{1}C_{k}(\phi)+x_{2}S_{k}(\phi))+\\\
\qquad\qquad+c_{1}\hat{\lambda}^{k}\xi_{1k}^{1}(x,y,\mu))+c_{2}\lambda^{k}(x_{1}\mu_{2}S_{k}(\phi)+x_{2}C_{k}(\phi))+\\\
\qquad\qquad+c_{2}\hat{\lambda}^{k}\xi_{1k}^{2}(x,y,\mu)+d(y-y^{-})^{2}+\\\
\qquad\qquad+O(\lambda^{2k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y-y^{-}|+(y-y^{-})^{3}),\\\
\end{array}$ (3.1)

Make a coordinate shift $x_{1new}=x_{1}-x_{1}^{+}+\psi_{k}^{1}$,
$x_{2new}=x_{2}-x_{2}^{+}+\psi_{k}^{2}$, $y_{new}=y-y^{-}+\psi_{k}^{3}$, where
$\psi_{k}^{1,2,3}=O(\lambda^{k})$. With that, the nonlinearity function in the
left part of the third equation of (3.1) can be expressed as a Taylor
expansion
$\xi_{2k}(\bar{x}_{1}+x_{1}^{+}-\psi_{k}^{1},\bar{x}_{2}+x_{2}^{+}-\psi_{k}^{2},\bar{y}+y^{-}-\psi_{k}^{3},\mu))=\xi_{k}^{0}+\xi_{k}^{1}\bar{y}+\xi_{k}^{2}(\bar{x},\bar{y})+\xi_{k}^{3}(\bar{y})$,
where coefficients $\xi_{k}^{0}$, $\xi_{k}^{1}$ are uniformly bounded in $k$
for all small $\mu$ and $\xi_{k}^{2}(\bar{x},\bar{y})=O(\bar{x})$,
$\xi_{k}^{3}(\bar{y})=O(\bar{y}^{2})$. We select constants $\psi_{k}^{1}$,
$\psi_{k}^{2}$, $\psi_{k}^{3}$ in such a way that all constant terms in the
first two equations and the linear in $y$ term in the last equation of (3.1)
vanish. Next, we substitute the right hand sides of first two equations to
term $O(\bar{x})$ in the last one. All coefficients then get additions of
order $\hat{\gamma}^{-k}$ including the appearance of one linear in $y$ in the
third equation. The latter can be again made zero by an additional shift of
$(x_{1},x_{2},y)$ coordinates by the value of order $\hat{\gamma}^{-k}$. The
system is rewritten as:

$\begin{array}[]{l}\bar{x}_{1}=\lambda^{k}(a_{11}C_{k}(\phi)+a_{12}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(a_{11}S_{k}(\phi)+a_{12}C_{k}(\phi))x_{2}+\\\
\qquad\qquad+b_{1}y+\hat{\lambda}^{k}O(\|x\|^{2})+\lambda^{k}O(\|x\|\cdot|y|)+O(y^{2}),\\\
\bar{x}_{2}=\lambda^{k}(a_{21}C_{k}(\phi)+a_{22}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(a_{21}S_{k}(\phi)+a_{22}C_{k}(\phi))x_{2}+\\\
\qquad\qquad+b_{2}y+\hat{\lambda}^{k}O(\|x\|^{2})+\lambda^{k}O(\|x\|\cdot|y|)+O(y^{2}),\\\
\gamma^{-k}(1+q_{k})\bar{y}+\hat{\gamma}^{-k}O(\bar{y}^{2})=M^{1}+\lambda^{k}(c_{1}C_{k}(\phi)+c_{2}\mu_{2}S_{k}(\phi))x_{1}+\\\
\qquad\qquad+\lambda^{k}(c_{1}S_{k}(\phi)+c_{2}C_{k}(\phi))x_{2}+dy^{2}+\\\
\qquad\qquad+\hat{\lambda}^{k}O(\|x\|^{2})+\lambda^{k}O(\|x\|\cdot|y|)+O(y^{3}),\\\
\end{array}$ (3.2)

where $q_{k}=O\left(\left(\hat{\gamma}/\gamma\right)^{-k}\right)$ and the
following expression is valid for $M^{1}$:

$M^{1}=\mu_{1}+\lambda^{k}c_{1}S_{k}(\phi)x_{2}^{+}-\gamma^{-k}y^{-}+O(\hat{\gamma}^{-k}+\lambda^{k}).$
(3.3)

Next, as $b_{2}\neq 0$, we make the following change of coordinates in order
to eliminate terms that depend on $y$ only in the first equation:
$\displaystyle x_{1new}=x_{1}-\frac{b_{1}}{b_{2}}x_{2}+O(\|x\|^{2})$. System
(3.2) takes the following form:

$\begin{array}[]{l}\bar{x}_{1}=\lambda^{k}A_{11k}(\phi)x_{1}+\lambda^{k}A_{12k}(\phi)x_{2}+\hat{\lambda}^{k}O(\|x\|^{2})+\lambda^{k}O(\|x\|\cdot|y|),\\\
\bar{x}_{2}=\lambda^{k}A_{21k}(\phi)x_{1}+\lambda^{k}A_{22k}(\phi)x_{2}+b_{2}y+\hat{\lambda}^{k}O(\|x\|^{2})+\\\
\qquad\qquad+\lambda^{k}O(\|x\|\cdot|y|)+O(y^{2}),\\\
\gamma^{-k}(1+q_{k})\bar{y}+\hat{\gamma}^{-k}O(\bar{y}^{2})=M^{1}+\lambda^{k}C_{1k}(\phi)x_{1}+\lambda^{k}C_{2k}(\phi)x_{2}+\\\
\qquad\qquad+dy^{2}+\hat{\lambda}^{k}O(\|x\|^{2})+\lambda^{k}O(\|x\|\cdot|y|)+O(y^{3}),\\\
\end{array}$ (3.4)

where

$\begin{array}[]{l}\displaystyle
A_{11k}(\phi)=a_{11}C_{k}(\phi)+a_{12}\mu_{2}S_{k}(\phi)-\frac{b_{1}}{b_{2}}(a_{21}C_{k}(\phi)+a_{22}\mu_{2}S_{k}(\phi))\\\
\displaystyle
A_{12k}(\phi)=a_{11}\left(\frac{b_{1}}{b_{2}}C_{k}(\phi)+S_{k}(\phi)\right)+a_{12}\left(\frac{b_{1}}{b_{2}}\mu_{2}S_{k}(\phi)+C_{k}(\phi)\right)-\\\
\displaystyle\qquad-
a_{21}\frac{b_{1}}{b_{2}}\left(\frac{b_{1}}{b_{2}}C_{k}(\phi)+S_{k}(\phi)\right)-a_{22}\frac{b_{1}}{b_{2}}\left(\frac{b_{1}}{b_{2}}\mu_{2}S_{k}(\phi)+C_{k}(\phi)\right)\\\
\displaystyle A_{21k}(\phi)=a_{21}C_{k}(\phi)+a_{22}\mu_{2}S_{k}(\phi)\\\
\displaystyle
A_{22k}(\phi)=a_{21}\left(\frac{b_{1}}{b_{2}}C_{k}(\phi)+S_{k}(\phi)\right)+a_{22}\left(\frac{b_{1}}{b_{2}}\mu_{2}S_{k}(\phi)+C_{k}(\phi)\right)\\\
\displaystyle C_{1k}(\phi)=c_{1}C_{k}(\phi)+c_{2}\mu_{2}S_{k}(\phi)\\\
\displaystyle
C_{2k}(\phi)=c_{1}\left(\frac{b_{1}}{b_{2}}C_{k}(\phi)+S_{k}(\phi)\right)+c_{2}\left(\frac{b_{1}}{b_{2}}\mu_{2}S_{k}(\phi)+C_{k}(\phi)\right)\end{array}$
(3.5)

Now consider $C_{2k}(\phi)$ for $\mu_{2}<0$. It can be represented in the form
$C_{2k}(\phi)=L_{1}(\mu)\cos k\phi+L_{2}(\mu)\sin
k\phi=L(\mu)\sin(k\phi+\psi)$. We will vary $\mu_{2}$ near those values, for
which $C_{2k}(\phi)$ vanishes (such values are dense on any interval from
$(0,\pi)$), we can make this coefficient asymptotically small for
$k\to\infty$. Note that for such values of $\mu_{2}$ expressions
$\displaystyle\frac{b_{1}}{b_{2}}C_{k}(\phi)+S_{k}(\phi)$ and
$\displaystyle\frac{b_{1}}{b_{2}}\mu_{2}S_{k}(\phi)+C_{k}(\phi)$ are non-
zeros, because otherwise coefficients $A_{12k}(\phi)$, $A_{21k}(\phi)$ and
$C_{2k}(\phi)$ will vanish simultaneously, and the Jacobian of the first
return map $T_{k}$ will become zero, which is impossible because it is a
diffeomorphism. Then, it is easy to check that

$\displaystyle
A_{12k}(\phi)=\frac{J_{T1}}{b_{2}c_{1}}\left(\frac{b_{1}}{b_{2}}\mu_{2}S_{k}(\phi)+C_{k}(\phi)\right)+O(C_{2k}(\phi))\neq
0.$ (3.6)

Now we make the following scaling of coordinates:
$x_{1}=\alpha_{1}X_{1},\;x_{2}=\alpha_{2}X_{2},\;y=\beta Y$, with

$\displaystyle\alpha_{1}=-\frac{A_{12k}b_{2}\gamma^{-k}\lambda^{k}}{d}(1+q_{k}),\;\;\alpha_{2}=-\frac{b_{2}\gamma^{-k}}{d}(1+q_{k}),\;\;\beta=-\frac{\gamma^{-k}}{d}(1+q_{k}).$
(3.7)

The system becomes the following:

$\begin{array}[]{l}\bar{X}_{1}=X_{2}+O(\lambda^{k})\\\
\bar{X}_{2}=Y+O(k\lambda^{k})\\\
\bar{Y}=M_{1}+J_{k}X_{1}+M_{2}X_{2}-Y^{2}+O(\lambda^{k}),\end{array}$ (3.8)

where $\displaystyle M_{1}=-d\gamma^{2k}M^{1}$,
$M_{2}=\lambda^{k}\gamma^{k}b_{2}C_{2k}(\phi)$ and
$J_{k}=(\lambda^{2}\gamma)^{k}J_{1}+O(k\lambda^{k})$ coincides with the
Jacobian of the first return map $T_{k}$ up to asymptotically small terms. For
arbitrary large $k$ multiplier $\gamma^{2k}$ unboundedly grows, so small
variations of $\mu_{1}$ will provide arbitrary finite values for $M_{1}$.
Then, as $|\lambda^{2}\gamma|=1+\mu_{3}$ at the bifurcation moment, the
product $\lambda^{k}\gamma^{k}\sim\lambda^{-k}$ grows unboundedly as
$k\to+\infty$. Varying parameter $\mu_{2}$ such that coefficient
$C_{2k}(\phi)$ is kept small, we can make parameter $M_{2}$ to take arbitrary
finite values. The third parameter $J_{k}$ can take arbitrary finite positive
values when $\mu_{3}$ gets small perturbations. System (3.8) is asymptotically
close to (0.1) as $k,j\to\infty$.

###### Remark 2

When $\mu_{2}>0$, scaling (3.7) does not bring map $T_{k}$ to the 3D Henon
form because in that case coefficient $C_{2k}(\phi)$ becomes growing in $k$
for all small $\mu_{2}$, so that parameter $M_{2}$ will be asymptotically
large. In order to understand the dynamics in this case, in map (3.4) we
perform an additional change of coordinates, which is valid because
$C_{2k}(\phi)\neq 0$:

$\displaystyle
x_{2new}=x_{2}+\frac{C_{1k}(\phi)}{C_{2k}(\phi)}x_{1}+O\left((\hat{\lambda}/\lambda)^{k}\right)O(\|x\|^{2})$
(3.9)

in order to eliminate in the last equation the linear in $x_{1}$ term together
with higher-order terms depending on $x$ only. The system rewrites as:

$\begin{array}[]{l}\displaystyle\bar{x}_{1}=\lambda^{k}\frac{J_{1}}{b_{2}C_{2k}(\phi)}x_{1}+\lambda^{k}A_{12k}(\phi)x_{2}+\hat{\lambda}^{k}O(\|x\|^{2})+\lambda^{k}O(\|x\|\cdot|y|),\\\
\bar{x}_{2}=O(k\lambda^{k})O(\|x\|)+b_{2}y+\hat{\lambda}^{k}O(\|x\|^{2})+\lambda^{k}O(\|x\|\cdot|y|)+O(y^{2}),\\\
(1+q_{k})\bar{y}+\hat{\gamma}^{-k}O(\bar{y}^{2})=\gamma^{k}M^{1}+\lambda^{k}\gamma^{k}C_{2k}(\phi)x_{2}+d\gamma^{k}y^{2}+\\\
\qquad\qquad+\lambda^{k}\gamma^{k}O(\|x\|\cdot|y|)+\gamma^{k}O(y^{3}).\\\
\end{array}$ (3.10)

Finally, we scale the variables as follows:
$x_{1}=\alpha_{1}X_{1},\;x_{2}=\alpha_{2}X_{2},\;y=\beta Y$, where

$\displaystyle\alpha_{1}=\alpha_{2}=-\frac{b_{2}\gamma^{-k}}{d}(1+q_{k}),\;\;\beta=-\frac{\gamma^{-k}}{d}(1+q_{k}).$
(3.11)

Map (3.10) rewrites as follows:

$\begin{array}[]{l}\bar{X}_{1}=O(k\lambda^{k})\\\
\bar{X}_{2}=Y+O(k\lambda^{k})\\\
\bar{Y}=M_{1}+M_{2}X_{2}-Y^{2}+O(\lambda^{k}),\end{array}$ (3.12)

where $\displaystyle M_{1}=-d\gamma^{2k}M^{1}$,
$M_{2}=\lambda^{k}\gamma^{k}b_{2}C_{2k}(\phi)$. From formula (3.12) it is
clear that the first return map has an invariant manifold of the form
$X_{1}=k\lambda^{k}\psi(X_{2},Y)$ on each $\sigma$ and the restriction on it
of system (3.12) is the well-known two-dimensional Henon map with an arbitrary
large Jacobian $M_{2}$ for large $k$, so it expands volumes and does not have
attractors.

###  3.2 Heteroclinic Case IIa.

The first return map is a composition of two local and two global maps,
defined above. We substitute local maps (2.11) and (2.2) to global maps (2.15)
and (2.16). Note that for simplicity we omit indices $0$, $k$ and $j$ at the
variables:

$\begin{array}[]{l}u_{1}-u_{1}^{+}=\lambda^{k}(a_{11}C_{k}(\phi)+a_{12}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(a_{11}S_{k}(\phi)+a_{12}C_{k}(\phi))x_{2}+\\\
\qquad+b_{1}(y-y^{-})+\hat{\lambda}^{k}\tilde{\xi}_{1k}(x,y,\mu)+O(\lambda^{2k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y-y^{-}|+(y-y^{-})^{2}),\\\
u_{2}-u_{2}^{+}=\lambda^{k}(a_{21}C_{k}(\phi)+a_{22}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(a_{21}S_{k}(\phi)+a_{22}C_{k}(\phi))x_{2}+\\\
\qquad+b_{2}(y-y^{-})+\hat{\lambda}^{k}\tilde{\xi}_{2k}(x,y,\mu)+O(\lambda^{2k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y-y^{-}|+(y-y^{-})^{2}),\\\
\gamma_{2}^{-j}v+\hat{\gamma}_{2}^{-j}\eta_{3j}(u,v,\mu)=\lambda^{k}(c_{1}C_{k}(\phi)+c_{2}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(c_{1}S_{k}(\phi)+c_{2}C_{k}(\phi))x_{2}+\\\
\qquad+d_{1}(y-y^{-})+\hat{\lambda}^{k}\tilde{\xi}_{3k}(x,y,\mu)+O(\lambda^{2k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y-y^{-}|+(y-y^{-})^{2}),\end{array}$
(3.13)
$\begin{array}[]{l}\bar{x}_{1}-x_{1}^{+}=a_{31}\nu_{1}^{j}u_{1}+\hat{\nu}^{j}\tilde{\eta}_{1j}(u,v,\mu)+b_{3}(v-v^{-})+O(\nu_{1}^{2j}\|u\|^{2}+\nu_{1}^{j}|u|\cdot|v-v^{-}|+(v-v^{-})^{2}),\\\
\bar{x}_{2}-x_{2}^{+}=a_{41}\nu_{1}^{j}u_{1}+\hat{\nu}^{j}\tilde{\eta}_{2j}(u,v,\mu)+b_{4}(v-v^{-})+O(\nu_{1}^{2j}\|u\|^{2}+\nu_{1}^{j}|u|\cdot|v-v^{-}|+(v-v^{-})^{2}),\\\
\gamma_{1}^{-k}\bar{y}+\hat{\gamma}_{1}^{-k}\xi_{2k}(\bar{x},\bar{y},\mu)=\mu_{1}+c_{3}\nu_{1}^{j}u_{1}+\hat{\nu}^{j}\tilde{\eta}_{3j}(u,v,\mu)+d_{2}(v-v^{-})^{2}+\\\
\qquad+O(\nu_{1}^{2j}\|u\|^{2}+\nu_{1}^{j}\|u\|\cdot|v-v^{-}|+(v-v^{-})^{3}).\end{array}$
(3.14)

Make a coordinate shift $u_{new}=u-u^{+}+\varphi_{kj}^{1}$,
$v_{new}=v-v^{-}+\varphi_{kj}^{2}$, $x_{new}=x-x^{+}+\psi_{kj}^{1}$,
$y_{new}=y-y^{-}+\psi_{kj}^{2}$, where
$\varphi_{kj}^{1},\;\psi_{kj}^{2}=O(\gamma_{2}^{-j}+k\lambda^{k})$ and
$\varphi_{kj}^{2},\;\psi_{kj}^{1}=O(\nu_{1}^{j})$. With that, the nonlinearity
functions in the left parts of the third equations of (3.13) and (3.14) can be
expressed as Taylor expansions
$\eta_{3j}(u+u^{+}+\varphi_{kj}^{1},v+v^{-}+\varphi_{kj}^{2},\mu))=\eta_{3j}^{0}+\eta_{3j}^{1}v+\eta_{3j}^{2}(u,v)+\eta_{3j}^{3}(v)$,
$\xi_{2k}(\bar{x}+x^{+}+\psi_{ij}^{1},\bar{y}+y^{-}+\psi_{ij}^{2},\mu))=\xi_{2k}^{0}+\xi_{2k}^{1}\bar{y}+\xi_{2k}^{2}(\bar{x},\bar{y})+\xi_{2k}^{3}(\bar{y})$
respectively, where coefficients $\eta_{3j}^{0}$, $\eta_{3j}^{1}$,
$\xi_{2k}^{0}$, $\xi_{2k}^{1}$ are uniformly bounded in $k$ and $j$ for all
small $\mu$ and $\eta_{3j}^{2}(u,v)=O(u)$,
$\xi_{2k}^{2}(\bar{x},\bar{y})=O(\bar{x})$, $\eta_{3j}^{3}(v)=O(v^{2})$,
$\xi_{2k}^{3}(\bar{y})=O(\bar{y}^{2})$. We select constants
$\varphi_{kj}^{1},\;\varphi_{kj}^{2},\;\psi_{kj}^{1},\;\psi_{kj}^{2}$ in such
a way that all constant terms in equations (3.13), the constant terms in the
first two equations and the linear in $v_{new}$ term in the last equation of
(3.14) vanish. In addition, we plug the expressions for $u$ coordinates from
the first two equations of (3.13) into the third one, this will cause
additions of order $\hat{\gamma}_{2}^{-j}$ to all the coefficients. The system
is rewritten as:

$\begin{array}[]{l}u_{1}=\lambda^{k}(a_{11}C_{k}(\phi)+a_{12}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(a_{11}S_{k}(\phi)+a_{12}C_{k}(\phi))x_{2}+b_{1}y+\\\
\qquad+O(\hat{\lambda}^{k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y|+y^{2}),\\\
u_{2}=\lambda^{k}(a_{21}C_{k}(\phi)+a_{22}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(a_{21}S_{k}(\phi)+a_{22}C_{k}(\phi))x_{2}+b_{2}y+\\\
\qquad+O(\hat{\lambda}^{k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y|+y^{2}),\\\
\gamma_{2}^{-j}(1+q_{kj}^{2})v+\hat{\gamma}_{2}^{-j}O(v^{2})=\lambda^{k}(c_{1}C_{k}(\phi)+c_{2}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(c_{1}S_{k}(\phi)+c_{2}C_{k}(\phi))x_{2}+\\\
\qquad+d_{1}y+O(\hat{\lambda}^{k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y|+y^{2}),\end{array}$
(3.15)
$\begin{array}[]{l}\bar{x}_{1}=a_{31}\nu_{1}^{j}u_{1}+\tilde{a}_{32}\hat{\nu}^{j}u_{2}+b_{3}v+O(\hat{\nu}^{j}\|u\|^{2}+\nu_{1}^{j}\|u\|\cdot|v|+v^{2}),\\\
\bar{x}_{2}=a_{41}\nu_{1}^{j}u_{1}+\tilde{a}_{42}\hat{\nu}^{j}u_{2}+b_{4}v+O(\hat{\nu}^{j}\|u\|^{2}+\nu_{1}^{j}\|u\|\cdot|v|+v^{2}),\\\
\gamma_{1}^{-k}(1+q_{kj}^{1})\bar{y}+\hat{\gamma}_{1}^{-k}O(\bar{x})+\hat{\gamma}_{1}^{-k}O(\bar{y}^{2})=M^{1}+c_{3}\nu_{1}^{j}u_{1}+\tilde{c}_{4}\hat{\nu}^{j}u_{2}+d_{2}v^{2}+\\\
\qquad+O(\hat{\nu}^{j}\|u\|^{2}+\nu_{1}^{j}\|u\|\cdot|v|+|v|^{3}),\end{array}$
(3.16)

where
$q_{kj}^{1}=O\left(\left(\hat{\gamma}_{1}/\gamma_{1}\right)^{-k}\right)$,
$q_{kj}^{2}=O\left(\left(\hat{\gamma}_{2}/\gamma_{2}\right)^{-j}\right)$,
coefficients marked with ‘‘tilde’’ are uniformly bounded for small $\mu$ and
the following expression is valid for $M^{1}$:

$M^{1}=\mu_{1}+\nu_{1}^{j}c_{3}u_{1}^{+}-\gamma_{1}^{-k}y^{-}+O(\hat{\gamma}_{1}^{-k}+\hat{\nu}^{j}+\gamma_{1}^{-k}\gamma_{2}^{-j}).$
(3.17)

Next, we take the right-hand side of the third equation of (3.15) divided by
the factor $(1+q_{kj}^{2})$ from the left-hand side as the new variable $y$ –
the equation becomes $\gamma_{2}^{-j}v+\hat{\gamma}_{2}^{-j}O(v^{2})=y$, and
we substitute it into the third equation of (3.16). We also define
$u_{1new}=u_{1}-\frac{b_{1}}{d_{1}}\gamma_{2}^{-j}v+\hat{\gamma}_{2}^{-j}O(v^{2})$
and
$u_{2new}=u_{2}-\frac{b_{2}}{d_{1}}\gamma_{2}^{-j}v+\hat{\gamma}_{2}^{-j}O(v^{2})$
such that we remove all terms in the equations for $u$ which depend on $y$
alone. In addition, we substitute the expressions for $\bar{x}$ to the last
equation of (3.16). These actions cause the linear in $v$ term of order
$O(\hat{\gamma}_{1}^{-k}+\nu_{1}^{j}\gamma_{2}^{-j})$ to appear in the
equation for $\bar{v}$. We will make it zero again later. Thus, we obtain

$\begin{array}[]{l}u_{1}=\lambda^{k}(A_{11}C_{k}(\phi)+A_{12}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(A_{11}S_{k}(\phi)+A_{12}C_{k}(\phi))x_{2}+\\\
\qquad+O(\hat{\lambda}^{k}\|x\|^{2}+\lambda^{k}\gamma_{2}^{-j}\|x\|\cdot|v|),\\\
u_{2}=\lambda^{k}(A_{21}C_{k}(\phi)+A_{22}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(A_{21}S_{k}(\phi)+A_{22}C_{k}(\phi))x_{2}+\\\
\qquad+O(\hat{\lambda}^{k}\|x\|^{2}+\lambda^{k}\gamma_{2}^{-j}\|x\|\cdot|v|),\\\
\bar{x}_{1}=a_{31}\nu_{1}^{j}u_{1}+\tilde{a}_{32}\hat{\nu}^{j}u_{2}+b_{3}v+O(\hat{\nu}^{j}\|u\|^{2}+\nu_{1}^{j}\|u\|\cdot|v|+v^{2}),\\\
\bar{x}_{2}=a_{41}\nu_{1}^{j}u_{1}+\tilde{a}_{42}\hat{\nu}^{j}u_{2}+b_{4}v+O(\hat{\nu}^{j}\|u\|^{2}+\nu_{1}^{j}\|u\|\cdot|v|+v^{2}),\\\
\displaystyle\frac{\gamma_{1}^{-k}\gamma_{2}^{-j}}{d_{1}}\bar{v}(1+q_{kj}^{3})+\gamma_{1}^{-k}\hat{\gamma}_{2}^{-j}O(\bar{v}^{2})=M^{1}+c_{3}\nu_{1}^{j}u_{1}+\tilde{c}_{4}\hat{\nu}^{j}u_{2}+\\\
\qquad\qquad\qquad\qquad+O(\hat{\gamma}_{1}^{-k}+\nu_{1}^{j}\gamma_{2}^{-j})v+d_{2}v^{2}+O(\hat{\nu}^{j}\|u\|^{2}+\nu_{1}^{j}\|u\|\cdot|v|+|v|^{3}),\end{array}$
(3.18)

where
$q_{ij}^{3}=O\left(\left(\hat{\gamma}_{1}/\gamma_{1}\right)^{-k}+\left(\hat{\gamma}_{2}/\gamma_{2}\right)^{-j}\right)$
and

$\begin{array}[]{c}A_{11}=a_{11}-b_{1}c_{1}/d_{1},\;A_{12}=a_{12}-b_{1}c_{2}/d_{1}\\\
A_{21}=a_{21}-b_{2}c_{1}/d_{1},\;A_{22}=a_{22}-b_{2}c_{2}/d_{1}.\end{array}$
(3.19)

We substitute $u$ as a function of $x$ and $v$ from the first two equations
into the last three ones. After this, in addition, we make a shift of $(x,v)$
coordinates by a constant of order
$O(\hat{\gamma}_{1}^{-k}+\nu_{1}^{j}\gamma_{2}^{-j})$ to nullify the linear in
$v$ term in the last equation. This gives us the following formula in
variables $(x,v)$:

$\begin{array}[]{l}\bar{x}_{1}=a_{31}\lambda^{k}\nu_{1}^{j}(A_{11}C_{k}(\phi)+A_{12}\mu_{2}S_{k}(\phi))x_{1}+a_{31}\lambda^{k}\nu_{1}^{j}(A_{11}S_{k}(\phi)+A_{12}C_{k}(\phi))x_{2}+b_{3}v+\\\
\qquad\qquad+O(\hat{\lambda}^{k}\nu_{1}^{j}+\lambda^{k}\nu_{1}^{j}\gamma_{2}^{-j})O(\|x\|^{2})+\lambda^{k}\nu_{1}^{j}O(\|x\|\cdot|v|)+O(v^{2}),\\\
\bar{x}_{2}=a_{41}\lambda^{k}\nu_{1}^{j}(A_{11}C_{k}(\phi)+A_{12}\mu_{2}S_{k}(\phi))x_{1}+a_{41}\lambda^{k}\nu_{1}^{j}(A_{11}S_{k}(\phi)+A_{12}C_{k}(\phi))x_{2}+b_{4}v+\\\
\qquad\qquad+O(\hat{\lambda}^{k}\nu_{1}^{j}+\lambda^{k}\nu_{1}^{j}\gamma_{2}^{-j})O(\|x\|^{2})+\lambda^{k}\nu_{1}^{j}O(\|x\|\cdot|v|)+O(v^{2}),\\\
\displaystyle\frac{\gamma_{1}^{-k}\gamma_{2}^{-j}}{d_{1}}\bar{v}(1+q_{kj}^{3})+\gamma_{1}^{-k}\hat{\gamma}_{2}^{-j}O(\bar{v}^{2})=M^{1}+c_{3}\lambda^{k}\nu_{1}^{j}(A_{11}C_{k}(\phi)+A_{12}\mu_{2}S_{k}(\phi))x_{1}+\\\
\qquad\qquad+c_{3}\lambda^{k}\nu_{1}^{j}(A_{11}S_{k}(\phi)+A_{12}C_{k}(\phi))x_{2}+d_{2}v^{2}+\\\
\qquad\qquad+O(\hat{\lambda}^{k}\nu_{1}^{j}+\lambda^{k}\nu_{1}^{j}\gamma_{2}^{-j})O(\|x\|^{2})+\lambda^{k}\nu_{1}^{j}O(\|x\|\cdot|v|)+O(v^{3}),\end{array}$
(3.20)

In order to eliminate the terms that depend on $v$ only in the first equation,
we make the following coordinate change: $x_{1new}=x_{1}-b_{3}/b_{4}x_{2}$.
The first return map takes the form:

$\begin{array}[]{l}\bar{x}_{1}=\tilde{A}_{11}(\phi)\lambda^{k}\nu_{1}^{j}x_{1}+\tilde{A}_{12}(\phi)\lambda^{k}\nu_{1}^{j}x_{2}+O(\hat{\lambda}^{k}\nu_{1}^{j}+\lambda^{k}\nu_{1}^{j}\gamma_{2}^{-j})O(\|x\|^{2})+\lambda^{k}\nu_{1}^{j}O(\|x\|\cdot|v|)\\\
\bar{x}_{2}=\tilde{A}_{21}(\phi)\lambda^{k}\nu_{1}^{j}x_{1}+\tilde{A}_{22}(\phi)\lambda^{k}\nu_{1}^{j}x_{2}+b_{4}v+O(\hat{\lambda}^{k}\nu_{1}^{j}+\lambda^{k}\nu_{1}^{j}\gamma_{2}^{-j})O(\|x\|^{2})+\lambda^{k}\nu_{1}^{j}O(\|x\|\cdot|v|)+O(v^{2})\\\
\displaystyle\frac{\gamma_{1}^{-k}\gamma_{2}^{-j}}{d_{1}}\bar{v}(1+q_{kj}^{3})+\gamma_{1}^{-k}\hat{\gamma}_{2}^{-j}O(\bar{v}^{2})=M^{1}+\tilde{C}_{1}(\phi)\lambda^{k}\nu_{1}^{j}x_{1}+\tilde{C}_{2}\lambda^{k}\nu_{1}^{j}x_{2}+d_{2}v^{2}+\\\
\qquad\qquad+O(\hat{\lambda}^{k}\nu_{1}^{j}+\lambda^{k}\nu_{1}^{j}\gamma_{2}^{-j})O(\|x\|^{2})+\lambda^{k}\nu_{1}^{j}O(\|x\|\cdot|v|)+O(v^{3}),\par\end{array}$
(3.21)

where

$\begin{array}[]{l}\displaystyle\tilde{A}_{11}(\phi)=\left(a_{31}-\frac{b_{3}}{b_{4}}a_{41}\right)(A_{11}C_{k}(\phi)+A_{12}\mu_{2}S_{k}(\phi))+O\left(\left(\hat{\lambda}/\lambda\right)^{k}+\left(\hat{\nu}/\nu_{1}\right)^{j}\right)\\\
\displaystyle\tilde{A}_{12}(\phi)=\frac{b_{3}}{b_{4}}\left(a_{31}-\frac{b_{3}}{b_{4}}a_{41}\right)(A_{11}C_{k}(\phi)+A_{12}\mu_{2}S_{k}(\phi))+\\\
\displaystyle\qquad+\left(a_{31}-\frac{b_{3}}{b_{4}}a_{41}\right)(A_{11}S_{k}(\phi)+A_{12}C_{k}(\phi))+O\left(\left(\hat{\lambda}/\lambda\right)^{k}+\left(\hat{\nu}/\nu_{1}\right)^{j}\right)\\\
\displaystyle\tilde{A}_{21}(\phi)=a_{41}(A_{11}C_{k}(\phi)+A_{12}\mu_{2}S_{k}(\phi))+O\left(\left(\hat{\lambda}/\lambda\right)^{k}+\left(\hat{\nu}/\nu_{1}\right)^{j}\right)\\\
\displaystyle\tilde{A}_{22}(\phi)=a_{41}\left(A_{11}C_{k}(\phi)+A_{12}\mu_{2}S_{k}(\phi)+\frac{b_{3}}{b_{4}}\left(A_{11}S_{k}(\phi)+A_{12}C_{k}(\phi)\right)\right)+\\\
\qquad+O\left(\left(\hat{\lambda}/\lambda\right)^{k}+\left(\hat{\nu}/\nu_{1}\right)^{j}\right)\\\
\displaystyle\tilde{C}_{1}(\phi)=c_{3}(A_{11}C_{k}(\phi)+A_{12}\mu_{2}S_{k}(\phi))+O\left(\left(\hat{\lambda}/\lambda\right)^{k}+\left(\hat{\nu}/\nu_{1}\right)^{j}\right)\\\
\displaystyle\tilde{C}_{2}(\phi)=c_{3}\left(A_{11}C_{k}(\phi)+A_{12}\mu_{2}S_{k}(\phi)+\frac{b_{3}}{b_{4}}\left(A_{11}S_{k}(\phi)+A_{12}C_{k}(\phi)\right)\right)+\\\
\qquad+O\left(\left(\hat{\lambda}/\lambda\right)^{k}+\left(\hat{\nu}/\nu_{1}\right)^{j}\right).\end{array}$
(3.22)

Take $\mu_{2}<0$. Coefficient $\tilde{C}_{2}(\phi)$ can be then represented as
$\tilde{C}_{2}(\phi)=L_{1}(\mu)\cos k\phi+L_{2}(\mu)\sin
k\phi=L(\mu)\sin(k\phi+\psi)$. We will vary $\mu_{2}$ near those values, for
which $\tilde{C}_{2}(\phi)$ vanishes, so that it becomes asymptotically small
for $k,j\to\infty$. Note that $\tilde{C}_{1}(\phi)$ and $\tilde{A}_{12}(\phi)$
will remain bounded from zero at these parameter values, because otherwise the
first return map will be not a diffeomorphism. Finally, we rescale the
coordinates as follows:

$\begin{array}[]{c}\displaystyle
v=-\frac{\gamma_{1}^{-k}\gamma_{2}^{-j}}{d_{1}d_{2}}(1+q_{kj}^{3})\;Y\;,\;x_{1}=-\frac{b_{4}\tilde{A}_{12}\lambda^{k}\nu_{1}^{j}\gamma_{1}^{-k}\gamma_{2}^{-j}}{d_{1}d_{2}}(1+q_{kj}^{3})\;X_{1}\;,\;x_{2}=-\frac{b_{4}\gamma_{1}^{-k}\gamma_{2}^{-j}}{d_{1}d_{2}}(1+q_{kj}^{3})\;X_{2}\;.\end{array}$
(3.23)

.

The resulting map is asymptotically close to three-dimensional Henon map (0.1)
as $k,j\to\infty$, and formulas (2.30) are valid for the rescaled parameters.
For small variations of $\mu_{1}$ parameter $M_{1}$ takes arbitrary finite
values. We vary $\mu_{2}<0$ near a zero of function $\sin(k\phi+\psi)$, and
being multiplied by
$(\lambda\gamma_{1})^{k}(\nu_{1}\gamma_{2})^{j}\sim\lambda^{-k}\nu_{2}^{-j}$
we obtain that parameter $M_{2}$ also takes arbitrary finite values. We also
select arbitrary large $k$ and $j$ in such a way that the Jacobian of the
first return map
$B=J_{12}J_{21}(\lambda^{2}\gamma_{1})^{k}(\nu_{1}\nu_{2}\gamma_{2})^{j}$
stays finite. Then, by small variations of $\mu_{3}$ it will also take
arbitrary finite values, positive or negative.

###  3.3 Heteroclinic Case IIb.

Now we construct the first return map substituting local maps (2.11) and (2.2)
to global maps (2.22) and (2.23):

$\begin{array}[]{l}u_{1}-u_{1}^{+}=\lambda^{k}(a_{11}C_{k}(\phi)+a_{12}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(a_{11}S_{k}(\phi)+a_{12}C_{k}(\phi))x_{2}+\\\
\qquad+b_{1}(y-y^{-})+\hat{\lambda}^{k}\tilde{\xi}_{1k}(x,y,\mu)+O(\lambda^{2k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y-y^{-}|+(y-y^{-})^{2}),\\\
u_{2}-u_{2}^{+}=\lambda^{k}(a_{21}C_{k}(\phi)+a_{22}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(a_{21}S_{k}(\phi)+a_{22}C_{k}(\phi))x_{2}+\\\
\qquad+b_{2}(y-y^{-})+\hat{\lambda}^{k}\tilde{\xi}_{2k}(x,y,\mu)+O(\lambda^{2k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y-y^{-}|+(y-y^{-})^{2}),\\\
\gamma_{2}^{-j}v+\hat{\gamma}_{2}^{-j}\eta_{3j}(u,v,\mu)=\mu_{1}+\lambda^{k}(c_{1}C_{k}(\phi)+c_{2}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(c_{1}S_{k}(\phi)+c_{2}C_{k}(\phi))x_{2}+\\\
\qquad+d_{1}(y-y^{-})^{2}+\hat{\lambda}^{k}\tilde{\xi}_{3k}(x,y,\mu)+O(\lambda^{2k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y-y^{-}|+(y-y^{-})^{3}),\end{array}$
(3.24)
$\begin{array}[]{l}\bar{x}_{1}-x_{1}^{+}=a_{31}\nu_{1}^{j}u_{1}+\hat{\nu}^{j}\tilde{\eta}_{1j}(u,v,\mu)+b_{3}(v-v^{-})+O(\nu_{1}^{2j}\|u\|^{2}+\nu_{1}^{j}|u|\cdot|v-v^{-}|+(v-v^{-})^{2}),\\\
\bar{x}_{2}-x_{2}^{+}=a_{41}\nu_{1}^{j}u_{1}+\hat{\nu}^{j}\tilde{\eta}_{2j}(u,v,\mu)+b_{4}(v-v^{-})+O(\nu_{1}^{2j}\|u\|^{2}+\nu_{1}^{j}|u|\cdot|v-v^{-}|+(v-v^{-})^{2}),\\\
\gamma_{1}^{-k}\bar{y}+\hat{\gamma}_{1}^{-k}\xi_{2k}(\bar{x},\bar{y},\mu)=c_{3}\nu_{1}^{j}u_{1}+\hat{\nu}^{j}\tilde{\eta}_{3j}(u,v,\mu)+d_{2}(v-v^{-})+\\\
\qquad+O(\nu_{1}^{2j}\|u\|^{2}+\nu_{1}^{j}\|u\|\cdot|v-v^{-}|+(v-v^{-})^{2}).\end{array}$
(3.25)

Make a coordinate shift $u_{new}=u-u^{+}+\varphi_{kj}^{1}$,
$v_{new}=v-v^{-}+\varphi_{kj}^{2}$, $x_{new}=x-x^{+}+\psi_{kj}^{1}$,
$y_{new}=y-y^{-}+\psi_{kj}^{2}$, where
$\varphi_{kj}^{1},\;\psi_{kj}^{2}=O(\gamma_{2}^{-j}+k\lambda^{k})$ and
$\varphi_{kj}^{2},\;\psi_{kj}^{1}=O(\nu_{1}^{j})$. With that, the nonlinearity
functions in the left parts of the third equations of (3.13) and (3.14) can be
expressed as Taylor expansions
$\eta_{3j}(u+u^{+}+\varphi_{kj}^{1},v+v^{-}+\varphi_{kj}^{2},\mu))=\eta_{3j}^{0}+\eta_{3j}^{1}v+\eta_{3j}^{2}(u,v)+\eta_{3j}^{3}(v)$,
$\xi_{2k}(\bar{x}+x^{+}+\psi_{ij}^{1},\bar{y}+y^{-}+\psi_{ij}^{2},\mu))=\xi_{2k}^{0}+\xi_{2k}^{1}\bar{y}+\xi_{2k}^{2}(\bar{x},\bar{y})+\xi_{2k}^{3}(\bar{y})$
respectively, where coefficients $\eta_{3j}^{0}$, $\eta_{3j}^{1}$,
$\xi_{2k}^{0}$, $\xi_{2k}^{1}$ are uniformly bounded in $k$ and $j$ for all
small $\mu$ and $\eta_{3j}^{2}(u,v)=O(u)$,
$\xi_{2k}^{2}(\bar{x},\bar{y})=O(\bar{x})$, $\eta_{3j}^{3}(v)=O(v^{2})$,
$\xi_{2k}^{3}(\bar{y})=O(\bar{y}^{2})$. We select constants
$\varphi_{kj}^{1},\;\varphi_{kj}^{2},\;\psi_{kj}^{1},\;\psi_{kj}^{2}$ in such
a way that all constant terms in equations (3.25), the constant terms in the
first two equations and the linear in $y_{new}$ term in the last equation of
(3.24) vanish. In addition, we plug the expressions for $\bar{x}$ coordinates
from the first two equations of (3.25) into the third one, this will cause
additions of order $\hat{\gamma}_{1}^{-k}$ to all the coefficients. The system
is rewritten as:

$\begin{array}[]{l}u_{1}=\lambda^{k}(a_{11}C_{k}(\phi)+a_{12}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(a_{11}S_{k}(\phi)+a_{12}C_{k}(\phi))x_{2}+b_{1}y+\\\
\qquad+O(\hat{\lambda}^{k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y|+y^{2}),\\\
u_{2}=\lambda^{k}(a_{21}C_{k}(\phi)+a_{22}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(a_{21}S_{k}(\phi)+a_{22}C_{k}(\phi))x_{2}+b_{2}y+\\\
\qquad+O(\hat{\lambda}^{k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y|+y^{2}),\\\
\gamma_{2}^{-j}(1+q_{kj}^{2})v+\hat{\gamma}_{2}^{-j}O(u)+\hat{\gamma}_{2}^{-j}O(v^{2})=M^{1}+\lambda^{k}(c_{1}C_{k}(\phi)+c_{2}\mu_{2}S_{k}(\phi))x_{1}+\\\
\qquad+\lambda^{k}(c_{1}S_{k}(\phi)+c_{2}C_{k}(\phi))x_{2}+d_{1}y^{2}+O(\hat{\lambda}^{k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y|+|y|^{3}),\end{array}$
(3.26)
$\begin{array}[]{l}\bar{x}_{1}=a_{31}\nu_{1}^{j}u_{1}+\tilde{a}_{32}\hat{\nu}^{j}u_{2}+b_{3}v+O(\hat{\nu}^{j}\|u\|^{2}+\nu_{1}^{j}\|u\|\cdot|v|+v^{2}),\\\
\bar{x}_{2}=a_{41}\nu_{1}^{j}u_{1}+\tilde{a}_{42}\hat{\nu}^{j}u_{2}+b_{4}v+O(\hat{\nu}^{j}\|u\|^{2}+\nu_{1}^{j}\|u\|\cdot|v|+v^{2}),\\\
\gamma_{1}^{-k}(1+q_{kj}^{1})\bar{y}+\hat{\gamma}_{1}^{-k}O(\bar{y}^{2})=c_{3}\nu_{1}^{j}u_{1}+\tilde{c}_{4}\hat{\nu}^{j}u_{2}+d_{2}v+\\\
\qquad+O(\hat{\nu}^{j}\|u\|^{2}+\nu_{1}^{j}\|u\|\cdot|v|+v^{2}),\end{array}$
(3.27)

where
$q_{kj}^{1}=O\left(\left(\hat{\gamma}_{1}/\gamma_{1}\right)^{-k}\right)$,
$q_{kj}^{2}=O\left(\left(\hat{\gamma}_{2}/\gamma_{2}\right)^{-j}\right)$,
coefficients marked with ‘‘tilde’’ are uniformly bounded for small $\mu$ and
the following expression is valid for $M^{1}$:

$M^{1}=\mu_{1}+\lambda^{k}((c_{1}C_{k}(\phi)+c_{2}\mu_{2}S_{k}(\phi))x_{1}^{+}+(c_{1}S_{k}(\phi)+c_{2}C_{k}(\phi))x_{2}^{+})-\gamma_{2}^{-j}v^{-}+O(\hat{\gamma}_{2}^{-j}+\hat{\lambda}^{k}+\gamma_{1}^{-k}\gamma_{2}^{-j}).$
(3.28)

Next, we take the right-hand side of the third equation of (3.27) divided by
the factor $(1+q_{kj}^{1})$ from the left-hand side as the new variable $v$ –
the equation becomes
$\gamma_{1}^{-k}\bar{y}+\hat{\gamma}_{1}^{-k}O(\bar{y}^{2})=v$, and we
substitute it into the third equation of (3.26). We also define
$x_{1new}=x_{1}-\frac{b_{3}}{d_{2}}\gamma_{1}^{-k}y+\hat{\gamma}_{1}^{-k}O(y^{2})$
and
$x_{2new}=x_{2}-\frac{b_{4}}{d_{2}}\gamma_{1}^{-k}y+\hat{\gamma}_{1}^{-k}O(y^{2})$
such that we remove all terms in the equations for $x$ which depend on $v$
alone. In addition, we substitute the expressions for $\bar{x}$ to the last
equation of (3.27). These actions cause the linear in $y$ term of order
$O(\hat{\gamma}_{2}^{-j}+\lambda^{k}\gamma_{1}^{-k})$ to appear in the
equation for $\bar{y}$. We will make it zero again later. Thus, we obtain

$\begin{array}[]{l}u_{1}=\lambda^{k}(a_{11}C_{k}(\phi)+a_{12}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(a_{11}S_{k}(\phi)+a_{12}C_{k}(\phi))x_{2}+b_{1}v\\\
\qquad+O(\hat{\lambda}^{k}\|x\|^{2}+\lambda^{k}\gamma_{2}^{-j}\|x\|\cdot|v|+v^{2}),\\\
u_{2}=\lambda^{k}(a_{21}C_{k}(\phi)+a_{22}\mu_{2}S_{k}(\phi))x_{1}+\lambda^{k}(a_{21}S_{k}(\phi)+a_{22}C_{k}(\phi))x_{2}+b_{2}v\\\
\qquad+O(\hat{\lambda}^{k}\|x\|^{2}+\lambda^{k}\gamma_{2}^{-j}\|x\|\cdot|v|+v^{2}),\\\
\bar{x}_{1}=A_{31}\nu_{1}^{j}u_{1}+\tilde{A}_{32}\hat{\nu}^{j}u_{2}+O(\hat{\nu}^{j}\|u\|^{2}+\nu_{1}^{j}\|u\|\cdot|v|),\\\
\bar{x}_{2}=A_{41}\nu_{1}^{j}u_{1}+\tilde{A}_{42}\hat{\nu}^{j}u_{2}+O(\hat{\nu}^{j}\|u\|^{2}+\nu_{1}^{j}\|u\|\cdot|v|),\\\
\displaystyle\frac{\gamma_{1}^{-k}\gamma_{2}^{-j}}{d_{2}}\bar{y}(1+q_{kj}^{3})+\hat{\gamma}_{1}^{-k}\gamma_{2}^{-j}O(\bar{y}^{2})=M^{1}+\lambda^{k}(c_{1}C_{k}(\phi)+c_{2}\mu_{2}S_{k}(\phi))x_{1}+\\\
\qquad+\lambda^{k}(c_{1}S_{k}(\phi)+c_{2}C_{k}(\phi))x_{2}+O(\hat{\gamma}_{2}^{-j}+\lambda^{k}\gamma_{1}^{-k})y+d_{1}y^{2}+\\\
\qquad+O(\hat{\lambda}^{k}\|x\|^{2}+\lambda^{k}\|x\|\cdot|y|+|y|^{3}),\end{array}$
(3.29)

where
$q_{ij}^{3}=O\left(\left(\hat{\gamma}_{1}/\gamma_{1}\right)^{-k}+\left(\hat{\gamma}_{2}/\gamma_{2}\right)^{-j}\right)$
and $A_{31}=a_{31}-b_{3}c_{3}/d_{2},\;A_{41}=a_{41}-b_{4}c_{3}/d_{2}.\;$

We substitute $u$ as a function of $x$ and $y$ from the first two equations
into the last three ones. After this, in addition, we make a shift of $(x,y)$
coordinates by a constant of order
$O(\hat{\gamma}_{2}^{-j}+\lambda^{k}\gamma_{1}^{-k})$ to nullify the linear in
$y$ term in the last equation. This gives us the following formula in
variables $(x,y)$:

$\begin{array}[]{l}\bar{x}_{1}=A_{31}\lambda^{k}\nu_{1}^{j}(a_{11}C_{k}(\phi)+a_{12}\mu_{2}S_{k}(\phi))x_{1}+A_{31}\lambda^{k}\nu_{1}^{j}(a_{11}S_{k}(\phi)+a_{12}C_{k}(\phi))x_{2}+A_{31}b_{1}\nu_{1}^{j}y+\\\
\qquad\qquad+O(\lambda^{k}\hat{\nu}^{j}+\lambda^{k}\nu_{1}^{j}\gamma_{1}^{-k})O(\|x\|^{2})+\lambda^{k}\nu_{1}^{j}O(\|x\|\cdot|y|)+O(y^{2}),\\\
\bar{x}_{2}=A_{41}\lambda^{k}\nu_{1}^{j}(a_{11}C_{k}(\phi)+a_{12}\mu_{2}S_{k}(\phi))x_{1}+A_{41}\lambda^{k}\nu_{1}^{j}(a_{11}S_{k}(\phi)+a_{12}C_{k}(\phi))x_{2}+A_{41}b_{1}\nu_{1}^{j}y+\\\
\qquad\qquad+O(\lambda^{k}\hat{\nu}^{j}+\lambda^{k}\nu_{1}^{j}\gamma_{1}^{-k})O(\|x\|^{2})+\lambda^{k}\nu_{1}^{j}O(\|x\|\cdot|y|)+O(y^{2}),\\\
\displaystyle\frac{\gamma_{1}^{-k}\gamma_{2}^{-j}}{d_{2}}\bar{y}(1+q_{kj}^{3})+\hat{\gamma}_{1}^{-k}\gamma_{2}^{-j}O(\bar{y}^{2})=M^{1}+\lambda^{k}(c_{1}C_{k}(\phi)+c_{2}\mu_{2}S_{k}(\phi))x_{1}+\\\
\qquad+\lambda^{k}(c_{1}S_{k}(\phi)+c_{2}C_{k}(\phi))x_{2}+d_{1}y^{2}+\\\
\qquad\qquad+O(\lambda^{k}\hat{\nu}^{j}+\lambda^{k}\nu_{1}^{j}\gamma_{1}^{-k})O(\|x\|^{2})+\lambda^{k}\nu_{1}^{j}O(\|x\|\cdot|y|)+O(y^{3}),\end{array}$
(3.30)

In order to eliminate the terms that depend on $y$ only in the first equation,
we make the following coordinate change: $x_{1new}=x_{1}-A_{31}/A_{41}x_{2}$.
The first return map takes the form:

$\begin{array}[]{l}\bar{x}_{1}=\tilde{A}_{11}(\phi)\lambda^{k}\nu_{1}^{j}x_{1}+\tilde{A}_{12}(\phi)\lambda^{k}\nu_{1}^{j}x_{2}+O(\lambda^{k}\hat{\nu}^{j}+\lambda^{k}\nu_{1}^{j}\gamma_{1}^{-k})O(\|x\|^{2})+\lambda^{k}\nu_{1}^{j}O(\|x\|\cdot|y|)\\\
\bar{x}_{2}=\tilde{A}_{21}(\phi)\lambda^{k}\nu_{1}^{j}x_{1}+\tilde{A}_{22}(\phi)\lambda^{k}\nu_{1}^{j}x_{2}+A_{41}b_{1}\nu_{1}^{j}y+O(\lambda^{k}\hat{\nu}^{j}+\lambda^{k}\nu_{1}^{j}\gamma_{1}^{-k})O(\|x\|^{2})+\lambda^{k}\nu_{1}^{j}O(\|x\|\cdot|y|)+O(y^{2})\\\
\displaystyle\frac{\gamma_{1}^{-k}\gamma_{2}^{-j}}{d_{2}}\bar{y}(1+q_{kj}^{3})+\gamma_{1}^{-k}\hat{\gamma}_{2}^{-j}O(\bar{y}^{2})=M^{1}+\tilde{C}_{1}(\phi)\lambda^{k}x_{1}+\tilde{C}_{2}\lambda^{k}x_{2}+d_{1}y^{2}+\\\
\qquad\qquad+O(\lambda^{k}\hat{\nu}^{j}+\lambda^{k}\nu_{1}^{j}\gamma_{1}^{-k})O(\|x\|^{2})+\lambda^{k}\nu_{1}^{j}O(\|x\|\cdot|y|)+O(y^{3})\par\end{array}$
(3.31)

where

$\begin{array}[]{l}\displaystyle\tilde{A}_{11}(\phi)=O\left(\left(\hat{\lambda}/\lambda\right)^{k}+\left(\hat{\nu}/\nu_{1}\right)^{j}\right)\\\
\displaystyle\tilde{A}_{12}(\phi)=O\left(\left(\hat{\lambda}/\lambda\right)^{k}+\left(\hat{\nu}/\nu_{1}\right)^{j}\right)\\\
\displaystyle\tilde{A}_{21}(\phi)=A_{41}(a_{11}C_{k}(\phi)+a_{12}\mu_{2}S_{k}(\phi))+O\left(\left(\hat{\lambda}/\lambda\right)^{k}+\left(\hat{\nu}/\nu_{1}\right)^{j}\right)\\\
\displaystyle\tilde{A}_{22}(\phi)=A_{31}(a_{11}C_{k}(\phi)+a_{12}\mu_{2}S_{k}(\phi))+A_{41}(a_{11}S_{k}(\phi)+a_{12}C_{k}(\phi))+O\left(\left(\hat{\lambda}/\lambda\right)^{k}+\left(\hat{\nu}/\nu_{1}\right)^{j}\right)\\\
\displaystyle\tilde{C}_{1}(\phi)=c_{1}C_{k}(\phi)+c_{2}\mu_{2}S_{k}(\phi)+O\left(\left(\hat{\lambda}/\lambda\right)^{k}+\left(\hat{\nu}/\nu_{1}\right)^{j}\right)\\\
\displaystyle\tilde{C}_{2}(\phi)=c_{1}S_{k}(\phi)+c_{2}C_{k}(\phi)+\frac{A_{31}}{A_{41}}(c_{1}C_{k}(\phi)+c_{2}\mu_{2}S_{k}(\phi))+O\left(\left(\hat{\lambda}/\lambda\right)^{k}+\left(\hat{\nu}/\nu_{1}\right)^{j}\right).\end{array}$
(3.32)

Take $\mu_{2}<0$. Coefficient $\tilde{C}_{2}(\phi)$ can be then represented as
$\tilde{C}_{2}(\phi)=L_{1}(\mu)\cos k\phi+L_{2}(\mu)\sin
k\phi=L(\mu)\sin(k\phi+\psi)$. We will vary $\mu_{2}$ near those values, for
which $\tilde{C}_{2}(\phi)$ vanishes so that it becomes asymptotically small
for $k,j\to\infty$. Note that $\tilde{C}_{1}(\phi)$ and $\tilde{A}_{12}(\phi)$
will remain bounded from zero at these parameter values because otherwise, the
first return map will be not a diffeomorphism. Finally, we rescale the
coordinates as follows:

$\begin{array}[]{c}\displaystyle
y=-\frac{\gamma_{1}^{-k}\gamma_{2}^{-j}}{d_{1}d_{2}}(1+q_{kj}^{3})\;Y\;,\;x_{1}=-\frac{b_{1}A_{41}\tilde{A}_{12}\lambda^{k}\nu_{1}^{2j}\gamma_{1}^{-k}\gamma_{2}^{-j}}{d_{1}d_{2}}(1+q_{kj}^{3})\;X_{1}\\\
\displaystyle
x_{2}=-\frac{b_{1}A_{41}\nu_{1}^{j}\gamma_{1}^{-k}\gamma_{2}^{-j}}{d_{1}d_{2}}(1+q_{kj}^{3})\;X_{2}\;.\end{array}$
(3.33)

.

The resulting map is asymptotically close to three-dimensional Henon map (0.1)
as $k,j\to\infty$, and formulas (2.30) are valid for the rescaled parameters.
For small variations of $\mu_{1}$ parameter $M_{1}$ takes arbitrary finite
values. We vary $\mu_{2}<0$ near a zero of function $\sin(k\phi+\psi)$, and
being multiplied by
$(\lambda\gamma_{1})^{k}(\nu_{1}\gamma_{2})^{j}\sim\lambda^{-k}\nu_{2}^{-j}$
we obtain that parameter $M_{2}$ also takes arbitrary finite values. We also
select arbitrary large $k$ and $j$ in such a way that the Jacobian of the
first return map
$B=J_{12}J_{21}(\lambda^{2}\gamma_{1})^{k}(\nu_{1}\nu_{2}\gamma_{2})^{j}$
stays finite. Then, by small variations of $\mu_{3}$ it will also take
arbitrary finite values, positive or negative.

## Acknowledgements

The author thanks Sergey Gonchenko for the formulation of the problem and
useful discussions, including the normal forms near saddles having eigenvalues
with multiplicity greater than one.

## Funding

This paper is a contribution to the project M7 (Dynamics of Geophysical
Problems in Turbulent Regimes) of the Collaborative Research Centre TRR 181
‘‘Energy Transfer in Atmosphere and Ocean’’ funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – Projektnummer
274762653. The paper is also supported by the grant of the Russian Science
Foundation 19-11-00280.

## Data availability statement

The original contributions presented in the study are included in the
article/supplementary material, further inquiries can be directed to the
corresponding author.

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