# The occurrence of riddled basins and blowout bifurcations in a parametric
nonlinear system

M. Rabiee  Department of Mathematics, Ferdowsi University of Mashhad, Mashhad,
Iran<EMAIL_ADDRESS>,  F. H. Ghane  Department of
Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.
<EMAIL_ADDRESS>,  M. Zaj  Department of Mathematics, Ferdowsi University
of Mashhad, Mashhad, Iran<EMAIL_ADDRESS>and  S. Karimi  Department of
Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.
<EMAIL_ADDRESS>
###### Abstract.

In this paper, a two parameters family $F_{\beta_{1},\beta_{2}}$ of maps of
the plane living two different subspaces invariant is studied. We observe
that, our model exhibits two chaotic attractors $A_{i}$, $i=0,1$, lying in
these invariant subspaces and identify the parameters at which $A_{i}$ has a
locally riddled basin of attraction or becomes a chaotic saddle. Then, the
occurrence of riddled basin in the global sense is investigated in an open
region of $\beta_{1}\beta_{2}$-plane. We semi-conjugate our system to a random
walk model and define a fractal boundary which separates the basins of
attraction of the two chaotic attractors, then we describe riddled basin in
detail. We show that the model undergos a sequence of bifurcations: “a blowout
bifurcation”, “a bifurcation to normal repulsion” and “a bifurcation by
creating a new chaotic attractor with an intermingled basin”. Numerical
simulations are presented graphically to confirm the validity of our results.

###### Key words and phrases:

riddled basin of attraction, blowout bifurcation, skew product, normal
Lyapunov exponent

###### 2010 Mathematics Subject Classification:

37C05,37C40, 37C70, 37H15, 37E05, 37D35

∗Corresponding author

##  1\. Introduction

There has been a lot of recent interest in the global dynamics of systems with
multiple attractors, with the recognition that the structure of basins of
attraction may be very complicated. The terminology of multiple attractors,
allows several attractors to coexist, has appeared in [Buescu(1997),
Ott(1993), Daza et al.(2016), Dudkowskia et al.(2016)]. It is very common for
dynamical systems to have more than one attractor. Among of them, we focus on
systems having multiple attractors may present basins of attraction densely
blended, a phenomenon called _riddling_. This means that for every initial
condition in the basin of one attractor there are arbitrary near conditions
which tend to the basin of any of the other attractors. Ott et al. [Ott et
al.(1993)] introduced non-linear dynamical systems with a simple symmetry
contain riddled basins. Also, the conditions of occurrence riddled basins are
defined in Alexander et al. [Alexander et al.(1992)] and then generalized by
Ashwin et al. [Ashwin et al.(1996)].

Riddling basins arise in systems that possess chaotic dynamics in a smooth
invariant manifold of lower dimension than that of the full phase space. This
complexity appears at the transition point between strong and weak stability
of the invariant subspace. When riddled basins occur, differing initial
conditions of the different replicates may asymptotic to different attractors.
Hence, the task of predicting what will be the final state of the system
becomes difficult. A detailed picture of multiple attractors with riddled
basins is available through works of several authors [Alexander et al.(1992),
Ashwin et al.(1996), Lai et al.(2005), Mohd Roslan & Ashwin(2016), Schultz et
al.(1993)]. Applications of riddling in complex dynamical systems of physical
and biological interest can be found: for instance, for a forced double-well
Duffing oscillator [Ott & Sommer(1993), Ott et al.(1993), Ott et al.(1994)],
ecological population models [Cazelles(2001), Viana et al.(2009), Karimi &
Ghane(2020)], learning dynamical systems [Nakajima & Ueda(1996)], coupled non-
linear electronic circuits [Ashwin et al.(1994), Heagy et al.(1995)], among
others.

Take a nonlinear dynamical system possesses a smooth invariant manifold $N$.
Suppose that the restriction of the system to $N$ has an attractor $A$, so it
is stable to perturbation within $N$. Assume the behaviour of the system near
$A$ is determined by combining the dynamics on $A$ and the dynamics transverse
to $N$. In the case that $A$ is chaotic, by the global nature of the effect of
perturbations transverse to $N$, considerable complexities in dynamics is
observed [Ashwin et al.(1996), Ashwin et al.(1994), Cazelles(2001), Viana et
al.(2009)]. In this situation, the local dynamic stability of the chaotic
attractor $A$ may be described in terms of normal Lyapunov exponents. When the
largest normal Lyapunov exponent is negative, there is a set of positive
measure which is forward asymptotic to $A$ [Alexander et al.(1992), Ashwin et
al.(1996)]. In general, when we discuss the occurrence of the riddled basin
for the attractor $A$, it is necessary to have a dense set of points with zero
Lebesgue measure in $A$ lying in the invariant subspace which are transversely
unstable, thus it is necessary that the attractor $A$ be chaotic. In most
chaotic attractors ergodic measures are not unique; for instance, they may
exhibit dirac ergodic measures whose support are periodic orbits. Each ergodic
measure carries its own Lyapunov exponents, so the stability in transverse
directions can be considered independently for every ergodic measure supported
in that attractor. For example, two different periodic orbits in the attractor
may have normal exponents of different signs. If there exists a natural
measure on $A$, then Lebesgue-almost all points have corresponding normal
exponents and manifolds, but there can still be a dense set in $A$ with the
opposite behaviour [Ashwin et al.(1996)].

When the full space contains two chaotic attractors lying in different
invariant subspaces, the system presents a complex fractal boundary between
the initial conditions leading to each of the two attractors. However, in a
riddled basin, small variations in initial conditions induce a switch between
the different asymptotic attractors but fractal boundary causes to predict,
from a given initial condition, what trajectory in phase-space the system will
follow.

Ott et al. [Ott et al.(1994)] observed the occurrence of riddling basin for a
certain nonlinear model of point particle motion subject to friction and
periodic forcing in a two-dimensional potential and tested this observation
numerically. However, they verified their results theoretically by
calculations leading to a simple piecewise-linear model.

In this article, we examine the behavior of a two parameter family of planar
systems $F_{\beta_{1},\beta_{2}}$. In our setting, each system
$F_{\beta_{1},\beta_{2}}$ exhibits two invariant subspaces $N_{i}$, $i=1,2$,
having chaotic attractors $A_{i}$. We demonstrate the emergence conditions of
locally riddled basin, chaotic saddle and riddled basin in the global sense.
The blowout bifurcations of chaotic attractors in these invariant subspaces
are illustrated. We give a detailed analysis to occurrence of these phenomena.
It can be done by semi-conjugating the system to a random walk model and
defining a fractal boundary which separates the basins of attraction of two
chaotic attractors. Using this approach, we investigate the system at riddled
basin and blowout in detail.

The setting of this paper is a family $\mathcal{F}$ of two parametric skew
product maps of the form

$\displaystyle
F_{\beta_{1},\beta_{2}}:\mathbb{I}\times\mathbb{I}\to\mathbb{I}\times\mathbb{I},\
F_{\beta_{1},\beta_{2}}(x,y):=(f(x),g_{\beta_{1},\beta_{2}}(x,y)),$ (1)

where $\mathbb{I}$ is the unit interval $[0,1]$, $f$ is an expanding Markov
map given by

$\displaystyle f(x)=\left\\{\begin{array}[]{cc}2x&$ for$\quad 0\leq x\leq
1/2,\\\ 2x-1&$ for$\quad 1/2<x\leq 1\end{array}\right.$ (4)

and

$\displaystyle
g_{\beta_{1},\beta_{2}}(x,y):=\left\\{\begin{array}[]{cc}g_{1,\beta_{1}}(y)&$
for$\quad 0\leq x\leq 1/2,\\\ g_{2,\beta_{2}}(y)&$ for$\quad 1/2<x\leq
1.\end{array}\right.$ (7)

We assume that the $C^{2}$ diffeomorphisms
$g_{i,\beta_{i}}:\mathbb{I}\to\mathbb{I}$, $i=1,2$, fulfill the following
conditions:

  1. (I1)

$g_{i,\beta_{i}}(0)=0,\quad g_{i,\beta_{i}}(1)=1,\quad
g_{i,\beta_{i}}(\beta_{i})=\beta_{i},\quad\text{for}\ i=1,2;$

  2. (I2)

$g_{1,\beta_{1}}(y)<y\quad\text{for}\ y<\beta_{1},\quad
g_{1,\beta_{1}}(y)>y\quad\text{for}\ y>\beta_{1};$

  3. (I3)

$g_{2,\beta_{2}}(y)>y\quad\text{for}\ y<\beta_{2},\quad
g_{2,\beta_{2}}(y)<y\quad\text{for}\ y>\beta_{2};$

In (1), the subspaces

$N_{0}:=\mathbb{I}\times\\{0\\},\quad N_{1}:=\mathbb{I}\times\\{1\\}$ (8)

play the role of the $F_{\beta_{1},\beta_{2}}$-invariant manifolds with
chaotic dynamics inside, for each $\beta_{1},\beta_{2}\in(0,1)$. Our objective
is to study some types of bifurcations by varying the parameters $\beta_{i}$
and characterize the different possible dynamics.

Particular example is given by

$\displaystyle
g_{\beta_{1},\beta_{2}}(x,y):=\left\\{\begin{array}[]{cc}g_{1,\beta_{1}}(y)=y+y(1-y)(y-\beta_{1})&$
for$\quad 0\leq x\leq 1/2,\\\ g_{2,\beta_{2}}(y)=y-y(1-y)(y-\beta_{2})&$
for$\quad 1/2<x\leq 1.\end{array}\right.$ (11)

The parameters $\beta_{i}$, $i=1,2$, vary the transverse dynamics without
changing the dynamics on the invariant subspaces $N_{0}$ and $N_{1}$. We will
show that, for some values of $\beta_{i}$, the parametric family
$F_{\beta_{1},\beta_{2}}$ exhibits two attractors lying in invariant subspaces
$N_{i}$ with a qualitative dynamics which depends on the initial conditions.
Attractors in our model, for some parameters values, exhibit a complex
attracting basin structure that is riddled by holes. This phenomenon produces
an unpredictability qualitatively greater than the traditional sensitive
dependence on initial conditions within a single chaotic attractor. The
dynamics of the system is described by two Lyapunov exponents. The first one
is the parallel Lyapunov exponent which describes the evolution on the
invariant subspaces and must be positive for emergence of riddled basins. The
second is the normal Lyapunov exponent that characterizes evolution transverse
to the subspaces [Ashwin et al.(1996), Cazelles(2001), Viana et al.(2009)].

In this paper, we estimate the range of values of parameters $\beta_{i}$,
$i=1,2$, such that the attractors $A_{i}$ have a locally riddled basin,
riddled basin in the global sense or becomes a chaotic saddle and provide
rigorous analysis for these complex behaviors. Also, we investigate that, when
$\beta_{1}=\beta_{2}$, a new chaotic attractor is born and its basin is
intermingled with the basins of both chaotic attractors $A_{i}$, $i=1,2$. We
show that by varying the parameters $\beta_{i}$, we are forced to undergo a
sequence of bifurcations: a “blowout bifurcation”, a “bifurcation to normal
repulsion” and “a bifurcation by creating a new chaotic attractor with an
intermingled basin”.

Here, we will show that by varying the parameters $\beta_{i}$, it is possible
one of the chaotic sets in the invariant subspaces looses the stability when
the parameters pass through critical values. This happens if the normal
Lyapunov exponent of one of the chaotic attractors crosses zero at these
critical values. If the normal Lyapunov exponent is negative, there is a set
of positive measure which is forward asymptotic to that attractor. However,
there can still be an infinite set of trajectories in the neighborhood of the
attractor that are repelled from it. In particular, if the normal Lyapunov
exponent is small and negative, the blowout bifurcation occurs and the
riddling basin can be observed near the bifurcation point.

This paper is organized as follows. In Section 2, we describe precisely the
notions and terminology used in this paper. In Section 3, we concentrate on
studying the two parameters family $F_{\beta_{1},\beta_{2}}$. Using the
results of [Ashwin et al.(1996)], we investigate the occurrence of locally
riddled basins and chaotic saddles for some values of parameters in an open
region of $\beta_{1}\beta_{2}$-plane. In Section 4, we introduce a random walk
model which is semi-conjugate to the skew product system
$F_{\beta_{1},\beta_{2}}$. This allows us to define a fractal boundary between
the initial conditions leading to each of the two attractors $A_{0}$ and
$A_{1}$. In Section 5, we demonstrate the emergence conditions of riddled
basins in the global sense and indicate the parameters values for which the
family undergos a sequence of bifurcations: a blowout bifurcation and a
bifurcation by creating a new chaotic attractor with intermingled basin.

##  2\. Terminology

In this section, we introduce the concepts and the notations which are basic
in this paper.

###  2.1.  Attractors and riddled basins

Let $M$ be a compact connected smooth Riemannian manifold and $m$ denotes the
normalized Lebesgue measure. First we recall some classical definitions
related to attractors.

Let $F:M\to M$ be a continuous map and $A\subset M$ is a compact $F$-invariant
set ( i.e. $F(A)=A$). We say $A$ is _transitive_ if there exists $x\in A$ such
that $\omega(x)=A$, where $\omega(x)$ is the set of limit points of the orbit
$\\{F^{n}(x)\\}_{n\geq 0}$. _The basin of attraction_ of $A$, we denote it by
$\mathcal{B}(A)$, is the set of points whose $\omega$-limit set is contained
in $A$. For non-empty $A$ the basin $\mathcal{B}(A)$ is always non-empty
because it includes $A$. For $A$ to be an attractor, we require that
$\mathcal{B}(A)$ is large in the appropriate sense. The compact invariant set
$A$ is called an _asymptotically stable attractor_ if it is Lyapunov stable
and the basin of attraction $\mathcal{B}(A)$ contains a neighbourhood of $A$.

Many variants for the definition of an attractor can be found the literature,
see [Milnor(1985)] for a discussion. In a weaker form [Milnor(1985)], we say
that a compact $F$-invariant set $A$ is an attractor in Milnor sense for $F$
if the basin of attraction $\mathcal{B}(A)$ has positive Lebesgue measure. To
be more precise, we say that A is a _Milnor attractor_ if $\mathcal{B}(A)$ has
non-zero Lebesgue measure and there is no compact proper subset $A^{\prime}$
of $A$ whose basin coincides with $\mathcal{B}(A)$ up to a set of zero
measure. Melbourne introduced a stronger form of a Milnor attractor
[Melbourne(1999)]. $A$ is called an _essential attractor_ if

$\lim_{\delta\to
0}\frac{m(B_{\delta}(A)\cap\mathcal{B}(A))}{m(B_{\delta}(A))}=1,$

where $B_{\delta}(A)$ is a $\delta$-neighbourhood of $A$ in $M$.

Here, we deal with chaotic attractors. A compact $F$-invariant set $A$ is a
_chaotic attractor_ if $A$ is a transitive Milnor attractor and supports an
ergodic measure $\mu$ but is not uniquely ergodic. In particular, at least one
of the Lyapunov exponents (with respect to $\mu$) is positive.

Some of dynamical systems having chaotic attractors with densely intertwined
basins of attraction, which we call it _riddled basin_. Riddled basins were
introduced in 1992 by [Alexander et al.(1992)]. In this case, a basin is
riddled with holes (in a measure theoretical sense) of another basin. To be
more precise, the basin of attraction $\mathcal{B}(A)$ of an attractor $A$ is
_riddled_ if its complement $\mathcal{B}(A)^{c}$ intersects every disk in a
set of positive measure.

This concept is generalized to the “locally riddled basin”. It considers the
case where the basin of $A$ is open but local normal unstable manifolds exist
in a dense set in $A$. Precisely, a Milnor attractor $A$ has a _locally
riddled basin_ if there exists a neighbourhood $U$ of $A$ such that, for all
$x\in A$ and $\varepsilon>0$

$m(B_{\varepsilon}(x)\cap(\bigcap_{n\geq 0}F^{-n}(U))^{c})>0.$ (12)

If there is another Milnor attractor $A^{\prime}$ such that
$\mathcal{B}(A)^{c}$ in the definition of locally riddled basin may be
replaced with $\mathcal{B}(A^{\prime})$, then we say that the basin of $A$ is
riddled with the basin of $A^{\prime}$. If $\mathcal{B}(A)$ and
$\mathcal{B}(A^{\prime})$ are riddled with each other, we say that they are
_intermingled_.

The corresponding concepts can be defined for repelling sets. An invariant
transitive set $A$ is a _chaotic saddle_ if there exists a neighborhood $U$ of
$A$ such that $\mathcal{B}(A)\cap U\neq\emptyset$ but $m(\mathcal{B}(A))=0$.

Consider the case that the invariant set $A$ is contained in an invariant
$n$-dimensional submanifold $N$ of $\mathbb{R}^{m}$, with $n<m$. We say that
$A$ is a _normally repelling chaotic saddle_ if $\mathcal{B}(A)\neq A$ and
$\mathcal{B}(A)\subset N$. Note that in this case, however, $A$ is an
attractor in the invariant subspace, but all points not lying on this subspace
eventually leave a neighborhood of $A$.

###  2.2. Lyapunov exponents

Assume $F$ is a smooth map defined on a smooth manifold $M$ and let $N\subset
M$ be an $n$-dimensional embedded submanifold and forward invariant by $F$,
with $n<m$. Hence, for $x\in N$, one has that $d_{x}F(T_{x}N)\subset
T_{x}F(N)$. We consider the restriction of $F$ to $N$, denoted by $F_{|N}$.
Moreover, we assume that $A$ is a chaotic attractor for $F$. We denote by
$\mathcal{M}_{F}(A)$ and $\mathcal{E}_{F}(A)$ the sets of invariant
probability measures and ergodic measures supported in $A$, respectively. It
is known that both $\mathcal{M}_{F}(A)$ and $\mathcal{E}_{F}(A)$ are non-empty
(see [Walters(1982)]).

For a vector $v\neq 0$ with base point $x$, _the Lyapunov exponent_
$\lambda(x,v)$ at the point $x$ in the direction of $v$ is defined to be

$\lambda(x,v)=\lim_{n\to\infty}\frac{1}{n}\log\|d_{x}F^{n}(v)\|_{T_{F^{n}(x)}M}$
(13)

whenever the limit exists. Here, we have two kind of Lyapunov exponents for
the chaotic invariant set $A$: the _parallel Lyapunov exponents_ which
indicate the exponential rate of stretching on $A$ when $F$ is restricted to
$N$ and the _normal Lyapunov exponents_ that present the exponential rate of
expansion on $A$ in the normal direction denoted by $\lambda_{\parallel}$ and
$\lambda_{\perp}$, respectively. Precisely, we define these Lyapunov exponents
as follows. Since $N$ is an embedded submanifold, we can take a smooth
splitting of the tangent bundle $TM$ in a neighbourhood of $N$ of the form
$T_{x}M=T_{x}N\oplus(T_{x}N)^{\bot}$, when $x\in N$. To simplify the notation,
we write $TM_{n}:=T_{F^{n}(x)}M$.

######  Definition 1.

Given $x\in A$; $v\in T_{x}M=T_{x}N\oplus T_{x}N^{\perp}$, we define the
_parallel Lyapunov exponent_ at $x$ in the direction of $v$ to be

$\displaystyle\lambda_{\parallel}(x,v)=\lim_{n\rightarrow\infty}\dfrac{1}{n}\ln\parallel\pi_{(TN_{n})}\circ
d_{x}F^{(n)}\circ\pi_{TN_{0}}(v)\parallel_{TM},$ (14)

where $\pi_{V}$ is the orthogonal projection onto a subspace $V$. Similarly,
we define the _normal Lyapunov exponent_ at $x$ in the direction of $v$ to be

$\displaystyle\lambda_{\perp}(x,v)=\lim_{n\rightarrow\infty}\dfrac{1}{n}\ln\parallel\pi_{(TN_{n})^{\perp}}\circ
d_{x}F^{(n)}\circ\pi_{(TN_{0})^{\perp}}(v)\parallel_{TM_{n}}.$ (15)

Here, we are interested in Sinai-Ruelle-Bowen (or SRB) measures which are a
special type of invariant measures, see [Mohd Roslan & Ashwin(2016)].

Let $A$ be an asymptotically stable attractor under $F_{|N}$. An invariant
ergodic probability measure $\mu$ is called an $SRB$ measure for $A$ if its
support is $A$ and has absolutely continuous conditional measures on unstable
manifolds (with respect to the Riemannian measure).

An attractor $A$ is an _SRB-attractor_ if it supports an SRB measure. Since
$A$ is an asymptotically stable attractor under $F_{|N}$, hence, it is the
closure of the union of unstable manifolds on $N$. Note that the existence of
an SRB measure supported on $A$ implies the absolute continuity of the stable
foliation of $A$, see [Pugh & Shub(1989)]. By a result from [Pugh &
Shub(1989)], we have the following:

Assume $\mu$ is an SRB measure for $F_{|N}$. Given a neighborhood $U$ of $A$
there is a set $B(\mu)\subset U$, we call it the _basin_ of $\mu$, with
positive Lebesgue measure such that for all $x\in B(\mu)$ and all continuous
functions $\phi:N\to\mathbb{R}$ one has that

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\phi(F^{i}|_{N}(x))=\int_{A}\phi
d\mu.$

######  Remark 2.

Take an open set $D$ with $\mu_{SRB}(D)=\mu_{SRB}(\text{Cl}(D))$, then

$\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\chi_{D}(F^{i}|_{N}(x))=\mu_{SRB}(D),$

for a.e. $x$ in the basin $B(\mu)$.

Given an ergodic invariant probability measure
$\mu\in\mathcal{E}_{F_{|N}}(A)$, the normal Lyapunov exponents
$\lambda_{\bot}^{1}(\mu)<\cdots<\lambda_{\bot}^{s}(\mu)$ exists and are
constants in a set $B_{\mu}$ of full $\mu$-measure. We define

$\lambda_{min}:=\inf\bigcup_{\mu\in\mathcal{E}_{F_{|N}}(A)}\\{\lambda_{\bot}^{i}(\mu)\\},\
\Lambda_{max}:=\sup\bigcup_{\mu\in\mathcal{E}_{F_{|N}}(A)}\\{\lambda_{\bot}^{i}(\mu)\\}.$
(16)

Let $\mu$ be an $F$-invariant ergodic probability measure supported in $A$,
with normal Lyapunov exponents
$\lambda_{\bot}^{1}(\mu)<\cdots<\lambda_{\bot}^{s}(\mu)$. The _normal
stability index_ $\Lambda_{\mu}$ of $\mu$ is

$\Lambda_{\mu}:=\lambda_{\bot}^{s}(\mu).$ (17)

We recall the next result from [Alexander et al.(1992)].

######  Theorem 3.

Assume $A$ is an SRB attractor for $F_{|N}$ with $\Lambda_{SRB}<0$, where
$\Lambda_{SRB}$ is defined by (17) for SRB measure $\mu_{SRB}$. Then
$m(\mathcal{B}(A))>0$. Furthermore, $A$ is an essential attractor provided
that $A$ is either uniformly hyperbolic or $\mu_{SRB}$ is absolutely
continuous with respect to Riemannian measure on $N$.

###  2.3. Blowout bifurcations

Let $M$ be a smooth Riemannian manifold and $N\subset M$ be an $n$-dimensional
embedded submanifold and forward invariant by $F$, with $n<m$. Assume that
there is a parameter $\beta$ that varies the transverse dynamics without
changing the dynamics on the invariant subspace $N$. We call the parameter
$\beta$ a _normal parameter_ [Ashwin et al.(1996)]. Moreover, we assume that
the normal Lyapunov exponents vary continuously with $\beta$ and that $A$ is
an asymptotically stable attractor for the restriction map $F_{|N}$.

Let $A$ be a chaotic attractor that supports an SRB measure $\mu_{SRB}$. Then,
the sign of $\Lambda_{SRB}$ determines whether $A$ attracts or repels
infinitesimal perturbations in the direction transverse to $N$. If
$\Lambda_{SRB}<0$, $A$ attracts trajectories transversely in the phase space
and hence, $A$ is also an attractor of the whole phase space. When
$\Lambda_{SRB}>0$, trajectories in the vicinity of $A$ are repelled away from
it. That is, $A$ is transversely unstable and is not an attractor of the whole
phase space. Thus a bifurcation occurs when $\Lambda_{SRB}$ crosses zero, the
so-called _blowout bifurcation_.

Blowout bifurcations are classified as either hysteretic (subcritical) or non-
hysteretic (supercritical). In a hysteretic blowout, riddled basins before the
blowout give rise to a hard loss of stability [Ashwin et al.(1998),
Pikovsky(1984)], after blowout, almost all points near the invariant subspace
eventually move away, never to return. However, a non-hysteretic blowout
giving a soft loss of stability to an on-off intermittent attractor [Ashwin et
al.(1998), Platt et al.(1993)].

It was shown that [Ashwin et al.(1996)] the occurrence of “locally riddled
basins” is exhibited near the blowout bifurcation.

###  2.4. Step skew products

Let $\Sigma_{2}^{+}=\\{1,2\\}^{\mathbb{N}}$ and equip $\Sigma_{2}^{+}$ with
the topology generated by the base of cylinder sets

$[\alpha_{0},\cdots,\alpha_{n-1}]=\\{\omega\in\Sigma_{2}^{+}:\omega_{j}=\alpha_{j},\
\text{for}\ j=0,\cdots,n-1\\}.$

Let $\sigma:\Sigma_{2}^{+}\to\Sigma_{2}^{+}$ be the (left) shift operator
defined by $(\sigma\omega)_{i}=\omega_{i+1}$ for
$\omega=(\omega_{i})_{0}^{\infty}$.

Here, we take the Bernoulli measure $\mathbb{P}^{+}$ on $\Sigma_{2}^{+}$ where
the symbols $1,2$ have probability $p_{1}=p_{2}=1/2$. Indeed, the Bernoulli
measure $\mathbb{P}^{+}$ on $\Sigma_{2}^{+}$ is determined by its values on
cylinders;

$\mathbb{P}^{+}([\alpha_{0},\cdots,\alpha_{n-1}])=\prod_{j=0}^{n-1}p_{\alpha_{j}}.$
(18)

Let $G^{+}$ be a step skew product map with the fiber maps $g_{1}$ and $g_{2}$
defined by

$G^{+}:\Sigma_{2}^{+}\times\mathbb{I}\to\Sigma_{2}^{+}\times\mathbb{I},\quad(\omega,x)\mapsto(\sigma\omega,g_{\omega_{0}}(x)).$
(19)

Denote by $\mathcal{B}$ the Borel sigma-algebra on $\mathbb{I}$. For a skew
product $G^{+}$ with fiber maps $g_{1}$ and $g_{2}$, a measure $m$ on
$\mathbb{I}$ and any $\mathcal{B}$-measurable set $A$, we denote the push-
forward measure of $m$ by $g_{i}m$, $i=1,2$, in which

$g_{i}m(A)=m(g_{i}^{-1}(A)).$

We recall that a probability measure $m$ on the interval $\mathbb{I}$ for
$G^{+}$ is _stationary_ if it satisfies

$m=\sum_{i=1}^{2}p_{i}g_{i}m.$

The natural extension of $G^{+}$ is obtained when the shift acts on two sided
time $\mathbb{Z}$; this yields a skew product system
$G:\Sigma_{2}\times\mathbb{I}\to\Sigma_{2}\times\mathbb{I}$ with
$\Sigma_{2}=\\{1,2\\}^{\mathbb{Z}}$ and given by the same expression

$G(\omega,y)=(\sigma\omega,g_{\omega_{0}}(y)).$ (20)

Write $\pi:\Sigma_{2}\to\Sigma_{2}^{+}$ for the natural projection
$(\omega_{n})_{-\infty}^{\infty}\mapsto(\omega_{n})_{0}^{\infty}$. The Borel
sigma-algebra $\mathcal{F}^{+}$ on $\Sigma_{2}^{+}$ yields a sigma-algebra
$\mathcal{F}=\pi^{-1}\mathcal{F}^{+}$ on $\Sigma_{2}$. Write $\mathbb{P}$ for
the Bernoulli measure on $\Sigma_{2}$, defined analogues to $\mathbb{P}^{+}$
(see (18)).

The invariant measure $\mu^{+}=\mathbb{P}^{+}\times m$ for $G^{+}$ gives rise
to an invariant measure $\mu$ for $G$, with marginal $\mathbb{P}$. Invariant
measures for $G^{+}$ with marginal $\mathbb{P}^{+}$ and invariant measures for
$G$ with marginal $\mathbb{P}$ are in one to one relationship, see
[Arnold(1998)].

A measure $\mu$ on $\Sigma_{2}\times\mathbb{I}$ with marginal $\mathbb{P}$ has
conditional measures $\mu_{\omega}$ on the fibers
$\\{\omega\\}\times\mathbb{I}$ such that

$\mu(A)=\int_{\Sigma_{2}}\mu_{\omega}A_{\omega}d\mathbb{P}(\omega)$ (21)

for each measurable set $A$, where
$A_{\omega}=A\cap(\\{\omega\\}\times\mathbb{I})$. For $\mathbb{P}$-almost all
$\omega$, the conditional measures are given as the weak star limit

$\mu_{\omega}=\lim_{n\to\infty}g_{\sigma^{-n}\omega}^{n}m$ (22)

of push-forwards of the stationary measure.

##  3\. The occurrence of locally riddled basin

In this section, we concentrate on studying the two parameters family
$F_{\beta_{1},\beta_{2}}:\mathbb{I}\times\mathbb{I}\to\mathbb{I}\times\mathbb{I}$
of skew product maps as defined in (1). By varying the parameters $\beta_{1}$
and $\beta_{2}$, we will investigate the occurrence of locally riddled basins
for the family $F_{\beta_{1},\beta_{2}}$.

We recall a normal parameter of the system—one that preserves the dynamics on
the invariant submanifold but varies it in the rest of the phase space, first
introduced in [Ashwin et al.(1996)]. We observe that the parameters
$\beta_{i}$ vary the transverse dynamics without changing the dynamics on the
invariant subspaces $N_{0}$ or $N_{1}$, so, they are normal parameters.

In our model, the expanding map $f$ exhibits a chaotic attractor which
supports an absolutely continuous invariant ergodic measure (a.c.i.m) whose
density is bounded and bounded away from zero (see [Adler & Flatto(1991)]). In
particular, this measure is absolutely continuous with respect to Lebesgue. By
this fact and invariance of the subspaces $N_{i}$, the restriction of
$F_{\beta_{1},\beta_{2}}$ to these invariant subspaces possess chaotic
attractors $A_{i}$, $i=0,1$, with the basin of attraction
$\mathcal{B}(A_{i})$. In particular, these attractors are $SRB$ attractors.

Note that, in our context, the normal dynamics is continuously dependent on
normal parameters $\beta_{i}$. Also, the invariant subspaces have codimension
1 in the phase space $\mathbb{I}\times\mathbb{I}$ and there is only one normal
direction. Hence, we can discuss the transitions between the invariant sets
$A_{i}$ being a locally riddled basin attractor and a chaotic saddle.

By Theorem 3, since $\mu_{SRB}$ is absolutely continuous with respect to an
Riemannian measure on the subspace $N_{i}$, $i=1,2$, the subset $A_{i}$ is an
essential attractor if the normal Lyapunov exponent is negative, see also
[Alexander et al.(1992)].

In general, for the occurrence of a riddled basin, we require that the
parallel Lyapunov exponent $\lambda_{\parallel}$ is positive but the maximal
normal Lyapunov exponent $\Lambda_{SRB}$ is slightly negative.

First, we describe the dynamics of the particular skew product
$F_{\beta_{1},\beta_{2}}$ whose fiber maps $g_{i,\beta_{1},\beta_{2}}$,
$i=1,2$, given by (11). For almost every point in $A_{i}$ the parallel
Lyapunov exponent for the dynamics on $N_{i}$ is

$\displaystyle L_{\|}=1/2\ln 2+1/2\ln 2=\ln 2.$ (23)

In particular, it is positive and does not depend on $\beta_{i}$, $i=1,2$. By
a straightforward computations, the normal Lyapunov exponents at typical
points in $A_{i}$ are given by

$\displaystyle
L_{\perp,\beta_{1},\beta_{2}}(0)=1/2\ln(g_{1,\beta_{1},\beta_{2}}^{\prime}(0))+1/2\ln(g_{2,\beta_{1},\beta_{2}}^{\prime}(0))=1/2\ln(1-\beta_{1})+1/2\ln(1+\beta_{2})$
(24)

and

$\displaystyle
L_{\perp,\beta_{1},\beta_{2}}(1)=1/2\ln(g_{1,\beta_{1},\beta_{2}}^{\prime}(1))+1/2\ln(g_{2,\beta_{1},\beta_{2}}^{\prime}(1))=1/2\ln\beta_{1}+1/2\ln(2-\beta_{2}).$
(25)

We observe that, by (24) and (25), these normal Lyapunov exponents vary
continuously with $\beta_{i}$. In Figure 1 the plot of
$L_{\perp,\beta_{1},\beta_{2}}(i)$, $i=0,1$, are illustrated by varying
$\beta_{1}$ and for fixed $\beta_{2}=1/2$. It demonstrates the continuous
dependence of normal Lyapunov exponents with respect to $\beta_{1}$ in this
case.

$\begin{array}[]{c}\includegraphics[scale={0.4}]{Lyapunov}\end{array}$

Figure 1. Normal Lyapunov exponents vary continuously with $\beta_{i}$. The
blue curve depicts $L_{\perp,\beta_{1},\beta_{2}}(0)$ by varying
$\beta_{1}=\beta$ and for fixed value $\beta_{2}=1/2$, while the red curve
depicts $L_{\perp,\beta_{1},\beta_{2}}(1)$.

Given an ergodic measure $\mu$, let $G_{\mu}$ be the set of generic points of
$\mu$. That is

$G_{\mu}=\\{x\in A:\frac{1}{n}\sum_{i=0}^{n-1}\delta_{f^{j}(x)}\to\mu\\}$

where convergence is in the weak∗ topology. For any $\alpha>0$ define

$G_{\alpha}:=\bigcup_{\mu\in\mathcal{E}_{f}(A),\Lambda_{\mu}\geq\alpha}G_{\mu}.$
(26)

######  Proposition 4.

[Ashwin et al.(1996), Proposition 3.19] Suppose $F:M\to M$ is a $C^{1+\alpha}$
map leaving the embedded submanifold $N$ invariant, and that $A$ is an
asymptotically stable chaotic attractor for $F|_{N}$. Let $\Lambda_{max}$,
$\lambda_{min}$ and $\Lambda_{SRB}$ be given by (16) and (17). Then, under
$F:M\to M$

  1. $(1)$

If $\Lambda_{SRB}<0<\Lambda_{max}$ then $A$ is a Milnor (essential) attractor.
If in addition there exists $\alpha>0$ with $G_{\alpha}$ dense in $A$, then
$A$ has a locally riddled basin.

  2. $(2)$

If $\lambda_{min}<0<\Lambda_{SRB}$, $\mu_{SRB}$-almost all Lyapunov exponents
are non-zero and $m(\bigcup_{\mu\neq\mu_{SRB}}G_{\mu})=0$, where $m$ is the
Riemannian volume on $N$, then $A$ is a chaotic saddle.

Note that, by [Ashwin et al.(1994), Remark 3.4], if $codim(N)=1$, as in our
setting, there is only one normal direction. In this case
$\lambda_{\mu}=\Lambda_{\mu}$ for all ergodic $\mu$, and the normal spectrum
depends smoothly on normal parameters.

Define $\rho:\mathbb{I}\to\\{-1,1\\}$ by

$\displaystyle\rho(x)=\left\\{\begin{array}[]{cc}-1&$ for$\quad 0\leq x\leq
1/2,\\\ 1&$ for$\quad 1/2<x\leq 1.\end{array}\right.$ (29)

Let $x$ be a periodic point of $f$ of period $n$ and let
$\mathcal{O}(x)=\\{x_{i}:f(x_{i-1})=x_{i},\ i=1,\dots,n,\quad\text{with}\quad
x_{0}=x\\}$. Take

$Per^{+}(f):=\\{x\in Per(f):\sum_{i=0}^{n-1}\rho(x_{i})>0\\},$ (30)

where $Per(f)$ denotes the set of all periodic orbits of $f$. By
[Gorodnik(2017), Proposition 3.1], there is a semi-conjugacy between the shift
map $\sigma:\Sigma_{2}^{+}\to\Sigma_{2}^{+}$ and the doubling map $f$. By this
fact, the following result is immediate.

######  Lemma 5.

The subset $Per^{+}(f)$ is dense in each chaotic attractor $A_{i}$, $i=0,1$,
within the invariant subspaces $N_{i}$.

######  Theorem 6.

Let $F_{\beta_{1},\beta_{2}}\in\mathcal{F}$ be a skew product of the form (1)
whose fiber maps $g_{i,\beta_{1},\beta_{2}}$, $i=1,2$, given by (11) and
$\beta_{1},\beta_{2}\in(0,1)$. Then the following holds:

  1. $(a)$

if $\beta_{1}>1/2$ or ($\beta_{1}\leq 1/2$ $\&$
$\beta_{2}<\frac{1}{1-\beta_{1}}-1$), then $A_{0}$ is a Milnor (essential)
attractor and has a locally riddled basin;

  2. $(b)$

if $\beta_{1}<1/2$ or ($\beta_{1}\geq 1/2$ $\&$
$\beta_{2}>2-\frac{1}{\beta_{1}}$), then $A_{1}$ is a Milnor (essential)
attractor and has a locally riddled basin;

  3. $(c)$

if $\beta_{1}<1/2$ and $\beta_{2}>\frac{1}{1-\beta_{1}}-1$ then $A_{0}$ is a
chaotic saddle;

  4. $(d)$

if $\beta_{1}>1/2$, $\beta_{2}<2-\frac{1}{\beta_{1}}$ then $A_{1}$ is a
chaotic saddle.

###### Proof.

Let $N_{i}$, $i=0,1$, be given by (8) and consider the restriction
$F_{\beta_{1},\beta_{2}}|_{N_{i}}$. By definition, since there is only one
normal direction, the normal stability index $\Lambda_{\mu}(A_{i})$ of an
ergodic invariant probability measure
$\mu\in\mathcal{E}_{F_{\beta_{1},\beta_{2}}|_{N_{i}}}(A_{i})$ is equal to the
normal Lyapunov exponent $\lambda_{\mu}(A_{i})$.

Note that, for $0\leq x\leq 1/2$, we have

$d_{(x,0)}F_{\beta_{1},\beta_{2}}=\begin{pmatrix}2&\quad 0\\\ 0&\quad
dg_{1,\beta_{1}}(0)\\\ \end{pmatrix}=\begin{pmatrix}2&\quad 0\\\ 0&\quad
1-\beta_{1}\\\ \end{pmatrix}$

and for $1/2<x\leq 1$, we have

$d_{(x,0)}F_{\beta_{1},\beta_{2}}=\begin{pmatrix}2&\quad 0\\\ 0&\quad
dg_{2,\beta_{2}}(0)\\\ \end{pmatrix}=\begin{pmatrix}2&\quad 0\\\ 0&\quad
1+\beta_{2}\\\ \end{pmatrix}$

where $g_{i,\beta_{i}}$, $i=1,2$, given by (11).

Hence, the normal stability index is computed as follows:

$\displaystyle\lambda_{\mu}(A_{0})=\Lambda_{\mu}(A_{0})=\int_{A_{0}\cap[0,1/2]}\log(1-\beta_{1})d\mu(x)+\int_{A_{0}\cap(1/2,1]}\log(1+\beta_{2})d\mu(x).$
(31)

Therefore,

$\displaystyle\lambda_{\mu}(A_{0})=\Lambda_{\mu}(A_{0})=\mu(A_{0}\cap[0,1/2])\log(1-\beta_{1})+\mu(A_{0}\cap(1/2,1])\log(1+\beta_{2}).$
(32)

Note that, by (34), for each invariant measure $\mu$, $\Lambda_{\mu}$ is
finite. Additionally, $\Lambda_{\mu}$ is smoothly dependent on normal
parameters $\beta_{1}$ and $\beta_{2}$.

The base map $f$ given by (4) is a piecewise expanding map. By definition of
$F_{\beta_{1},\beta_{2}}$ and since $N_{0}$ is one dimensional, we conclude
that $F_{\beta_{1},\beta_{2}}|_{A_{0}}$ is also piecewise expanding. This fact
implies that $F_{\beta_{1},\beta_{2}}|_{A_{0}}$ has a Lebesgue-equivalent
ergodic invariant measure (see [Ashwin et al.(1994), Walters(1982)]); this
corresponds to the desired $\mu_{SRB}(A_{0})$ (see Subsection 4.3 of [Ashwin
et al.(1994)]). By this fact and (34),

$\Lambda_{SRB}(A_{0})=1/2\log(1-\beta_{1})+1/2\log(1+\beta_{2}).$

Note that $\Lambda_{SRB}(A_{0})=L_{\perp,\beta_{1},\beta_{2}}(0)$, where
$L_{\perp,\beta_{1},\beta_{2}}(0)$ is the normal Lyapunov exponent given by
(24). Also, it characterizes [Alexander et al.(1992)] evolution transverse to
the $x$-axis. If it is negative, the invariant set $A_{0}$ is a Milnor
attractor. Simple computations show that, for $\beta_{1}>1/2$ or
($\beta_{1}\leq 1/2$ $\&$ $\beta_{2}<\frac{1}{1-\beta_{1}}-1$),
$\Lambda_{SRB}(A_{0})<0$ and hence $A_{0}$ is a Milnor (essential) attractor.
Take the invariant dirac measure $\mu_{1}$ supported on the fixed point at
$(1,0)$. Using (34), $\Lambda_{\mu_{1}}(A_{0})=\log(1+\beta_{2})$ which is
positive. By this fact, $0<\Lambda_{\mu_{1}}(A_{0})\leq\Lambda_{max}(A_{0})$.
These computations show that $\Lambda_{SRB}(A_{0})<0<\Lambda_{max}(A_{0})$.

By Lemma 4, the set $Per(f)^{+}$ is dense in $A_{0}$. By definition of
$Per(f)^{+}$ and (34), the dirac measure supported on $\mathcal{O}(x)$, for
$x\in Per(f)^{+}$, has positive normal stability index. By this fact and
taking suitable dirac measures supported on $\mathcal{O}(x)$, for $x\in
Per(f)^{+}$, we may find $\alpha>0$ such that $G_{\alpha}$ given by (26) is
dense in $A_{0}$. By these observations and statement (1) of Proposition 4,
$A_{0}$ has a locally riddled basin and the proof of $(a)$ is finished.

Let $\mu_{2}$ be the invariant dirac measure supported on the fixed point at
$(0,0)$. Using (34), $\Lambda_{\mu_{2}}(A_{0})=\log(1-\beta_{1})$ which is
negative. By this fact, $\lambda_{min}(A_{0})\leq\Lambda_{\mu_{2}}(A_{0})<0$.
Also, it is easy to see that for $\beta_{1}<1/2$ and
$\beta_{2}>\frac{1}{1-\beta_{1}}-1$, $\Lambda_{SRB}(A_{0})>0$. By these facts,
for $\beta_{1}<1/2$ and $\beta_{2}>\frac{1}{1-\beta_{1}}-1$, we get
$\lambda_{min}(A_{0})<0<\Lambda_{SRB}(A_{0})$. Since $\mu_{SRB}(A_{0})$ is
equivalent to Lebesgue and whose support is $A_{0}$,
$m(\bigcup_{\mu\neq\mu_{SRB}(A_{0})}G_{\mu})=0$, where $m$ is the Lebesgue
measure on $N_{0}$. Clearly, $\mu_{SRB}(A_{0})$-almost all Lyapunov exponents
are non-zero. By these facts and statement (2) of Proposition 4, $A_{0}$ is a
chaotic saddle which verifies statement $(c)$.

We apply similar arguments to prove $(b)$ and $(d)$. Indeed, for $0\leq x\leq
1/2$, we have

$d_{(x,1)}F_{\beta_{1},\beta_{2}}=\begin{pmatrix}2&\quad 0\\\ 0&\quad
dg_{1,\beta_{1}}(1)\\\ \end{pmatrix}=\begin{pmatrix}2&\quad 0\\\
0&\quad\beta_{1}\\\ \end{pmatrix}$

and for $1/2<x\leq 1$, we have

$d_{(x,1)}F_{\beta_{1},\beta_{2}}=\begin{pmatrix}2&\quad 0\\\ 0&\quad
dg_{2,\beta_{2}}(1)\\\ \end{pmatrix}=\begin{pmatrix}2&\quad 0\\\ 0&\quad
2-\beta_{2}\\\ \end{pmatrix}$

where $g_{i,\beta_{i}}$, $i=1,2$, given by (11). Hence, the normal stability
index is computed as follows:

$\displaystyle\lambda_{\mu}(A_{1})=\Lambda_{\mu}(A_{1})=\int_{A_{1}\cap[0,1/2]}\log(\beta_{1})d\mu(x)+\int_{A_{1}\cap(1/2,1]}\log(2-\beta_{2})d\mu(x).$
(33)

As a consequence,

$\displaystyle\lambda_{\mu}(A_{1})=\Lambda_{\mu}(A_{1})=\mu(A_{1}\cap[0,1/2])\log(\beta_{1})+\mu(A_{1}\cap(1/2,1])\log(2-\beta_{2}).$
(34)

As above, $F_{\beta_{1},\beta_{2}}|_{A_{1}}$ has a Lebesgue-equivalent ergodic
invariant measure; this corresponds to the desired $\mu_{SRB}(A_{1})$. By this
fact and (34), for the attractor $A_{1}$,
$\Lambda_{SRB}(A_{1})=1/2\log(\beta_{1})+1/2\log(2-\beta_{2})$. Note that
$\Lambda_{SRB}(A_{1})=L_{\perp,\beta_{1},\beta_{2}}(1)$, where
$L_{\perp,\beta_{1},\beta_{2}}(1)$ is given by (25). Simple computations show
that, for $\beta_{1}<1/2$ or ($\beta_{1}\geq 1/2$ $\&$
$\beta_{2}>2-\frac{1}{\beta_{1}}$), $\Lambda_{SRB}(A_{1})<0$ and hence $A_{1}$
is a Milnor (essential) attractor. Take the invariant dirac measure $\nu_{1}$
supported on the fixed point at $(1,1)$. Using (34),
$\Lambda_{\nu_{1}}(A_{1})=\log(2-\beta_{2})$ which is positive. By this fact,
$0<\Lambda_{\nu_{1}}(A_{1})\leq\Lambda_{max}(A_{1})$. These computations show
that $\Lambda_{SRB}(A_{1})<0<\Lambda_{max}(A_{1})$. By argument applied in
$(a)$, we may find $\alpha>0$ such that $G_{\alpha}$ given by (26) is dense in
$A_{1}$. By these observations and statement (1) of Proposition 4, $A_{1}$ has
a locally riddled basin and the proof of $(b)$ is finished.

Let $\nu_{2}$ be the invariant dirac measure supported on the fixed point at
$(0,1)$. Using (34), $\Lambda_{\nu_{2}}(A_{1})=\log(\beta_{1})$ which is
negative. By this fact, $\lambda_{min}(A_{1})\leq\Lambda_{\nu_{2}}(A_{1})<0$.
Also, it is easy to see that for $\beta_{1}>1/2$,
$\beta_{2}<2-\frac{1}{\beta_{1}}$, $\Lambda_{SRB}(A_{1})>0$. By these facts,
if $\beta_{1}>1/2$, $\beta_{2}<2-\frac{1}{\beta_{1}}$, then
$\lambda_{min}(A_{1})<0<\Lambda_{SRB}(A_{1})$. Since $\mu_{SRB}(A_{1})$ is
equivalent to Lebesgue and whose support is $A_{1}$,
$m(\bigcup_{\mu\neq\mu_{SRB}(A_{1})}G_{\mu})=0$. By these facts and statement
(2) of Proposition 4, $A_{1}$ is a chaotic saddle which verifies statement
$(d)$. ∎

$\begin{array}[]{cc}\includegraphics[scale={0.4}]{9}\includegraphics[scale={0.4}]{10}\end{array}$

Figure 2. The basins of attraction for the attractors $A_{0}$ and $A_{1}$. The
blue region in each figure corresponds to the basin of attraction
$\mathcal{B}(A_{0})$ while the yellow region corresponds to the basin of
attraction $\mathcal{B}(A_{1})$. The left frame depicts the basins of
attraction of $A_{0}$ and $A_{1}$, for $\beta_{1}=0.4$, $\beta_{2}=0.5$. In
this case $A_{i}$, $i=0,1$, are Milnor attractors and $A_{0}$ has locally
riddled basin. The right frame shows the basins of attraction of $A_{0}$ and
$A_{1}$ for $\beta_{1}=0.3$, $\beta_{2}=0.5$. In this case $A_{0}$ is a
chaotic saddle and the basin $\mathcal{B}(A_{0})$ has zero measure, while
$A_{1}$ is a Milnor attractor.

$\begin{array}[]{cc}\includegraphics[scale={0.4}]{2}\includegraphics[scale={0.4}]{3-1_1}\end{array}$

Figure 3. The basins of attraction for the attractors $A_{0}$ and $A_{1}$. The
blue region in each figure corresponds to the basin of attraction
$\mathcal{B}(A_{0})$ while the yellow region corresponds to the basin of
attraction $\mathcal{B}(A_{1})$. The left frame depicts the basins of
attraction of $A_{0}$ and $A_{1}$, for $\beta_{1}=0.7$, $\beta_{2}=0.65$. In
this case $A_{i}$, $i=0,1$, are Milnor attractors and $A_{1}$ has locally
riddled basin. The right frame shows the basins of attraction of $A_{0}$ and
$A_{1}$ for $\beta_{1}=0.6$, $\beta_{2}=0.25$. In this case $A_{1}$ is a
chaotic saddle and the basin $\mathcal{B}(A_{1})$ has zero measure, while
$A_{0}$ is a Milnor attractor.

Let

$\Gamma_{S}:=\\{(\beta_{1},\beta_{2}):\beta_{1},\beta_{2}\in(0,1),\
2-\frac{1}{\beta_{1}}<\beta_{2}<\frac{1}{1-\beta_{1}}-1\\}.$ (35)

If we set

$\displaystyle\Gamma_{\frac{1}{2}}=\\{(\beta_{1},\beta_{2}):\beta_{2}=\frac{1}{2},\quad\beta_{1}\in(\frac{1}{3},\frac{2}{3})\\}$
(36)

then $\Gamma_{\frac{1}{2}}\subset\Gamma_{S}$. Therefore, $\Gamma_{S}$ is a
nonempty open region.

######  Corollary 7.

For each $(\beta_{1},\beta_{2})\in\Gamma_{S}$, the both invariant sets $A_{0}$
and $A_{1}$ are Milnor attractors and have locally riddled basins.

$\begin{array}[]{c}\includegraphics[scale={0.5}]{ss}\end{array}$

Figure 4. The open region $\Gamma_{S}$, where for each
$(\beta_{1},\beta_{2})\in\Gamma_{S}$, the both invariant sets $A_{0}$ and
$A_{1}$ are Milnor attractors.

We extend the above results to the general case. Let
$F_{\beta_{1},\beta_{2}}\in\mathcal{F}$ be a skew product of the form (1)
whose fiber maps $g_{i,\beta_{1},\beta_{2}}$, $i=1,2$, satisfy conditions
$(I1)$-$(I3)$. For $j=0,1$, we take

$\alpha_{\beta_{1}\beta_{2}}^{j}:=dg_{1,\beta_{1}}(j)dg_{2,\beta_{2}}(j),$
(37)

where $dg_{i,\beta_{i}}(j)$, $i=1,2$, are the derivatives of the fiber maps
$g_{i,\beta_{i}}$ at the point $j$.

######  Theorem 8.

Let $F_{\beta_{1},\beta_{2}}\in\mathcal{F}$ be a skew product of the form (1)
whose fiber maps $g_{i,\beta_{1},\beta_{2}}$, $i=1,2$, satisfy conditions
$(I1)$-$(I3)$. Then the following statements hold:

  1. $(a)$

If there exists $\beta_{1}^{r}\in(0,1)$ such that for each
$\beta_{1}<\beta_{1}^{r}$, the sign of $\log(\alpha_{\beta_{1}\beta_{2}}^{0})$
changes from negative to positive by varying $\beta_{2}$, then, there exists a
smooth function $\beta_{1}\mapsto\xi(\beta_{1})$ with $\xi(\beta_{1}^{r})=1$
such that

    1. $(i)$

if $\beta_{1}<\beta_{1}^{r}$ and $\beta_{2}<\xi(\beta_{1})$, then $A_{0}$ is a
Milnor (essential) attractor and has a locally riddled basin,

    2. $(ii)$

if $\beta_{1}<\beta_{1}^{c}$ and $\beta_{2}>\xi(\beta_{1})$, then $A_{0}$ is a
chaotic saddle;

  2. $(b)$

If there exists $\beta_{1}^{\ell}\in(0,1)$ such that for each
$\beta_{1}>\beta_{1}^{\ell}$, the sign of
$\log(\alpha_{\beta_{1}\beta_{2}}^{1})$ changes from positive to negative by
varying $\beta_{2}$, then, there exists a smooth function
$\beta_{1}\mapsto\zeta(\beta_{1})$ with $\zeta(\beta_{1}^{\ell})=0$ such that

    1. $(i)$

if $\beta_{1}>\beta_{1}^{\ell}$ and $\beta_{2}>\zeta(\beta_{1})$, then $A_{1}$
is a Milnor (essential) attractor and has a locally riddled basin;

    2. $(ii)$

if $\beta_{1}>\beta_{1}^{\ell}$ and $\beta_{2}<\zeta(\beta_{1})$, then $A_{1}$
is a chaotic saddle.

###### Proof.

To prove, we closely follow the proof of Theorem 6 and omit some details. As
we have seen before there is only one normal direction, so the normal
stability index $\Lambda_{\mu^{i}}$, $i=0,1$, of an ergodic invariant
probability measure
$\mu^{i}\in\mathcal{E}_{F_{\beta_{1},\beta_{2}}|_{N_{i}}}(A_{i})$ is equal to
the normal Lyapunov exponent $\lambda_{{\mu}^{i}}$.

For $0\leq x\leq 1/2$,

$d_{(x,y)}F_{\beta_{1},\beta_{2}}=\begin{pmatrix}2&\quad 0\\\ 0&\quad
dg_{1,\beta_{1}}(y)\\\ \end{pmatrix}$

and for $1/2<x\leq 1$,

$d_{(x,y)}F_{\beta_{1},\beta_{2}}=\begin{pmatrix}2&\quad 0\\\ 0&\quad
dg_{2,\beta_{2}}(y)\\\ \end{pmatrix}$

where $g_{i,\beta_{i}}$, $i=1,2$, are the fiber maps of
$F_{\beta_{1},\beta_{2}}$.

Hence, the normal stability index $\Lambda_{\mu^{i}}$ for the attractor
$A_{i}$, $i=0,1$, is computed as follows:

$\displaystyle\lambda_{\mu^{i}}=\Lambda_{{\mu}^{i}}=\int_{A_{i}\cap[0,1/2]}\log(dg_{1,\beta_{1}}(i))d\mu^{i}(x)+\int_{A_{i}\cap(1/2,1]}\log(dg_{2,\beta_{2}}(i))d\mu^{i}(x).$
(38)

Therefore,

$\displaystyle\lambda_{\mu^{i}}=\Lambda_{{\mu}^{i}}=\mu^{i}(A_{i}\cap[0,1/2])\log(dg_{1,\beta_{1}}(i))+\mu^{i}(A_{i}\cap(1/2,1])\log(dg_{2,\beta_{2}}(i)).$
(39)

Note that $F_{\beta_{1},\beta_{2}}|_{A_{i}}$, $i=0,1$, is piecewise expanding,
hence, it has a Lebesgue-equivalent ergodic invariant measure which is the
desired $\mu_{SRB}^{i}$. By this fact and (39),

$\Lambda_{SRB}^{i}=1/2\log(dg_{1,\beta_{1}}(i))+1/2\log(dg_{2,\beta_{2}}(i)).$

By (39), for each invariant measure $\mu^{i}$, $\Lambda_{{\mu}^{i}}$ is
smoothly dependent on the normal parameters $\beta_{1}$ and $\beta_{2}$. This
fact and the hypothesis of statements $(a)$ and $(b)$ imply that there exist
smooth functions $\beta_{1}\mapsto\xi(\beta_{1})$ and
$\beta_{1}\mapsto\zeta(\beta_{1})$ such that they satisfy the following
properties:

$\Lambda_{SRB}^{o}<0\quad\text{if}\quad\beta_{1}<\beta_{1}^{r}\ \&\
\beta_{2}<\xi(\beta_{1}),\quad\text{and}\quad\Lambda_{SRB}^{o}>0\quad\text{if}\quad\beta_{1}<\beta_{1}^{r}\
\&\ \beta_{2}>\xi(\beta_{1}),$ (40)
$\Lambda_{SRB}^{1}<1\quad\text{if}\quad\beta_{1}>\beta_{1}^{\ell}\ \&\
\beta_{2}>\zeta(\beta_{1}),\quad\text{and}\quad\Lambda_{SRB}^{1}>0\quad\text{if}\quad\beta_{1}>\beta_{1}^{\ell}\
\&\ \beta_{2}<\zeta(\beta_{1}).$ (41)

Take the invariant dirac measure $\nu_{1}^{0}$ supported on the fixed point at
$(1,0)$. Using (39), $\Lambda_{\nu_{1}^{0}}=\log(dg_{2,\beta_{2}}(0))$ which
is positive by condition $(I3)$. By this fact,
$0<\Lambda_{\nu_{1}^{0}}\leq\Lambda_{max}^{0}$. These computations show that
$\Lambda_{SRB}^{0}<0<\Lambda_{max}^{0}$ for the attractor $A_{0}$. Using (39)
and Lemma 4, we apply the argument used in the proof of Theorem 6 to find
$\alpha^{0}>0$ such that the subset $G_{\alpha^{0}}$ given by (26) is dense in
$A_{0}$. By these observations and statement (1) of Proposition 4, $A_{0}$ has
a locally riddled basin and the proof of the first statement of $(a)$ is
finished.

Similarly, we take the invariant dirac measure $\nu_{0}^{1}$ supported on the
fixed point at $(0,1)$. Using (39),
$\Lambda_{\nu_{0}^{1}}=\log(dg_{1,\beta_{1}}(1))$ which is positive by
condition $(I2)$. By this fact,
$0<\Lambda_{\nu_{0}^{1}}\leq\Lambda_{max}^{1}$. These computations show that
$\Lambda_{SRB}^{1}<0<\Lambda_{max}^{1}$. As above, using Lemma 4 and (39), we
can apply the argument used in the proof of Theorem 6 to find $\alpha^{1}>0$
such that the subset $G_{\alpha^{1}}$ given by (26) is dense in $A_{1}$. By
these observations and statement (1) of Proposition 4, $A_{1}$ has a locally
riddled basin and the proof of the first statement of $(b)$ is finished.

Let $\nu_{0}^{0}$ be the invariant dirac measure supported on the fixed point
at $(0,0)$. Using (39), $\Lambda_{\nu_{0}^{0}}=\log(dg_{1,\beta_{1}}(0))$
which is negative by condition $(I2)$. By this fact,
$\lambda_{min}^{0}\leq\Lambda_{\nu_{0}^{0}}<0$. Also, by (40) for
$\beta_{2}>\xi(\beta_{1})$, $\Lambda_{SRB}^{0}>0$. By these facts, for
$\beta_{2}>\xi(\beta_{1})$, we get $\lambda_{min}^{0}<0<\Lambda_{SRB}^{0}$.
Since $\mu_{SRB}^{0}$ is equivalent to Lebesgue and whose support is $A_{0}$,
$m(\bigcup_{\mu\neq\mu_{SRB}^{0}}G_{\mu})=0$, where $m$ is the Lebesgue
measure on $N_{0}$. Clearly, $\mu_{SRB}^{0}$-almost all Lyapunov exponents are
non-zero. By these facts and statement (2) of Proposition 4, $A_{0}$ is a
chaotic saddle which verifies the second statement of $(a)$.

Let $\nu_{0}^{1}$ be the invariant dirac measure supported on the fixed point
at $(0,1)$. Using (34), $\Lambda_{\nu_{0}^{1}}=\log(dg_{1,\beta_{1}}(0))$
which is negative by condition $(I2)$. By this fact,
$\lambda_{min}^{1}\leq\Lambda_{\mu_{0}^{1}}<0$. Also, by (41), for
$\beta_{2}<\zeta(\beta_{1})$, $\Lambda_{SRB}^{1}>0$. By these facts, if
$\beta_{2}>\zeta(\beta_{1})$, then $\lambda_{min}^{1}<0<\Lambda_{SRB}^{1}$.
Since $\mu_{SRB}^{1}$ is equivalent to Lebesgue and whose support is $A_{1}$,
$m(\bigcup_{\mu\neq\mu_{SRB}^{1}}G_{\mu})=0$. By these facts and statement (2)
of Proposition 4, $A_{1}$ is a chaotic saddle which verifies the second
statement of $(b)$. ∎

Let

$\Gamma_{G}:=\\{(\beta_{1},\beta_{2}):\beta_{1},\beta_{2}\in(0,1),\quad\zeta(\beta_{2})<\beta_{2}<\xi(\beta_{1})\\}.$
(42)

Then, $\Gamma_{S}\subset\Gamma_{G}$, and hence, $\Gamma_{G}$ is a nonempty
open region in $\beta_{1}\beta_{2}$-plane.

######  Corollary 9.

For each $(\beta_{1},\beta_{2})\in\Gamma_{G}$, the both invariant sets $A_{0}$
and $A_{1}$ are Milnor attractors and have locally riddled basins.

##  4\. Random walk model

In this section, we introduce a random walk model which is topologically semi-
conjugate to the skew product system $F_{\beta_{1},\beta_{2}}$ given by (1).
This allows us to define a fractal boundary between the initial conditions
leading to each of the two attractors $A_{1}$ and $A_{2}$.

######  Definition 10.

Let $\mathcal{S}$ be the set of step skew product systems of the form

$G_{\beta_{1},\beta_{2}}^{+}:\Sigma_{2}^{+}\times\mathbb{I}\to\Sigma_{2}^{+}\times\mathbb{I},\
\
G_{\beta_{1},\beta_{2}}^{+}(\omega,y)=(\sigma\omega,g_{\omega_{0},\beta_{\omega_{0}}}(y)),$
(43)

where $\mathbb{I}=[0,1]$, $\beta_{1},\beta_{2}\in(0,1)$ such that the fiber
maps $g_{1,\beta_{1}}$ and $g_{2,\beta_{2}}$ are strictly increasing $C^{2}$
diffeomorphisms given by (7) satisfying conditions $(I1)-(I3)$.

We will pick the diffeomorphisms $g_{1,\beta_{1}}$ and $g_{2,\beta_{2}}$
randomly, independently at each iterate, with positive probabilities $p_{1}$
and $p_{2}=1-p_{1}$. This corresponds to taking a Bernoulli measure on
$\Sigma_{2}^{+}$ from which we pick $\omega$. The obtained random compositions

$g_{\omega,\beta_{1},\beta_{2}}^{n}(y)=g_{\omega_{n-1},\beta_{\omega_{n-1}}}\circ\dots\circ
g_{\omega_{0},\beta_{\omega_{0}}}(y),\quad\text{for}\ n\geq 1,\quad
g_{\omega,\beta_{1},\beta_{2}}^{0}(y)=y$ (44)

form a random walk on the interval.

We define the _standard measure_ $s$ on $\Sigma_{2}^{+}\times\mathbb{I}$ by
the product of Bernoulli measure $\mathbb{P}^{+}$ and the Lebesgue measure on
$\mathbb{I}$.

Given a skew product $G_{\beta_{1},\beta_{2}}^{+}\in\mathcal{S}$, the _normal
Lyapunov exponent_ of $G_{\beta_{1},\beta_{2}}^{+}$ at a point
$(\omega,y)\in\Sigma_{2}^{+}\times\mathbb{I}$ is

$\lim_{n\to\infty}\frac{1}{n}\ln(g^{\prime}_{\omega_{n-1},\beta_{\omega_{n-1}}}(g_{\omega,\beta_{1},\beta_{2}}^{n-1}(y))\dots
g_{\omega_{0},\beta_{\omega_{0}}}^{\prime}(y))=\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\ln(g_{\omega_{i},\beta_{\omega_{i}}}^{\prime}(g_{\omega,\beta_{1},\beta_{2}}^{i}(y))),$
(45)

in case the limit exists. Since $y=0,1$ are fixed points of $g_{i,\beta_{i}}$,
$i=1,2$, by Birkhoff’s ergodic theorem, we obtain for $y=0,1$ that

$L_{\beta_{1},\beta_{2}}(y)=\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\ln(g_{\omega_{i},\beta_{\omega_{i}}}^{\prime}(g_{\omega,\beta_{1},\beta_{2}}^{i}(y)))=\int_{\Sigma_{2}^{+}}\ln(g_{\omega_{0},\beta_{\omega_{0}}}^{\prime}(y))d\nu^{+}(\omega)=\sum_{i=1}^{2}p_{i}\ln
g_{i,\beta_{i}}^{\prime}(y)$ (46)

for $\mathbb{P}^{+}$-almost all $\omega\in\Sigma_{2}^{+}$.

The subspaces

$\mathbb{A}_{0}:=\Sigma_{2}^{+}\times\\{0\\},\ \ \text{and}\ \
\mathbb{A}_{1}:=\Sigma_{2}^{+}\times\\{1\\}$

are invariants by $G^{+}_{\beta_{1},\beta_{2}}$, for each
$\beta_{1},\beta_{2}\in(0,1)$. The basins are

$\mathcal{B}_{\beta_{1},\beta_{2}}(\mathbb{A}_{0}):=\\{(\omega,y):d((G_{\beta_{1},\beta_{2}}^{+})^{n}(\omega,y),\mathbb{A}_{0})\to
0\ \text{as}\ n\to\infty\\},$
$\mathcal{B}_{\beta_{1},\beta_{2}}(\mathbb{A}_{1}):=\\{(\omega,y):d((G_{\beta_{1},\beta_{2}}^{+})^{n}(\omega,y),\mathbb{A}_{1})\to
0\ \text{as}\ n\to\infty\\},$

where $d(y,A)=\inf_{z\in A}\|y-z\|$, for each subset $A\subset\mathbb{I}$.

First, we consider a specific example of a step skew product system from
$\mathcal{S}$ with the fiber maps

$g_{1,\beta_{1}}(y)=y+y(1-y)(y-\beta_{1}),\ \ g_{2,1/2}(y)=y-y(1-y)(y-1/2),$
(47)

with $\beta_{2}=\frac{1}{2}$ and illustrate time series of random walks for
some values of $\beta_{1}$. Using (46), we get

$L_{\beta_{1},1/2}(0)=1/2\ln(1-\beta_{1})+1/2/ln3/2,\ \text{and}\
L_{\beta_{1},1/2}(1)=1/2\ln\beta_{1}+1/2/ln3/2.$ (48)

Note that $L_{\beta_{1},1/2}(0)<0$ and $L_{\beta_{1},1/2}(1)<0$ for each
$1/3<\beta_{1}<2/3$.

$\begin{array}[]{cc}\includegraphics[scale={0.4}]{fbeta4}\includegraphics[scale={0.4}]{32}\end{array}$

Figure 5. The left frame depicts the graphs of $g_{1,\beta_{1}}$, $g_{2,1/2}$,
for $\beta_{1}=2/5$. The right frame shows a time series of the random walk
generated by $g_{1,\beta_{1}}$, $g_{2,1/2}$, for $\beta_{1}=2/5$, both picked
with probability $1/2$.

$\begin{array}[]{cc}\includegraphics[scale={0.4}]{fbeta5}\includegraphics[scale={0.4}]{22}\end{array}$

Figure 6. The left frame depicts the graphs of $g_{1,\beta_{1}}$, $g_{2,1/2}$,
for $\beta_{1}=1/2$. The right frame shows a time series of the random walk
generated by $g_{1,\beta_{1}}$, $g_{2,1/2}$, for $\beta_{1}=1/2$, both picked
with probability $1/2$.

$\begin{array}[]{cc}\includegraphics[scale={0.4}]{fbeta6}\includegraphics[scale={0.4}]{11}\end{array}$

Figure 7. The left frame depicts the graphs of $g_{1,\beta_{1}}$, $g_{2,1/2}$,
for $\beta_{1}=3/5$. The right frame shows a time series of the random walk
generated by $g_{1,\beta_{1}}$, $g_{2,1/2}$, for $\beta_{1}=3/5$, both picked
with probability $1/2$.

The graphs of $g_{1,\beta_{1}}$, $g_{2,1/2}$ are illustrated in the left
frames of Fig. 5, Fig. 6 and Fig. 7, for parameter values
$\beta_{1}=2/5,1/2,3/5$, respectively. The right panels of these figures show
time series of the random walk generated by $g_{1,\beta_{1}}$, $g_{2,1/2}$,
for $\beta_{1}=2/5,1/2,2/3$, both picked with probability $1/2$. For these
values of $\beta_{1}$, the normal Lyapunov exponents at $0$ and $1$ are
negative.

######  Theorem 11.

Let $G^{+}_{\beta_{1},\beta_{2}}\in\mathcal{S}$ be a step skew product whose
fiber maps defined by (11) and $(\beta_{1},\beta_{2})\in\Gamma_{S}$, where
$\Gamma_{S}$ is given by (35). Then, the sets $\mathbb{A}_{0}$ and
$\mathbb{A}_{1}$ are Milnor attractors for $G^{+}_{\beta_{1},\beta_{2}}$. They
attract sets of positive standard measure and the union of the basins
$\mathcal{B}_{\beta_{1},\beta_{2}}(\mathbb{A}_{i})$, $i=0,1$, has full
standard measure.

###### Proof.

By (46) and the definition of $\Gamma_{S}$, $L_{\beta_{1},\beta_{2}}(0)<0$ and
$L_{\beta_{1},\beta_{2}}(1)<0$.

######  Lemma 12.

Let $(\beta_{1},\beta_{2})\in\Gamma_{S}$ and take

$r_{\beta_{1},\beta_{2}}(\omega)=\sup\\{y\in\mathbb{I}:\lim_{n\to\infty}g_{\omega,\beta_{1},\beta_{2}}^{n}(y)=0\\},\quad
s_{\beta_{1},\beta_{2}}(\omega)=\inf\\{y\in\mathbb{I}:\lim_{n\to\infty}g_{\omega,\beta_{1},\beta_{2}}^{n}(y)=1\\},$

where $g_{\omega,\beta_{1},\beta_{2}}^{n}(y)$ is given by (44). Then
$r_{\beta_{1},\beta_{2}}(\omega)>0$ and $s_{\beta_{1},\beta_{2}}(\omega)<1$
for $\mathbb{P}^{+}$ almost all $\omega\in\Sigma_{2}^{+}$.

###### Proof.

The lemma is a counterpart of [Gharaei & Homburg(2017), Lemma 3.1] to our
setting. In the proof of [Gharaei & Homburg(2017), Lemma 3.1], the authors
only uses the fact that the normal Lyapunov exponents at the point 0 and 1 are
negative and applying Birkhoff’s ergodic theorem. Since
$L_{\beta_{1},\beta_{2}}(0)<0$ and $L_{\beta_{1},\beta_{2}}(1)<0$, the lemma
is proved. ∎

Since, by the previous lemma, for each $(\beta_{1},\beta_{2})\in\Gamma_{S}$,
the function $r_{\beta_{1},\beta_{2}}$ is positive almost everywhere, thus,
the basin $\mathcal{B}_{\beta_{1},\beta_{2}}(\mathbb{A}_{0})$ has positive
standard measure. Since, the function $s_{\beta_{1},\beta_{2}}$ is less than 1
almost everywhere, the same holds for the basin of
$\mathcal{B}_{\beta_{1},\beta_{2}}(\mathbb{A}_{1})$. By these facts, the
basins $\mathcal{B}_{\beta_{1},\beta_{2}}(\mathbb{A}_{i})$ of
$G_{\beta_{1},\beta_{2}}^{+}$-invariant sets $\mathbb{A}_{i}$, $i=0,1$, have
positive standard measures. In particular, they are Milnor attractors. This
proves the first statement of the theorem.

Write $\mathcal{P}$ for the space of probability measures on $\mathbb{I}$,
equipped with the weak star topology and define
$\mathcal{T}:\mathcal{P}\to\mathcal{P}$ by

$\mathcal{T}m=\sum_{i=1}^{2}p_{i}g_{i,\beta_{i}}m.$

We recall that a stationary measure is a fixed point of $\mathcal{T}$. Note
that a probability measure $m$ is a stationary measure if and only if
$\mu^{+}=\mathbb{P}^{+}\times m$ is an invariant measure of
$G_{\beta_{1},\beta_{2}}^{+}$ with marginal $\mathbb{P}^{+}$ on
$G_{\beta_{1},\beta_{2}}^{+}$, see [Gharaei & Homburg(2017), Lemma A.2]. We
say that $m$ is ergodic if $\mathbb{P}^{+}\times m$ is ergodic for
$G_{\beta_{1},\beta_{2}}^{+}$.

To proceed proving Theorem 11, we prove that there is a boundary between
basins $\mathcal{B}_{\beta_{1},\beta_{2}}(\mathbb{A}_{0})$ and
$\mathcal{B}_{\beta_{1},\beta_{2}}(\mathbb{A}_{1})$. Indeed, we show that
there is an invariant measurable graph
$\psi_{\beta_{1},\beta_{2}}:\Sigma_{2}\to\mathbb{I}$ that separates the
basins: for $\mathbb{P}$-almost all $\omega$,

$\displaystyle\lim_{n\to\infty}g_{\omega,\beta_{1},\beta_{2}}^{n}(y)=\left\\{\begin{array}[]{cc}0&$
if$\quad y<\psi_{\beta_{1},\beta_{2}}(\omega),\\\ 1&$ if$\quad
y>\psi_{\beta_{1},\beta_{2}}(\omega)\end{array}\right.$ (51)

where $g_{\omega,\beta_{1},\beta_{2}}^{n}(y)$ is given by (44). To prove the
theorem, we closely follow [Gharaei & Homburg(2017), Theorem 3.1] and omit
some details. Let $H_{\beta_{1},\beta_{2}}^{+}$ be the skew product map whose
fiber maps are defined by $h_{i,\beta_{i}}=g_{i,\beta_{i}}^{-1}$. Then, it has
positive normal Lyapunov exponents along $\mathbb{A}_{0}$ and
$\mathbb{A}_{1}$. This means that $L_{\beta_{1},\beta_{2}}(0)>0$ and
$L_{\beta_{1},\beta_{2}}(1)>0$ for the skew product map
$H_{\beta_{1},\beta_{2}}^{+}$.

######  Lemma 13.

For the skew product $H_{\beta_{1},\beta_{2}}^{+}$ defined above, there exists
an ergodic stationary measure $m$ with $m(\\{0\\}\cup\\{1\\})=0$.

###### Proof.

The lemma is just a reformulation of [Gharaei & Homburg(2017), Lemma 3.2] in
our context. Note that to prove [Gharaei & Homburg(2017), Lemma 3.2] the
authors use only the fact that the normal Lyapunov exponents at both fixed
points $0$ and $1$ are positive. So the result holds for our setting. ∎

Take the extension skew product $H_{\beta_{1},\beta_{2}}$ of
$H_{\beta_{1},\beta_{2}}^{+}$ given by (20). Note that the stationary measure
$m$ gives an invariant measure $\mu_{m}$ for the extension skew product system
$H_{\beta_{1},\beta_{2}}$ (see [Arnold(1998)]). Its conditional measures on
fibers $\\{\omega\\}\times\mathbb{I}$ are denoted by $\mu_{m,\omega}$ (see
(21) and (22)). By [Gharaei & Homburg(2017), Lemma 3.3], the conditional
measure $\mu_{m,\omega}$ of $\mu_{m}$ is a $\delta$-measure for
$\mathbb{P}$-almost every $\omega\in\Sigma_{2}$.

By [Gharaei & Homburg(2017), Lemma 3.4], the stationary measure $m$ obtained
by Lemma 13 is unique. Thus, for each $(\beta_{1},\beta_{2})\in\Gamma_{S}$,
there is a unique stationary measure $m_{\beta_{1},\beta_{2}}$ with
$m_{\beta_{1},\beta_{2}}(\\{0\\}\cup\\{1\\})=0$. So, by these facts, there
exists a measurable function
$\psi_{\beta_{1},\beta_{2}}:\Sigma_{2}\to\mathbb{I}$ such that for
$\mathbb{P}$-almost all $\omega$,

$\lim_{n\to\infty}h^{n}_{\sigma^{-n}\omega,\beta_{1},\beta_{2}}m_{\beta_{1},\beta_{2}}=\delta_{\psi_{\beta_{1},\beta_{2}}(\omega)},$

where $h^{n}_{\sigma^{-n}\omega,\beta_{1},\beta_{2}}$ is given by (44). Note
that a function increasing if and only if its inverse is increasing. As the
convex hull of the support of $m_{\beta_{1},\beta_{2}}$ equals $\mathbb{I}$
and $h_{1,\beta_{1}}$, $h_{2,\beta_{2}}$ are increasing, this implies that for
every $y\in(0,1)$

$\lim_{n\to\infty}h^{n}_{\sigma^{-n}\omega,\beta_{1},\beta_{2}}(y)=\psi_{\beta_{1},\beta_{2}}(\omega).$

By the fact that $h_{1,\beta_{1}}$, $h_{2,\beta_{2}}$ are increasing, we get

$\lim_{n\to\infty}(h^{n}_{\sigma^{-n}\omega,\beta_{1},\beta_{2}})^{-1}(y)=1$

if $y>\psi_{\beta_{1},\beta_{2}}(\omega)$ and

$\lim_{n\to\infty}(h^{n}_{\sigma^{-n}\omega,\beta_{1},\beta_{2}})^{-1}(y)=0$

if $y<\psi_{\beta_{1},\beta_{2}}(\omega)$. Thus,

$\lim_{n\to\infty}g^{n}_{\omega,\beta_{1},\beta_{2}}(y)=1$

if $y>\psi_{\beta_{1},\beta_{2}}(\omega)$ and

$\lim_{n\to\infty}g^{n}_{\omega,\beta_{1},\beta_{2}}(y)=0$

if $y<\psi_{\beta_{1},\beta_{2}}(\omega)$.

This observation shows that the union of the basins of attraction of
$\mathbb{A}_{0}$ and $\mathbb{A}_{1}$ has full standard measure and hence,
Theorem 11 is proved. ∎

The following result describes intermingled basins for step skew product
systems $G^{+}_{\beta_{1},\beta_{2}}\in\mathcal{S}$.

######  Theorem 14.

Let $G^{+}_{\beta_{1},\beta_{2}}\in\mathcal{S}$ be a step skew product whose
fiber maps defined by (11) and $(\beta_{1},\beta_{2})\in\Gamma_{S}$, where
$\Gamma_{S}$ is given by (35). Let $\beta_{1}=\beta_{2}=\beta$. Then, the
invariant subset $\mathbb{A}_{\beta}=\Sigma_{2}^{+}\times\\{\beta\\}$ is a
Milnor attractor and the basins of $\mathbb{A}_{i}$ and $\mathbb{A}_{\beta}$
are intermingled, for each $i=0,1$.

###### Proof.

Let $(\beta_{1},\beta_{2})\in\Gamma_{S}$ and $\beta_{1}=\beta_{2}=\beta$.
Then, the subset $\mathbb{A}_{\beta}=\Sigma_{2}^{+}\times\\{\beta\\}$ is
invariant. By (46) and the definition of $\Gamma_{S}$, the normal Lyapunov
exponents at the points 0, 1 and $\beta$ are negative. Thus, by Theorem 11,
$\mathbb{A}_{\beta}$ is a Milnor attractor. Now, we take the restriction skew
products

$G^{+}_{\beta_{1},\beta_{2}}|_{\Sigma_{2}\times[0,\beta]}\quad\text{and}\quad
G^{+}_{\beta_{1},\beta_{2}}|_{\Sigma_{2}\times[\beta,1]}.$ (52)

The restriction skew products given by (52) satisfy the hypothesis of [Gharaei
& Homburg(2017), Theorem 3.1], thus, the basins of $\mathbb{A}_{i}$ and
$\mathbb{A}_{\beta}$ are intermingled, for each $i=0,1$. This finishes the
proof. ∎

Let us define $\tau:\mathbb{I}\to\\{1,2\\}$ by

$\displaystyle\tau(x)=\left\\{\begin{array}[]{cc}1&$ for$\quad 0\leq x\leq
1/2,\\\ 2&$ for$\quad 1/2<x\leq 1.\end{array}\right.$ (55)

Let $\Sigma_{12}\subset\Sigma_{2}$ be the sets of bi-infinite sequences of 1’s
and 2’s which do not end with a tail of 1s or 2s. The metric and measure is
inherited from the space $\Sigma_{2}$. Note that $\Sigma_{12}$ is closed and
invariant by the shift map. By [Gorodnik(2017)], there exists a map $K$ that
semi-conjugates the restrictions $\sigma|_{\Sigma_{12}}$ and
$f|_{\mathbb{I}\setminus D}$, where $D$ is the set of dyadic rationals (i.e.
rational numbers of the form $k/2^{n}$, whose denominator is a power of 2). By
definition, the subset $\Sigma_{12}$ has the full Bernoulli measure and $D$
has zero Lebesgue measure (see [Gorodnik(2017)]). Then, the composition of the
semi-conjugating map $K$ and the map $\psi_{\beta_{1},\beta_{2}}$ given by
(51) yields a measurable map $\Phi:\mathbb{I}\to\mathbb{I}$ that separates the
basins.

######  Corollary 15.

There exists a measurable graph map
$\Phi_{\beta_{1},\beta_{2}}:\mathbb{I}\to\mathbb{I}$ such that for Lebesgue-
almost all $x\in\mathbb{I}$,

$\displaystyle\lim_{n\to\infty}g_{x,\beta_{1},\beta_{2}}^{n}(y)=\left\\{\begin{array}[]{cc}0&$
if$\quad y<\Phi_{\beta_{1},\beta_{2}}(x),\\\ 1&$ if$\quad
y>\Phi_{\beta_{1},\beta_{2}}(x)\end{array}\right.$ (58)

where
$g_{x,\beta_{1},\beta_{2}}^{n}(y)=g_{\tau(f^{n-1}(x)),\beta_{\tau(f^{n-1}(x))}}\circ\dots\circ
g_{\tau(x),\beta_{\tau(x)}}(y)$.

The following is immediate from Theorem 11 and Corollary 15.

######  Corollary 16.

Let $F_{\beta_{1},\beta_{2}}\in\mathcal{F}$ be a skew product of the form (1)
whose fiber maps $g_{i,\beta_{1},\beta_{2}}$, $i=1,2$, given by (11). If both
normal Lyapunov exponents $L_{\bot,\beta_{1},\beta_{2}}(i)$, $i=0,1$, are
negative,then the attractors $A_{0}$ and $A_{1}$ attract sets of positive
Lebesgue measure and the union of the basins
$\mathcal{B}_{\beta_{1},\beta_{2}}(A_{i})$, $i=0,1$, has full Lebesgue
measure.

##  5\. The existence of riddled basin and blowout bifurcation

In this section, we demonstrate the emergence conditions of riddled basins in
the global sense.
Note that a general set of conditions under which riddled basins can occur are
as follows (see [Ott et al.(1993)], [Alexander et al.(1992)], [Cazelles(2001)]
and [Viana et al.(2009)]):

  1. $(H1)$

There exists an invariant subspace $N$ whose dimension is less than the
dimension of the full-phase space.

  2. $(H2)$

The dynamics on the invariant subspace $N$ has a chaotic attractor $A$.

  3. $(H3)$

For typical orbits on $A$ the Lyapunov exponents for infinitesimal
perturbations in the direction transverse to $N$ are all negative.

  4. $(H4)$

There is another attractor $A^{\prime}$ not belonging to $N$.

  5. $(H5)$

At least one of the normal Lyapunov exponents, although negative for almost
any orbits on $A$, has finite time fluctuations that are positive.

######  Remark 17.

Note that for occurrence the riddled basin, it is necessary to have a dense
set of points with zero Lebesgue measure in the attractor lying in the
invariant subspace which are transversely unstable, thus it is necessary that
this attractor be chaotic.

Let $F_{\beta_{1},\beta_{2}}\in\mathcal{F}$ be a skew product of the form (1)
whose fiber maps $g_{i,\beta_{1},\beta_{2}}$, $i=1,2$, given by (11) and
$\beta_{1},\beta_{2}\in(0,1)$. In Theorem 6, we estimated the values of
parameters $\beta_{i}$, $i=1,2$, for which the locally riddled basin occurs
for both attractors $A_{0}$ and $A_{1}$. In particular, for each
$(\beta_{1},\beta_{2})\in\Gamma_{S}$, the verification of condition
$(H1)-(H4)$ was done. Condition $(H5)$ is verified in the next theorem. To do
that we show the existence a set of unstable periodic orbits embedded in
$A_{i}$ which is transversely unstable. This implies that at least one of the
Lyapunov exponents along the directions transverse to invariant subspaces
experiences positive finite-type fluctuations (see [Viana et al.(2009)]).

Here, we prove the riddling in a more direct way. Indeed, in our setting, the
full space contains two chaotic attractors $A_{i}$, $i=0,1$, lying in
different invariant subspaces $N_{i}$. By Corollary 15, the system
$F_{\beta_{1},\beta_{2}}$ presents a complex fractal boundary between the
initial conditions leading to each of the two attractors. This fractal
boundary is the graph of $\Phi_{\beta_{1},\beta_{2}}$ which separates the
basin of attraction. Note that in a riddled basin, small variations in initial
conditions induce a switch between the different chaotic attractors but the
fractal boundary causes to predict, from a given initial condition, what
trajectory in phase-space the system will follow. These facts allow us to
verify the occurrence of riddled basin (in the global sense).

######  Theorem 18.

Let $F_{\beta_{1},\beta_{2}}\in\mathcal{F}$ be a skew product of the form (1)
whose fiber maps $g_{i,\beta_{1},\beta_{2}}$, $i=1,2$, given by (11) and let
$(\beta_{1},\beta_{2})\in\Gamma_{S}$, where $\Gamma_{S}$ is given by (35).
Consider the chaotic attractors $A_{i}$, $i=0,1$. Then the following holds:

  * $(a)$

if $\beta_{1}\leq 1/2$ and $\beta_{2}<\frac{1}{1-\beta_{1}}-1$ then
$\mathcal{B}(A_{0})$ is riddled with $\mathcal{B}(A_{1})$;

  * $(b)$

if $\beta_{1}\geq 1/2$ and $\beta_{2}>2-\frac{1}{\beta_{1}}$, then
$\mathcal{B}(A_{1})$ is riddled with $\mathcal{B}(A_{0})$;

where $\mathcal{B}(A_{i})$ is the basin of attraction of $A_{i}$, for $i=0,1$.

###### Proof.

As we have seen in Section 3, the two chaotic attractors $A_{i}$, $i=0,1$, are
$SRB$ attractors for the restriction of $F_{\beta_{1},\beta_{2}}$ to the
invariant subspaces $N_{i}$. Since $(\beta_{1},\beta_{2})\in\Gamma_{S}$, by
Corollary 7, both normal Lyapunov exponents $L_{\perp,\beta_{1},\beta_{2}}(0)$
and $L_{\perp,\beta_{1},\beta_{2}}(1)$ are negative. This fact ensures that
$A_{i}$, $i=0,1$, are (essential) attractors (in the Milnor sense) in the
whole phase space (see Theorem 6).

To prove $(a)$, consider the fixed point $Q=(1,0)\in A_{0}$. We closely follow
[Alexander et al.(1992), Section 4] and construct an open set near the fixed
point $Q$ which is not contained in the basin $\mathcal{B}(A_{0})$. Indeed,
consider the graph $x\mapsto(1-x)^{\log\frac{(1+\beta_{2})}{2}}$, a subset of
$A_{0}\times[0,1]$ having a cusp at $Q=(1,0)$. This graph is strictly monotone
and concave on each side of its cusp point. Since the mapping $y\mapsto
dg_{2,\beta_{2}}(y)$ is continuous, there exists a real
$\gamma(\beta_{1},\beta_{2})>0$ sufficiently close to 1 such that if we take

$W_{\beta_{1},\beta_{2}}^{+}=\\{(x,y)\in[0,1]\times[0,1]:\gamma(\beta_{1},\beta_{2})<x<1\
\text{and}\ y>(1-x)^{\log(1+\beta_{2})}\\}$

then for $(x,y)\in W_{\beta_{1},\beta_{2}}^{+}$, $dg_{2,\beta_{2}}(y)>1$. Thus
any point in $W_{\beta_{1},\beta_{2}}^{+}$ escapes from the attractor $A_{0}$.
By Corollary 16, the union of the basins
$\mathcal{B}_{\beta_{1},\beta_{2}}(A_{i})$, $i=0,1$, has full Lebesgue
measure. Also, the graph of the map $\Phi_{\beta_{1},\beta_{2}}$ obtained in
Corollary 15 is the fractal boundary which separates the basin of attraction.
By these facts, if $\gamma(\beta_{1},\beta_{2})$ is close enough to 1, for any
point $(x,y)\in W_{\beta_{1},\beta_{2}}^{+}$, there is $n>1$ such that
$g_{x,\beta_{1},\beta_{2}}^{n}(y)>\Phi_{\beta_{1},\beta_{2}}(x)$ and hence
$(x,y)$ escapes to $\mathcal{B}(A_{1})$ after some iterates.

Let $W_{\beta_{1},\beta_{2}}=\cup_{n\geq
0}F_{\beta_{1},\beta_{2}}^{-n}(W_{\beta_{1},\beta_{2}}^{+})$. Then
$W_{\beta_{1},\beta_{2}}$ is an open set and its boundary has a cusp at each
point of the set $\\{f^{-n}(1)\\}$. The set $\\{f^{-n}(1)\\}$ is dense in
$\mathbb{I}$ and thus every neighborhood of any point in $A_{0}$ intersects
$W_{\beta_{1},\beta_{2}}$. Thus the basin $\mathcal{B}(A_{0})$ is riddled with
the basin $\mathcal{B}(A_{1})$.

To prove ($b$), we construct an open set near the fixed point $Z=(1,1)\in
A_{1}$ which is not contained in the basin $\mathcal{B}(A_{1})$. Consider the
graph $x\mapsto x^{\log(2-\beta_{2})}$, a subset of $A_{1}\times[0,1]$ which
has a cusp at $Z=(1,1)$. This graph is strictly monotone and concave on each
side of its cusp point. Since the mapping $y\mapsto dg_{2,\beta_{2}}(y)$ is
continuous, there exists a real $\eta(\beta_{1},\beta_{2})>0$ sufficiently
close to 0 such that if we take

$U_{\beta_{1},\beta_{2}}^{+}=\\{(x,y)\in[0,1]\times[0,1]:0<x<\eta(\beta_{1},\beta_{2})\
\text{and}\ y>x^{log(2-\beta_{2})}\\},$

then for $(x,y)\in U^{+}$, $dg_{2,\beta_{2}}(y)>1$. Thus any point in
$U_{\beta_{1},\beta_{2}}^{+}$ escapes from the attractor $A_{1}$. By applying
an argument similar to statement ($a$), any point $(x,y)\in
U_{\beta_{1},\beta_{2}}^{+}$ escapes to $\mathcal{B}(A_{0})$ after some
iterates. Let $U=\cup_{n\geq 0}F^{-n}(U^{+})$. Then $U$ is an open set and its
boundary has a cusp at each point of the set $\\{f^{-n}(1)\\}$. The set
$\\{f^{-n}(1)\\}$ is dense in $[0,1]$ and thus every neighborhood of any point
in $A_{1}$ intersects $U$. Thus, $\mathcal{B}(A_{1})$ is riddled with the
basin $\mathcal{B}(A_{0})$. ∎

$\begin{array}[]{cc}\includegraphics[scale={0.4}]{f3_2.eps}\includegraphics[scale={0.4}]{f4}\end{array}$

Figure 8. The left frame depicts the basins of attraction of $A_{0}$ and
$A_{1}$, for $\beta_{1}=.3$ and $\beta_{2}=.42$. The basin of $A_{0}$ is
riddled by the basin of $A_{1}$. The right frame shows the basins of
attraction of $A_{0}$ and $A_{1}$ for $\beta_{1}=.7$ and $\beta_{2}=.58$. The
basin of $A_{1}$ is riddled by the basin of $A_{0}$. The blue region in both
figures corresponds to the basin of attraction $\mathcal{B}(A_{0})$, while the
yellow region corresponds to the basin of attraction $\mathcal{B}(A_{1})$.

Let $(\beta_{1},\beta_{2})\in\Gamma_{S}$. Take $\beta:=\beta_{1}=\beta_{2}$
and

$N_{\beta}:=\\{(x,y):\quad 0\leq x\leq 1,\quad y=\beta\\}.$ (59)

Then $N_{\beta}$ is an invariant subspace by $F_{\beta,\beta}$ and the
restriction of $F_{\beta,\beta}$ to this invariant subspace possesses a
chaotic attractor $A_{\beta}$.

The normal Lyapunov exponent for $F_{\beta,\beta}$ is given by

$\displaystyle
L_{\perp,\beta,\beta}(y)=\frac{1}{2}\ln(-\beta^{2}+\beta+1)+\frac{1}{2}\ln(\beta^{2}-\beta+1).$
(60)

Simple computation shows that $L_{\perp,\beta,\beta}$ is negative for each
$\beta\in(0,1)$. Note that the intervals $[0,\beta]$ and $[\beta,1]$ are
invariants by both fiber maps $g_{1,\beta}$ and $g_{2,\beta}$. By these facts,
Theorem 11 and the comments before Corollary 15, we get the next result.

######  Corollary 19.

$F_{\beta,\beta}$ exhibit three Milnor attractors $A_{0}$, $A_{1}$ and
$A_{\beta}$. Moreover, the basins of $A_{i}$ and $A_{\beta}$ are intermingled,
for each $i=0,1$.

$\begin{array}[]{c}\includegraphics[scale={0.5}]{int}\end{array}$

Figure 9. The basins of attraction for the attractors $A_{0}$, $A_{1}$ and
$A_{\beta}$, for parameters values $\beta_{1}=\beta_{2}=\beta=0.5$. The blue
region in the figure corresponds to the basin of attraction
$\mathcal{B}(A_{0})$, the green region in the figure corresponds to the basin
of attraction $\mathcal{B}(A_{\beta})$, while yellow region corresponds to the
basin of attraction $\mathcal{B}(A_{1})$. The intermingled basin is observed.

We recall that the blowout bifurcation [Ott & Sommer(1994), Ashwin et
al.(1996)] occurs when $L_{\perp,\beta_{1},\beta_{2}}$ crosses zero.

######  Corollary 20.

Let $F_{\beta_{1},\beta_{2}}\in\mathcal{F}$ be a skew product of the form (1)
whose fiber maps $g_{i,\beta_{1},\beta_{2}}$, $i=1,2$, given by (11). Then the
following holds:

  * $(a)$

$F_{\beta_{1},\beta_{2}}$ exhibits a (subcritical) hysteretic blowout
bifurcation on passing through any $\beta_{1}<1/2$ and
$\beta_{2}=\frac{1}{1-\beta_{1}}-1$;

  * $(b)$

$F_{\beta_{1},\beta_{2}}$ exhibits a (subcritical) hysteretic blowout
bifurcation on passing through any $\beta_{1}>1/2$ and
$\beta_{2}=2-\frac{1}{\beta_{1}}$.

###### Proof.

By the proof of Theorem 6, for each $(\beta_{1},\beta_{2})\in\Gamma_{S}$, the
normal Lyapunov exponents $L_{\perp,\beta_{1},\beta_{2}}(i)$, $i=0,1$, is
smoothly dependent on the normal parameters $\beta_{1}$ and $\beta_{2}$. By
this fact and Theorem 6, $L_{\perp,\beta_{1},\beta_{2}}(i)$ crosses zero at
any $\beta_{1}<1/2$ and $\beta_{2}=\frac{1}{1-\beta_{1}}-1$ and at any
$\beta_{1}>1/2$ and $\beta_{2}=2-\frac{1}{\beta_{1}}$. Thus, by definition of
a hysteretic blowout bifurcation and Theorem 6, the result is immediate. ∎

##  6\. Conclusion

We have investigated the formation of locally riddled basins of attraction and
chaotic saddle for a two parameter family $F_{\beta_{1},\beta_{2}}$,
$\beta_{1},\beta_{2}\in(0,1)$, of skew product systems defined on the plane.
Our model exhibits two distinct chaotic attractors $A_{0}$ and $A_{1}$ lying
in two different invariant subspaces. We have analyzed the model rigorously
using the results of [Ashwin et al.(1996)] and estimated the range of values
of parameters $\beta_{i}$, $i=1,2$, such that the attractor $A_{0}$ or $A_{1}$
has a locally riddled basin, or becomes a chaotic saddle. Then by varying the
parameters $\beta_{i}$, $i=1,2$, in an open region in the
$\beta_{1}\beta_{2}$-plane, we have shown the occurrence of riddled basin (in
the global sense) and hysteretic blowout bifurcation. To prove the riddling
basin, we have semi-conjugated the system to a random walk model and provided
a complex fractal boundary between the initial conditions leading to each of
the two attractors. This boundary separates the basin of attraction and causes
to predict, from a given initial condition, what trajectory in phase-space the
system will follow. Moreover, it was shown that, by varying the parameters in
an open region, a new chaotic attractor appears when $\beta_{1}=\beta_{2}$.
Also, the basin of this new attractor intermingled with the basins of both
attractors $A_{0}$ and $A_{1}$. Numerical simulations were presented
graphically to confirm the validity of our results.

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