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\begin{document}
\title{The discrete hyperchaotic double scroll}
\author{Zeraoulia Elhadj$^{1}$, J. C. Sprott$^{2}$ \\
%EndAName
$^{1}$Department of Mathematics, University of T\'{e}b\'{e}ssa, (12000),
Algeria.\\
E-mail: zeraoulia@mail.univ-tebessa.dz and zelhadj12@yahoo.fr.\\
$^{2}$ Department of Physics, University of Wisconsin, Madison, WI 53706,
USA.\\
E-mail: sprott@physics.wisc.edu.}
\maketitle
\begin{abstract}
In this paper we present and analyze a new piecewise linear map of
the plane capable of generating chaotic attractors with one and
two scrolls. Due to the shape of the attractor and its
hyperchaoticity, we call it the ``discrete hyperchaotic double
scroll." It has the same nonlinearity as used in the well-known
Chua circuit. A rigorous proof of the hyperchaoticity of this
attractor is given and numerically justified.
\end{abstract}
\textit{Keywords:} piecewise linear map, border collision
bifurcation, discrete hyperchaotic double scroll.
\bigskip PACS numbers: 05.45.-a, 05.45.Gg.
\section{\textbf{Introduction}}
It is well known that if two or more Lyapunov exponents of a
dynamical system are positive throughout a range of parameter
space, then the resulting attractors are hyperchaotic. The
importance of these attractors is that they are less regular and
are seemingly ``almost full'' in space, which explains their
importance in fluid mixing [10-11-12-13]. On the other hand, the
attractors generated by Chua's circuit [1] given by
$\dot{x}=\alpha \left( y-h\left( x\right) \right)
,\dot{y}=x-y+z,\mathring{z} =-\beta y$ are associated with
saddle-focus homoclinic loops and are not hyperchaotic, where
$h(x)=\frac{2m_{1}x+(m_{0}-m_{1})\left( \left\vert x+1\right\vert
-\left\vert x-1\right\vert \right) }{2}.$ The double scroll
attractor for this case is shown in Fig. 1.
The double scroll is more complex than the Lorenz-type and the
hyperbolic attractors [14], and thus it is not suitable for some
potential applications of chaos such as secure communications and
signal masking [2-3]. Hyperchaotic attractors make robust tools
for some applications, but this circuit does not exhibit
hyperchaos because of its limited dimensionality [1]. To resolve
this problem, several works have focused on the
hyperchaotification of Chua's circuit using several techniques
such as coupling many Chua circuits as in [2] where a 15-D
dynamical system is obtained. However, the resulting system is
complicated and difficult to construct. A simpler method
introduces an additional inductor in the canonical Chua circuit as
given in [3], where a 4-D dynamical system is obtained that
converges to a hyperchaotic attractor by a border collision
bifurcation [8]. On the other hand, the study of piecewise linear
maps [4-5-6-7] can contribute to the development of the theory of
dynamical systems, especially in finding new chaotic attractors
with applications in science and engineering [10-11]. Furthermore,
the techniques employed in the circuit realization of smooth maps
are simple, and the approach can be extended to other systems such
as piecewise linear or piecewise smooth maps [15]. Also, it seems
that the circuit realizations of low-dimensional maps is simpler
than with high-dimensional continuous systems. For this reason, we
present a discrete version of Chua's circuit attractor governed by
a simple 2-D piecewise linear map that is capable of producing
hyperchaotic attractors with the same shape as the classic double
scroll attractor, which is not hyperchaotic. We analytically show
the hyperchaoticity of the attractor and numerically show that the
proposed map behaves in a similar way to the 4-D dynamical system
given in [3], i.e., both hyperchaotic attractors are obtained by a
border collision bifurcation.
\FRAME{ftbpFU}{241.6875pt%
}{174.0625pt}{0pt}{\Qcb{The classic double scroll attractor
obtained for $\protect\alpha =9.35,$ $\protect\beta =14.79,$
$m_{0}=-\frac{1}{7},m_{1}=\frac{2}{7}$
[1].}}{}{Figure}{\special{language "Scientific Word";type
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\section{The Discrete hyperchaotic\textit{\ }Double scroll map}
In this section, we present the new map and show some of its basic
properties.
Consider the following 2-D piecewise linear map:
\begin{equation}
f\left( x,y\right) =\left(
\begin{array}{c}
x-ah\left( y\right) \\
bx%
\end{array}%
\right)
\end{equation}%
where $a$ and $b$ are the bifurcation parameters, $h$ is given
above by the characteristic function of the so-called double
scroll attractor [1], and $m_{0}$ and $m_{1}$ are respectively the
slopes of the inner and outer sets of the original Chua circuit.
Systems such as the one in Eq. (1) typically have no direct
application to particular physical systems, but they serve to
exemplify the kinds of dynamical behaviors, such as routes to
chaos, that are common in physical chaotic systems. Thus an
analytical and numerical study is warranted. Due to the shape of
the new attractor and its hyperchaoticity, we call it the
``discrete hyperchaotic double scroll'' because of its similarity
to the well-known Chua circuit [1].
One of the advantages of the map (1) is its extreme simplicity and
minimality in view of the number of terms and conservation of some
important properties of the classic double scroll. Firstly, the
associated function $f\left(x,y\right)$ is continuous in $
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}$, but it is not differentiable at the points $(x,-1)$ and $(x,1)$ for all $%
x\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
.$ Secondly, the map (1) is a diffeomorphism when
$abm_{1}m_{0}\neq 0$, since the determinant of its Jacobian is
nonzero if and only if $abm_{1}\neq 0$ or $abm_{0}\neq 0$, but it
does not preserve area and it is not a reversing twist map for all
values of the system parameters. Thirdly, the map (1) is symmetric
under the coordinate transformion $(x,y)\longrightarrow (-x,-y)$,
and this transformation persists for all values of the system
parameters. Therefore, the chaotic attractor obtained for map (1)
is symmetric just like the classic double scroll [1]. On the other
hand, and due to the shape of the vector field $f$ of the map (1),
the plane can be divided into three linear regions denoted by:
$R_{1}=\left\{ (x,y)\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}/\text{ }y\geq 1\right\} ,$ $R_{2}=\left\{ (x,y)\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}/\text{ }\left\vert y\right\vert \leq 1\right\} ,$ $R_{3}=\left\{
(x,y)\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2}/\text{ }y\leq -1\right\},$ where in each of these regions the
map (1) is linear. However, it is easy to verify that for all
values of the parameters $m_{0},m_{1}$ such that $m_{0}m_{1}>0$,
the map (1) has a single fixed point $\left( 0,0\right)$, while if
$m_{0}m_{1}<0,$ the map (1) has three fixed points, and they are
given by
$P_{1}=\left(\frac{m_{1}-m_{0}}{bm_{1}},\frac{m_{1}-m_{0}}{m_{1}}\right)$,
$P_{2}=\left(0,0\right),
P_{3}=\left(\frac{m_{0}-m_{1}}{bm_{1}},\frac{m_{0}-m_{1}}{m_{1}}\right)$.
Obviously, the Jacobian matrix of the map (1) evaluated at the
fixed points $P_{1}$ and $P_{3}$ is the same and is given by
$J_{1,3}=\left(
\begin{array}{cc}
1 & -abm_{1} \\
1 & 0%
\end{array}%
\right)$. Therefore, the two equilibrium points $P_{1}$ and
$P_{3}$ have the same stability type. The Jacobian matrix of the
map (1) evaluated at the fixed point $P_{2}$ is given by
$J_{2}=\left(
\begin{array}{cc}
1 & -abm_{0} \\
1 & 0%
\end{array}%
\right)$, and the characteristic polynomials for $J_{1,3}$ and
$J_{2}$ are given respectively by $\lambda ^{2}-\lambda
+abm_{1}=0$ and $\lambda ^{2}-\lambda +abm_{0}=0$, where the local
stability of these equilibria can be studied by evaluating the
eigenvalues of the corresponding Jacobian matrices given by the
solution of their characteristic polynomials.
\section{The hyperchaoticity of the attractor}
In this section, we give sufficient conditions for the
hyperchaoticity of the discrete hyperchaotic double scroll given
by the map (1). Note that this property is absent for the classic
double scroll [2-3].
It is shown in [9] that if a we consider a system $x_{k+1}=f\left(
x_{k}\right) ,x_{k}\in \Omega \subset
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{n},$ such that
\begin{equation}
\left\Vert \acute{f}(x)\right\Vert \leq N<+\infty
\end{equation}%
with a smallest eigenvalue of $f(x)^{T}f(x)$ that satisfies
\begin{equation}
\lambda _{\min }\left( f(x)^{T}f(x)\right) \geq \theta >0,
\end{equation}%
where $N^{2}\geq \theta ,$ then, for any $x_{0}\in \Omega $, all the
Lyapunov exponents at $x_{0}$ are located inside $\left[ \frac{\ln \theta }{2%
},\ln N\right] $, that is,%
\begin{equation}
\frac{\ln \theta }{2}\leq l_{i}\left( x_{0}\right) \leq \ln N,i=1,2,...,n,\
\end{equation}%
where $l_{i}\left( x_{0}\right) $ are the Lyapunov exponents for
the map $f.$ For the map (1), one has that$\allowbreak $
\begin{equation}
\ \left\Vert \acute{f}(x,y)\right\Vert =\left\{
\begin{array}{c}
\sqrt{\frac{b^{2}+a^{2}m_{1}^{2}+\sqrt{%
2b^{2}+b^{4}+2a^{2}m_{1}^{2}+a^{4}m_{1}^{4}-2a^{2}b^{2}m_{1}^{2}+1}+1}{2}},%
\text{ if }\left\vert y\right\vert \geq 1 \\
\sqrt{\frac{b^{2}+a^{2}m_{0}^{2}+\sqrt{%
2b^{2}+b^{4}+2a^{2}m_{0}^{2}+a^{4}m_{0}^{4}-2a^{2}b^{2}m_{0}^{2}+1}+1}{2}},%
\text{ if }\left\vert y\right\vert \leq 1%
\end{array}%
\right. <+\infty
\end{equation}
and
\begin{equation}
\lambda _{\min }\left( f(x)^{T}f(x)\right) =\left\{
\begin{array}{c}
\frac{b^{2}+a^{2}m_{1}^{2}-\sqrt{%
2b^{2}+b^{4}+2a^{2}m_{1}^{2}+a^{4}m_{1}^{4}-2a^{2}b^{2}m_{1}^{2}+1}+1}{2},%
\text{ if }\left\vert y\right\vert \geq 1 \\
\frac{b^{2}+a^{2}m_{0}^{2}-\sqrt{%
2b^{2}+b^{4}+2a^{2}m_{0}^{2}+a^{4}m_{0}^{4}-2a^{2}b^{2}m_{0}^{2}+1}+1}{2},%
\text{ if }\left\vert y\right\vert \leq 1.%
\end{array}%
\right.
\end{equation}
If
\begin{equation}
\left\vert a\right\vert >\max \left( \frac{1}{\left\vert m_{1}\right\vert },%
\frac{1}{\left\vert m_{0}\right\vert }\right) ,\left\vert b\right\vert >\max
\left( \frac{\left\vert am_{1}\right\vert }{\sqrt{a^{2}m_{1}^{2}-1}},\frac{%
\left\vert am_{0}\right\vert }{\sqrt{a^{2}m_{0}^{2}-1}}\right) ,
\end{equation}
then both Lyapunov exponents of the map (1) are positive for all
initial conditions $\left( x_{0},y_{0}\right) \in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
^{2},$ and hence the corresponding attractor is hyperchaotic. For
$m_{0}=-0.43$ and $m_{1}=0.41$, one has that $\left\vert
a\right\vert >\allowbreak 2.\,\allowbreak 439$, and for $b=1.4$,
one has that $\left\vert a\right\vert
>3.\,\allowbreak 323$. As a test of the previous analysis, Fig. 2 shows the
Lyapunov exponent spectrum for the map (1) for $m_{0}=-0.43,$
$m_{1}=0.41, b=1.4$, and $-3.365\leq a\leq 3.365.$ The regions of
hyperchaos are $-3.365\leq a\leq -3.\,\allowbreak 323$ and
$3.\,\allowbreak 323\leq a\leq 3.365.$
\FRAME{ftbpFU}{327.5pt}{201.9375pt}{0pt}{\Qcb{Variation of the
Lyapunov exponents of map (1) versus the parameter $-3.365\leq
a\leq 3.365$ with $b=1.4, m_{0}=-0.43$, and
$m_{1}=0.41.$}}{}{Figure}{\special{language "Scientific Word";type
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\FRAME{ftbpFU}{255.625pt}{195.3125pt}{0pt}{\Qcb{The discrete
hyperchaotic double scroll attractor obtained from the map (1) for
$a=-3.36, b=1.4, m_{0}=-0.43$, and $m_{1}=0.41$ with initial
conditions $x=y=0.1.$}}{}{Figure}{\special{language "Scientific
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On the other hand, the discrete hyperchaotic double scroll shown
in Fig. 3 results from a stable period-3 orbit transitioning to a
fully developed chaotic regime. This particular type of
bifurcation is called a border-collision bifurcation as shown in
Fig. 4, and it is the only observed scenario. If we fix parameters
$b=1.4, m_{0}=-0.43$, and $m_{1}=0.41$ and vary $a\in
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$, then the map (1) exhibits the following dynamical behaviors as
shown in Fig. 4: In the interval $a<-3.365$, the map (1) does not
converge. For $-3.365\leq a\leq 3.365,$ the map (1) begins with a
reverse border-collision bifurcation, leading to a stable period-3
orbit, and then collapses to a point that is reborn as a stable
period-3 orbit leading to fully developed chaos. For $a>3.365,$
the map (1) does not converge. However, it seems that the proposed
map behaves in a similar way to the 4-D dynamical system given in
[3], i.e., both hyperchaotic attractors are obtained by a
border-collision bifurcation [8].
\section{Conclusion}
We have described a new simple 2-D discrete piecewise linear
chaotic map that is capable of generating a hyperchaotic double
scroll attractor. Some important detailed dynamical behaviors of
this map were further investigated.
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\end{document}