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\begin{document}
\title{On the robustness of chaos in dynamical systems:Theories and
applications}
\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}
This paper offers an overview of some important issues concerning
the robustness of chaos in dynamical systems and their
applications to the real world.
\end{abstract}
\textit{Keywords:} Robust chaos, theories, methods, real
applications.
PACS numbers: 05.45.-a, 05.45.Gg.
\section{Introduction}
Chaotic dynamical systems display two kinds of chaotic attractors:
One type has fragile chaos (the attractors disappear with
perturbations of a parameter or coexist with other attractors),
and the other type has robust chaos, defined by the absence of
periodic windows and coexisting attractors in some neighborhood of
the parameter space. The existence of these windows in some
chaotic regions means that small changes of the parameters would
destroy the chaos, implying the fragility of this type of chaos.
Contrary to this situation, there are many practical applications,
such as in communication and spreading the spectrum of switch-mode
power supplies to avoid electromagnetic interference [15-16],
where it is necessary to obtain reliable operation in the chaotic
mode, and thus robust chaos is required. Another practical example
can be found in electrical engineering where robust chaos is
demonstrated in [13]. The occurrence of robust chaos in a smooth
system is proved and discussed in [2], which includes a general
theorem and a practical procedure for constructing S-unimodal maps
that generate robust chaos. This result contradicts the conjecture
that robust chaos cannot exist in smooth systems [13]. On the
other hand, there many methods used to search for a smooth and
robust chaotic map, for example in [1-2-3-4-5], where a
one-dimensional smooth map that generates robust chaos in a large
domain of the parameter space is presented. In [6], simple
polynomial unimodal maps that show robust chaos are constructed.
Other methods and algorithms are given in the discussion below.
This paper is organized as follows. In the following section, we
first discuss robust chaos, its theories and applications, and
then give in Section 3 several rigorous, numeric, and experimental
results that explain the different methods for the generation of
robust chaos in dynamical systems. The final section concludes the
paper.
\section{Why robust chaos?}
The past decade has seen heightened interest in the exploitation
of robust chaos for applications to engineering systems. Since
there are many areas for applications of robust chaos, we
concentrate on two examples of applications of robust chaos in the
real world. The first is given in [41], which suggests a new
approach (with experiments, statistical analysis, and key space
analysis) for image encryption based on a robust high-dimensional
chaotic map. The new scheme employs the so-called cat map to
shuffle the positions and then confuses the relationship between
the cipher-image and the plain-image by using the high-dimensional
preprocessed Lorenz chaotic map. This work shows that the proposed
image encryption scheme provides an efficient and secure means for
real-time image encryption and transmission. The second example is
given in [54], which uses the notion of robust chaos in another
new encryption scheme. A recent bibliography on the applications
of robust chaos in the real world are collected in these papers
[11-13-15-16-21-30-41-42-43-48-49-50-51-58-60-61-62-63-64-65-66-77-78-79-80].
\section{Theories and applications of robust chaos}
\subsection{Robust chaos in 1-D maps}
In this section, we give several methods to generate robust chaos
in 1-D maps.
\subsubsection{Robust chaos in 1-D piecewise-smooth maps}
As an example of the occurrence of robust chaos in 1-D
piecewise-smooth maps, the following model of networks of neurons
with the activation function $f(x)=|\tanh $ $s(x-c)|$ is studied
in [43], where it was shown
that in a certain range of $s$ and $c$ the dynamical system%
\begin{equation}
\text{ }x_{k+1}=|\tanh s(x_{k}-c)|
\end{equation}
cannot have stable periodic solutions, which proves the robustness
of chaos.
\subsubsection{Robust chaos in 1-D smooth maps}
The robustness of chaos in the sense of the above definition and
expected to be relevant for any practical applications of chaos
was shown to exist in a general family of piecewise-smooth
two-dimensional maps, but was conjectured to be impossible for
smooth unimodal maps [8-13]. However, it is shown in [1-2],
respectively, that the following 1-D maps have robust chaotic attractors:%
\begin{equation}
\text{ }f\left( x,\alpha \right) =\frac{1-x^{\alpha }-\left( 1-x\right)
^{\alpha }}{1-2^{1-\alpha }},\text{ }f\left( \phi \left( x\right) ,v\right) =%
\frac{1-v^{\pm \phi \left( x\right) }}{1-v^{\pm \phi \left( c\right) }},%
\text{ }
\end{equation}
where $\alpha, v$ are bifurcation parameters and $\phi \left(
x\right)$ is unimodal with a negative Schwarzian derivative (but
not necessarily chaotic), and $c$ is the critical point of $\phi
\left( x\right),$ i.e. $\dot{\phi}\left( c\right) = 0.$
Also in [5], the following map is shown to be robust:
\begin{equation}
f_{m,n}\left( x,a\right) = 1-2\left( ax^{m}+bx^{n}\right),
\end{equation}
where $0\leq a\leq 1, b=1-a,$ and $m, n$ are even and $>0.$ On
the other hand, it is known that a map $\varphi :I\longrightarrow
I$ is S-unimodal on the interval $I= \left[ a,b\right]$ if: (a)
The function $\varphi \left( x\right)$ is of class $C^{3}$, (b)
the point $a$ is a fixed point with $b$ its other preimage, i.e.
$\varphi (a)=\varphi (b)=a$, (c) there is a unique maximum at
$c\in $ $(a,b)$ such that $\varphi (x)$ is strictly increasing on
$x\in \left[ a,c\right) $ and strictly decreasing on $x\in \left(
c,b\right]$, and (d) $\varphi $ has a negative Schwarzian
derivative, i.e.
\begin{equation}
S\left( \varphi ,x\right) =\frac{\varphi ^{\prime \prime \prime }\left(
x\right) }{\varphi ^{\prime }\left( x\right) }-\frac{3}{2}\left( \frac{%
\varphi ^{\prime \prime }\left( x\right) }{\varphi ^{\prime
}\left( x\right) }\right) ^{2}<0
\end{equation}%
for all $\ x\in I-\left\{ y,\varphi ^{\prime }\left( y\right)
=0\right\}.$ Since what matters is only its sign, one may as well
work with the product:
\begin{equation}
\hat{S}\left( \varphi ,x\right) =2\varphi ^{\prime }\left( x\right) \varphi
^{\prime \prime \prime }\left( x\right) -3\left( \varphi ^{\prime \prime
}\left( x\right) \right) ^{2},
\end{equation}%
which has the same sign as $S\left( \varphi ,x\right).$
The importance of S-unimodal maps in chaos theory comes from the
theorem given in [7] that each attracting periodic orbit attracts
at least one critical point or boundary point. Thus, as a result,
an S-unimodal map can have at most one periodic attractor which
will attract the critical point. This result is used to formulate
the following theorem with its proof given in [2]:
\begin{theorem}
Let $\varphi _{v}(x)$: $I=[a;b]$ $\longrightarrow I$ be a
parametric S-unimodal map with the unique maximum at $c\in (a;b)$
and $\varphi _{v}(c)=b $, $\forall $ $v\in $ $(v_{\min },v_{\max
})$, then $\varphi _{v}(x)$ generates robust chaos for $v\in $
$(v_{\min },v_{\max })$.
\end{theorem}
This theorem gives the general conditions for the occurrence of
robust chaos in S-unimodal maps, but it does not give any
procedure for the construction of the S-unimodal map $\varphi
_{v}(x)$. A procedure for constructing S-unimodal maps that
generate robust from the composition of two S-unimodal maps is
given in [2-4-5-6].
A 1-dimensional generalization of the well known logistic map [7]
is proposed and studied in [53]. The generalized map is referred
to as the $\beta $-exponential map, and it is given by:
\begin{equation}
GL\left( \beta ,x\right) =\frac{\beta -x\beta ^{x}-\left( 1-x\right) \beta
^{1-x}}{\beta -\sqrt{\beta }},0\leq x\leq 1,\beta \geq 0,
\end{equation}
where $\beta $ is the adjustable parameter. It was proved that the
map (6) exhibits robust chaos for all real values of the parameter $\beta $ $%
\geq e^{-4}$.
Another example of robust chaos is found in [21], where the robust
chaos was identified in a family of discounted dynamic
optimization problems in economics (with verification of some
properties such as monotonicity and concavity of the return
functions and the aggregative production function) in which the
immediate return function depends on current consumption, capital
input, and a taste parameter. It was shown also that the optimal
transition functions are represented by the quadratic family,
well-studied in the literature on chaotic dynamical systems.
An hierarchy of many-parameter families of maps on the interval
$[0,1]$ having an analytic formula of the Kolmogorov--Sinai
entropy was introduced in [6]. These types of maps do not have
period-doubling or period-$n$tupling cascade bifurcations to
chaos, but they have single fixed-point attractors in certain
regions of parameters space where they bifurcate directly to chaos
at exact values of the parameters without the period-$n$tupling
scenario.
\subsubsection{Robust chaos in 1-D singular maps}
These types of 1-D singular maps play a key role in the theory of
up-embedability of graphs [76]. In [70], the critical behavior of
the Lyapunov exponent of a 1-D singular map (which has only one
face on a surface) near the transition to robust chaos via
type-III intermittency was determined for a family of
one-dimensional singular maps. The calculation of critical
boundaries separating the region of robust chaos from the region
of stable fixed points was given and discussed.
\subsection{Robust chaos in 2-D piecewise smooth maps}
Power electronics is an area with wide practical application
[11-12-13-14-40-47-77]. It is concerned with the problem of the
efficient conversion of electrical power from one form to another.
Power converters [40-77] exhibit several nonlinear phenomena such
as border-collision bifurcations, coexisting attractors
(alternative stable operating modes or fragile chaos), and chaos
(apparently random behavior). These phenomena are created by
switching elements [40]. Recently, several researchers have
studied border-collision bifurcations in piecewise-smooth systems
[11-12-13-14-40-47-77]. Piecewise-smooth systems can exhibit
classical smooth bifurcations, but if the bifurcation occurs when
the fixed point is on the border, there is a discontinuous change
in the elements of the Jacobian matrix as the bifurcation
parameter is varied. A variety of such border-collision
bifurcations have been reported [11-12-14] in this situation. In
[13-77] and under certain conditions, border-collision
bifurcations produce robust chaos.
Let us consider the following 2-D piecewise smooth system given by:
\begin{equation}
g(x,y;\rho )=\left(
\begin{array}{c}
g_{1}=\left(
\begin{array}{c}
f_{1}(x,y;\rho ) \\
f_{2}(x,y;\rho )%
\end{array}%
\right) ,\text{ if }x~~0,%
\end{array}%
\right. \ \
\end{equation}%
where $\mu$ is a parameter and $\tau _{i}, \delta _{i}, i=1,2$ are
the traces and determinants of the corresponding matrices of the
linearized map in the two subregion $R_{1}$ and $R_{2}$ evaluated
at $P_{1}$ (with eigenvalues $\lambda _{1,2}$) and $P_{2}$ (with
eigenvalues $\omega _{1,2}$), respectively. Now it is shown in
[13-77] that the resulting chaos from the 2-D map (7) is robust in
the following cases:
\subsubsection{Case 1}
\begin{equation}
\ \left\{
\begin{array}{c}
\tau _{1}>1+\delta _{1},\text{ and }\tau _{2}<-\left( 1+\delta
_{2}\right)
\\
0<\delta _{1}<1,\text{ and }0<\delta _{2}<1,%
\end{array}%
\right. \ \
\end{equation}
where the parameter range for boundary crisis is given by:
\begin{equation}
\delta _{1}\tau _{1}\lambda _{1}-\delta _{1}\lambda _{1}\lambda _{2}+\delta
_{2}\lambda _{2}-\delta _{1}\tau _{2}+\delta _{1}\tau _{1}-\delta
_{1}^{2}-\lambda _{1}\delta _{1}>0,
\end{equation}
where the inequality (12) determines the condition for stability
of the chaotic attractor. The robust chaotic orbit continues to
exist as $\tau _{1}$ is reduced below $1+\delta _{1}.$
\subsubsection{Case 2}
\begin{equation}
\ \left\{
\begin{array}{c}
\tau _{1}>1+\delta _{1},\text{ and }\tau _{2}<-\left( 1+\delta
_{2}\right)
\\
\delta _{1}<0,\text{ and }-1<\delta _{2}<0\\
\frac{\lambda _{1}-1}{\tau _{1}-1-\delta _{1}}>\frac{\omega _{2}-1}{\tau
_{2}-1-\delta _{2}},%
\end{array}%
\right. \ \
\end{equation}
The condition for stability of the chaotic attractor is also
determined by (12). However, if the third condition of (13) is not
satisfied, then the condition for existence of the chaotic
attractor changes to:
\begin{equation}
\text{ }\frac{\omega _{2}-1}{\tau _{2}-1-\delta _{1}}<\frac{\left(
\tau _{1}-\delta _{1}-\lambda _{2}\right) }{\left( \tau
_{1}-1-\delta _{1}\right) \left( \lambda _{2}-\tau _{2}\right) }.
\end{equation}
\subsubsection{Case 3}
The remaining ranges for the quantity $\tau _{i}, \delta _{i},
i=1,2,$ can be determined in some cases using the same logic as in
the above two cases, or there is no analytic condition for a
boundary crisis, and it has to be determined numerically.
\subsubsection{Example}
Several examples of robust chaos in 2-D piecewise smooth systems
can be found in [11-12-13-14-40-47]. In this overview, we give the
following example [69].
Consider the unified piecewise-smooth chaotic mapping that
contains the H\'{e}non [10] and the Lozi [9] systems defined by:
\begin{center}
\begin{equation}
U(x,y)=\left(
\begin{array}{c}
1-1.4f_{\alpha }\left( x\right) +y \\
0.3x%
\end{array}%
\right), \label{1}
\end{equation}
\end{center}
where $0\leq \alpha \leq 1$ is the bifurcation parameter and the function $%
f_{\alpha }$ is given by:
\begin{equation}
f_{\alpha }\left( x\right) =\alpha \left\vert x\right\vert +\left( 1-\alpha
\right) x^{2}.
\end{equation}
It is easy to see that for $\alpha =0,$ one has the original
H\'{e}non map, and for $\alpha =1,$ one has the original Lozi map,
and for $0<\alpha <1,$ the unified chaotic map (15) is chaotic
with different kinds of attractors. In this case it was shown
rigorously that the unified system (15) has robust chaotic
attractors for $0.493122734\leq \alpha <1$ as shown in Fig. 1.
Some corresponding robust chaotic attractors are shown in Fig. 2.
These chaotic attractors cannot be destroyed by small changes in
the parameters since small changes in the parameters can only
cause small changes in the Lyapunov exponents. Hence the range for
the parameter $0\leq \alpha <1,$ in which the map (15) converges
to a robust chaotic attractor, is approximately $50.6\,88$
percent. This result was also verified numerically by computing
Lyapunov exponents and bifurcation diagrams as shown in Figs. 1(a)
and (b). For $\alpha <0.493122734,$ the chaos is not robust in
some ranges of the variable $\alpha $ because there are numerous
small periodic windows as shown in Fig. 3(b) such as the
period-$8$ window at $\alpha =0.025.$ Also, for $\alpha =0.114,$
there are some periodic windows. We note also the existence of
some regions in $\alpha$ where the largest Lyapunov exponent is
positive, but this does not guarantee the unicity of the
attractor, contrary to the case of $\alpha \in \left[
0.493122734,1\right[$ where the attractor is guaranteed to be
unique due to the analytical expressions.
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Lyapunov exponents of the unified map (15) for $0\leq
\protect\alpha \leq 1$. (b) Bifurcation diagram for the unified
chaotic map (15) for $0\leq \protect\alpha \leq
1.$.}}{}{Figure}{\special{language "Scientific Word";type
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\FRAME{ftbpFU}{335.25pt}{337.0625pt}{0pt}{\Qcb{ (a) The transition
H\'{e}non-like chaotic attractor\textit{\ }obtained for the
unified chaotic map (15) with its basin of attraction (white) for
$\protect\alpha =0.2$. (b) Graph of the function $f_{0.2}.$ (c)
The transition Lozi-like chaotic attractor\textit{\ }obtained for
the unified chaotic map (15) with its basin of attraction (white)
for $\protect\alpha =0.8$. (d) Graph of the function
$f_{0.8}$.}}{}{Figure}{\special{language "Scientific Word";type
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Lyapunov exponents for the unified chaotic map (15) for $0.02\leq
\protect\alpha \leq 0.03$. (b) Bifurcation diagram for the unified
chaotic map (15) for $0.02\leq \protect\alpha \leq 0.03$ showing a
period-8 attractor obtained for $\protect\alpha
=0.025.$}}{}{Figure}{\special{language "Scientific Word";type
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\subsection{Robust chaos in non-smooth maps}
An example of robust chaos in non-smooth maps occurs with the
perceptron [24], which is the simplest kind of feedforward neural
network and is calculated as
\begin{equation}
\text{ }f\left( x\right) =\left\{
\begin{array}{c}
1,\text{ if }\omega\cdot x+b>0 \\
0,\text{ else,}%
\end{array}%
\right.
\end{equation}
where $\omega$ is a vector of real-valued weights, $\omega\cdot x$
is the dot product (which computes a weighted sum), $b$ is the
bias, and $a$ is a constant term that does not depend on any input
value. The value of $f\left( x\right) $ ($0$ or $1$) is used to
classify $x$ as either positive or negative in the case of a
binary classification problem. The properties of the time series
generated by a perceptron with monotonic and non-monotonic
transfer functions were examined in [71]. The results show that a
perceptron with a monotonic function can produce fragile chaos
only, whereas a non-monotonic function can generate robust chaos.
\subsection{Robust chaos in smooth continuous-time systems}
Strange attractors can be classified into three principal classes
[39-44-52]: hyperbolic, Lorenz-type, and quasi-attractors. The
hyperbolic attractors are the limit sets for which Smale's
\textquotedblleft axiom A\textquotedblright\ is satisfied and are
structurally stable. Periodic orbits and homoclinic orbits are
dense and are of the same saddle type, which is to say that they
have the same index (the same dimension for their stable and
unstable manifolds). However, the Lorenz-type attractors are not
structurally stable, although their homoclinic and heteroclinic
orbits are structurally stable (hyperbolic), and no stable
periodic orbits appear under small parameter variations, as for
example in the Lorenz system [17] . The quasi-attractors are the
limit sets enclosing periodic orbits of different topological
types (for example stable and saddle periodic orbits) and
structurally unstable orbits. For example, the attractors
generated by Chua's circuit [45] associated with saddle-focus
homoclinic loops are quasi-attractors. Note that this type is more
complex than the above two attractors, and thus are not suitable
for potential applications of chaos such as secure communications
and signal masking. For further information about these types of
chaotic attractors, see [39].
\subsubsection{Robust chaos in hyperbolic systems}
In strange attractors of the hyperbolic type all orbits in phase
space are of the saddle type, and the invariant sets of
trajectories approach the original one in forward or backward
time, i.e. the stable and unstable manifolds intersect
transversally. Generally, most known physical systems do not
belong to the class of systems with hyperbolic attractors [39].
The type of chaos in them is characterized by chaotic trajectories
and a set of stable orbits of large periods, not observable in
computations because of extremely narrow domains of attraction.
Hyperbolic strange attractors are robust (structurally stable)
[44]. Thus, both from the point of view of fundamental studies and
of applications, it would be interesting to find physical examples
of hyperbolic chaos. For example the Smale-Williams attractor [46]
is constructed for a three-dimensional map, and the composed
equations given by
\begin{equation}
\left\{
\begin{array}{c}
\dot{x}=-2\pi u+\left( h_{1}+A_{1}\cos 2\pi \tau /N\right) x-\frac{1}{3}%
x^{3} \\
\dot{u}=2\pi \left( x+\varepsilon _{2}y\cos 2\pi \tau \right) \\
\dot{y}=-4\pi v+\left( h_{2}-A_{2}\cos 2\pi \tau /N\right) y-\frac{1}{3}%
y^{3} \\
\dot{v}=4\pi \left( y+\varepsilon _{1}x^{2}\right)%
\end{array}%
\right.
\end{equation}
are obtained by applying the so-called equations of Kirchhoff
[23], where the variables $x$ and $u$ are normalized voltages and
currents in the $LC$ circuit of the first self-oscillator ($U_{1}$
and $I_{1}$, respectively), and $y$ and $v$ are normalized
voltages and currents in the second oscillator ($U_{2}$ and
$I_{2}$). Time is normalized to the period of oscillations of the
first $LC$ oscillator, and the parameters $A_{1}$ and $A_{2}$
determine the amplitude of the slow modulation of the parameter
responsible for the Andronov-Hopf bifurcation in both
self-oscillators. The parameters $h_{1}$ and $h_{2}$ determine a
map of the mean value of this parameter from the bifurcation
threshold, and $\varepsilon _{1}$ and $\varepsilon _{2}$ are
coupling parameters.
The system (18) has been constructed as a laboratory device [46],
and an experimental and numerical solution were found. This
example of a physical system with hyperbolic chaotic attractor is
of considerable significance since it opens the possibility for
real applications. For further details, see [46].
\subsubsection{Robust chaos in the Lorenz-type system}
As a Lorenz-type system, consider the original Lorenz system [17]
given by
\begin{equation}
\left\{
\begin{array}{c}
\dot{x}=\sigma \left( y-x\right) \\
\dot{y}=rx-y-xz \\
\mathring{z}=-bz+xy%
\end{array}%
\right.
\end{equation}
These equations have proved to be very resistant to rigorous
analysis and also present obstacles to numerical study. A very
successful approach was taken in [27-72] where they constructed
so-called geometric models (these models are flows in 3
dimensions) for the behavior observed by Lorenz for which one can
rigorously prove the existence of a robust attractor. Another
approach through rigorous numerics [28-73-74-75] showed that the
equations exhibit a suspended Smale horseshoe. In particular, they
have infinitely many closed solutions. A Computer assisted proof
of chaos for the Lorenz equations is given in [18-19-20-25-29-35].
In [18] a rigorous proof was provided that the geometric model
does indeed give an accurate description of the dynamics of the
Lorenz equations, i.e., it supports a strange attractor as
conjectured by Lorenz in 1963. This conjecture was listed by
Steven Smale as one of several challenging mathematical problems
for the 21st century [34]. Also a proof that the attractor is
robust, i.e., it persists under small perturbations of the
coefficients in the underlying differential equations was given.
This proof is based on a combination of normal form theory and
rigorous numerical computations. The robust chaotic Lorenz
attractor is shown in Fig. 4. As a general result, it was proved
in [38] that the so-called singular-hyperbolic (or Lorenz-like)
attractor of a $3$-dimensional flow is chaotic in two different
strong senses: Firstly, the flow is expansive: if two points
remain close for all times, possibly with time reparametrization,
then their orbits coincide. Secondly, there exists a physical (or
Sinai-Ruelle-Bowen) measure supported on the attractor whose
ergodic basin covers a full Lebesgue (volume) measure subset of
the topological basin of attraction. In particular, these results
show that both the flow defined by the Lorenz equations and the
geometric Lorenz flows are expansive.
\FRAME{ftbpFU}{289.5pt}{221.5pt}{0pt}{\Qcb{The robust Lorenz
chaotic attractor obtained from (19) for $\protect\sigma =10,r=28,b=\frac{8}{%
3}$ [17].}}{}{Figure}{\special{language "Scientific Word";type
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Another proof of the robustness of the Lorenz attractor is given
in [20] where the chaotic attractors of the Lorenz system
associated with $r=28$ and $r=60$ were characterized in terms of
their unstable periodic orbits and eigenvalues. While the
Hausdorff dimension is approximated with very good accuracy in
both cases, the topological entropy was computed in an exact sense
only for $r=28$. A general method for proving the robustness of
chaos in a set of systems called $C^{1}$-robust transitive sets
with singularities for flows on closed $3$-manifolds is given in
[22]. The elements of the set $C^{1}$ are partially hyperbolic
with a volume-expanding central direction and are either
attractors or repellers. In particular, any $C^{1}$-robust
attractor with singularities for flows on closed $3$-manifolds
always have an invariant foliation whose leaves are forward
contracted by the flow and has a positive Lyapunov exponent at
every orbit, showing that any $C^{1}$-robust attractor resembles a
geometric Lorenz attractor. A new topological invariant
(Lorenz-manuscript) leading to the existence of an uncountable set
of topologically various attractors is proposed in [57] where a
new definition of the hyperbolic properties of the Lorenz system
close to singular hyperbolicity is introduced, as well as a proof
that small non-autonomous perturbations do not lead to the
appearance of stable solutions.
Other than the Lorenz attractor, there are some works that focus
on the proof of the robustness of chaos in $3$-D continuous
systems, for example the set $C^{1}$ introduced in [22], and a
characterization of maximal transitive sets with singularities for
generic $C^{1}$-vector fields on closed $3$-manifolds in terms of
homoclinic classes associated with a unique singularity is given
and applied to some special cases.
\subsubsection{No robust chaos in quasi-attractor-type systems}
The complexity of quasi-attractors is essentially due to the
existence of structurally unstable homoclinic orbits in the system
itself, and in any system close to it. It results in a sensitivity
of the attractor structure to small variations of the parameters
of the generating dynamical equation, i.e., quasi-attractors are
structurally unstable. Then this type of system cannot generate
robust chaotic attractors in the sense of this paper [44].
Attractors generated by Chua's circuits [45], given by
\begin{equation}
\left\{
\begin{array}{c}
\dot{x}=\alpha \left( y-h\left( x\right) \right) \\
\dot{y}=x-y+z \\
\mathring{z}=-\beta y%
\end{array}%
\right.
\end{equation}%
where
\begin{equation}
h(x)=m_{1}x+\frac{1}{2}(m_{0}-m_{1})\left( \left\vert x+1\right\vert
-\left\vert x-1\right\vert \right)
\end{equation}
are associated with saddle-focus homoclinic loops and are
quasi-attractors. The corresponding non-robust double-scroll
attractor is shown in Fig.5.
\FRAME{ftbpFU}{318.5pt}{%
185.375pt}{0pt}{\Qcb{The non-robust double-scroll attractor
obtained from system (20) with $\protect\alpha =9.35,
\protect\beta =14.79, m_{0}=-\frac{1}{7}, m_{1}=\frac{2}{7}$
[45].}}{}{Figure}{\special{language "Scientific Word";type
"GRAPHIC";display "USEDEF";valid_file "T";width 318.5pt;height
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\section{Discussion}
In this paper we discuss the robustness of chaos in the sense that
there are no coexisting attractors and no periodic windows in some
neighborhood of the parameter space. As we saw, robust chaos
occurs in several types of dynamical systems: discrete,
continuous-time, autonomous, non-autonomous, smooth, and
non-smooth, with different topological dimensions, and it has been
confirmed either analytically, numerically, or experimentally, or
a combination of them. On the other hand, there is no robust chaos
in the quasi-attractor type because they are structurally
unstable.
There is several methods for proving the robustness of chaos in
dynamical systems. Some of these methods are collected and
summarized in the following:
\subsection{Normal form analysis}
This method was used essentially (with other technics) for both
2-D piecewise smooth maps [11-13-47-69-77] and 3-D continuous-time
systems [18]. In both cases the robustness was proved analytically
and confirmed numerically, and in some cases the results were
confirmed experimentally.
\subsection{Unimodality}
Unimodality is an analytic property defined for real functions.
This method was used for proving the robustness of chaos in smooth
1-D maps [1-2-3-4-5] with the investigation of their invariant
distributions and Lyapunov exponents. This idea came from the
analysis of the well-known logistic map.
\subsection{Metric entropy}
Metric entropy [67] measures the average rate of information loss
for a discrete measurable dynamical system. This method was used
to prove the robustness of chaos in a one-dimensional map [6]
using Kolmogrov-Sinai entropy, and in [20] the topological entropy
was computed in an exact sense for $r=28.$ Other examples can be
found in [53-54-70]
\subsection{Construction using basis of the robustness or the
non-robustness}
If a system is known that has robust chaos, then it is possible to
construct another model that has also robust chaos. This method
was applied to a hyperbolic-type system [46]. Another example of
this method was used to conclude that there is no robust chaos in
quasi-attractor-type systems [44-45].
\subsection{Geometric methods}
Geometric methods were used to prove the robustness of chaos in
the Lorenz system [18-19-22-25-26-27-28-29-36-38-57-72-73-74-75].
These methods employed the so-called geometric model, and a
computer-assisted proof was used leading to a rigorous numerical
study. Generally, these methods are the most useful for proving
chaos or robust chaos in dynamical systems.
\subsection{Detecting unstable periodic solutions}
This method was used both for discrete and continuous-time systems
[20-43-49-72].
\subsection{Ergodic theory}
Ergodic theory [67] was used to prove the existence of robust
chaos in several types of dynamical sysytems, for example in [21].
\subsection{Weight-space exploration}
The method of weight-space exploration is essentially based on two
concepts. The first is the concept of running, which is defined as
a mapping from the high-dimensional space of the neural network
structure (defined by the interconnection weights, the initial
conditions, and the nonlinear activation function) to one or two
scalar values (called dynamic descriptors) giving the essential
information about the dynamic behavior of the network as obtained
by running it on a manageable enough large time period. The second
concept is called descriptor map. This method was used to search
for robust chaos in a discrete CNN such as the example given in
[68].
\subsection{Numerical methods}
Other than the types of systems mentioned in the above gallery,
the regions of parameters space for multiple attractors (regions
of fragile chaos), were determined using a relatively large number
$N$ of different random initial conditions and looking for cases
where the distribution of the average value of the state variable
on the attractor is bimodal. Since there is no rigorous test for
bimodality, this was done by sorting the $N$ values of the state
variable and then dividing them into two equal groups.
\FRAME{ftbpFU}{328.0625pt}{206.5pt}{0pt}{\Qcb{The regions of $%
ab$-space for multiple attractors for the map given in [68].}}{}{Figure}{%
\special{language "Scientific Word";type "GRAPHIC";display
"USEDEF";valid_file "T";width 328.0625pt;height 206.5pt;depth
0pt;original-width 481.8125pt;original-height 391.4375pt;cropleft
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ARTICLE/IJBC/A new 2-D smooth quadratic
map/ijbc-3406-190407/JTZ27T00.wmf';tempfile-properties "XPR";}}
The group with the smallest range of the state variable was
assumed to represent one of the attractors, and a second attractor
was assumed to exist if the largest gap in the values of those in
the other group was twice the range of the first group. This
method allowed us to see regions of robust chaos (without multiple
attractors) as shown in [68] and in Fig. 11.
\subsection{Combination of several methods}
In most examples of systems that have robust chaos, it is easy to
see that the proof is a combination of several methods.
\section{Conclusion}
An overview on some issues of common concern related to the
robustness of chaos in dynamical systems with several examples in
the real world were given in this paper and discussed.
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\end{document}
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