\documentclass[12pt]{article}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{amsfonts}
\usepackage{amsmath}
\setcounter{MaxMatrixCols}{10}
%TCIDATA{OutputFilter=LATEX.DLL}
%TCIDATA{Version=4.00.0.2312}
%TCIDATA{Created=Tuesday, February 14, 2006 17:47:07}
%TCIDATA{LastRevised=Sunday, March 14, 2007 19:18:12}
%TCIDATA{}
%TCIDATA{}
%TCIDATA{CSTFile=40 LaTeX article.cst}
\newtheorem{theorem}{Theorem}
\newtheorem{acknowledgement}[theorem]{Acknowledgement}
\newtheorem{algorithm}[theorem]{Algorithm}
\newtheorem{axiom}[theorem]{Axiom}
\newtheorem{case}[theorem]{Case}
\newtheorem{claim}[theorem]{Claim}
\newtheorem{conclusion}[theorem]{Conclusion}
\newtheorem{condition}[theorem]{Condition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{criterion}[theorem]{Criterion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{exercise}[theorem]{Exercise}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{notation}[theorem]{Notation}
\newtheorem{problem}[theorem]{Problem}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{solution}[theorem]{Solution}
\newtheorem{summary}[theorem]{Summary}
\newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}}
\input{tcilatex}
\begin{document}
\title{A minimal 2-D quadratic map with quasi-periodicity route to chaos}
\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}
The aim of this paper is to present and analyze a minimal chaotic map of the
plane, then we describe in detail the dynamical behavior of this map, along
with some other dynamical phenomena.
\end{abstract}
\textit{Keywords:} Minimal 2-D quadratic chaotic map, quasi-periodicity
route to chaos.
PACS numbers: 05.45.-a, 05.45.Gg.
\section{\textbf{Introduction}}
The H\'{e}non map [1] given by
\begin{equation}
h(x,y)=\left(
\begin{array}{c}
1-ax^{2}+by \\
x%
\end{array}%
\right)
\end{equation}
has been widely studied because it is the simplest example of a dissipative map with chaotic solutions. It has a single quadratic nonlinearity and an area contraction that depends only on $b$ and is thus constant over the orbit in the $xy$-plane. It can also be written as a one-dimensional time-delayed map:
\begin{equation}
x_{n+1}=1-ax_{n}^{2}+bx_{n-1}\text{ }.
\end{equation}
Generalizations of this map [2-6] have contributed to the development of dynamical systems theory and have produced new chaotic attractors with applications in science and engineering [7-9]. Application areas include secure communication and information processing [8-9] where discrete-component electronic implementation is possible [10].
Here we propose and analyze an equally simple two-dimensional quadratic map given by
\begin{equation}
f(x,y)=\left(
\begin{array}{c}
1-ay^{2}+bx \\
x%
\end{array}%
\right),
\end{equation}
where $a$ and $b$ are bifurcation parameters. Equation (3) is an
interesting minimal system, similar to the H\'{e}non map, but with the time delay in the nonlinear rather than the linear term as evidenced by writing it in the time-delayed form:
\begin{equation}
x_{n+1}=1-ax_{n-1}^{2}+bx_{n}\text{ }.
\end{equation}
Despite its apparent similarity and simplicity, it differs from the H\'{e}non map in that it has a non-uniform dissipation, a more rich and varied route to chaos, and a much wider variety of attractors. Whereas the attractors for the H\'{e}non map have a maximum dimension of about 1.31, with all the attractors qualitatively similar, the map (3) has attractors covering the entire range of dimensions from 1 to 2 (as well as zero) with basins of attraction that are often much more complicated than for the H\'{e}non map. These systems are special cases of general 2-D quadratic maps, many examples of which are given by Sprott [5] but not extensively studied.
Equation (4) reduces to the time-delayed quadratic map for $b=0$, much as the H\'{e}non map in Eq. (2) reduces to the ordinary quadratic map for $b=0$. On the other hand, this system is different from other well-known 2-D maps such as the \textit{delayed logistic map} [11] given by
\begin{equation}
g\left( x,y\right) =\left(
\begin{array}{c}
ax\left( 1-y\right) \\
x%
\end{array}%
\right).
\end{equation}
Equation (3) is not topologically equivalent to Eq. (5) since the latter has two fixed points $\left( 0,0\right)$ and $\left(\frac{a-1}{a},\frac{a-1}{a}\right)$ that exist for all values of $a\neq 0$, whereas the former has none, one, or two fixed points, depending on the values of $a$ and $b$ as will be shown below.
Other minimal chaotic mappings that have been studied include the 2-D circle map [12] and the delayed H\'{e}non map [13]. As with the H\'{e}non map, these systems including the one in Eq. (3) typically have no direct application to particular physical systems, but they serve to exemplify the kinds of dynamical behavior, such as routes to chaos, that are common in physical chaotic systems. Thus an analytical and numerical study is warranted.
\section{Fixed points and their stability}
In this section, we begin by studying the existence of the fixed points of the $f$ mapping and determine their stability. The Jacobian matrix of the map (3) is $J\left( x,y\right) =\left(
\begin{array}{cc}
b & -2ay \\
1 & 0%
\end{array}%
\right),$ and its characteristic polynomial for a fixed point $\left(
x,x\right) $ is given by
\begin{equation}
\lambda ^{2}-b\lambda +2ax=0.
\end{equation}
The local stability of the two equilibria is studied by evaluating the
roots of Eq. (6). Hence if $a\geq -\left( \frac{-b+1}{2}\right) ^{2}$,
then the map (3) has two fixed points:
\begin{equation}
\left\{
\begin{array}{c}
P_{1}=\left( \frac{b-1-\sqrt{4a-2b+b^{2}+1}}{2a},\frac{b-1-\sqrt{%
4a-2b+b^{2}+1}}{2a}\right) \\
P_{2}=\left( \frac{b-1+\sqrt{4a-2b+b^{2}+1}}{2a},\frac{b-1+\sqrt{%
4a-2b+b^{2}+1}}{2a}\right) ,%
\end{array}%
\right.
\end{equation}
whereas if $a<-\left( \frac{-b+1}{2}\right) ^{2},$ then the map (3) has no
fixed point.
Thus, after some calculations, one can obtain the following results:
$P_{1}$ is unstable in the following cases:
(i) $a>\frac{1}{8}b^{2}-\frac{1}{8}b^{3}+\frac{1}{64}b^{4},b<-2$.
(ii) $a>\frac{1}{8}b^{2}-\frac{1}{8}b^{3}+\frac{1}{64}b^{4},a>\frac{1}{2}b+%
\frac{3}{4}b^{2}-\frac{1}{4},b\geq 0.$
(iii) $a<\frac{1}{8}b^{2}-\frac{1}{8}b^{3}+\frac{1}{64}b^{4},b>2$.
(iv) $a<\frac{1}{8}b^{2}-\frac{1}{8}b^{3}+\frac{1}{64}b^{4},a>-\frac{1}{2}b+%
\frac{3}{4},b<2.$
$P_{1}$ is a saddle point in the following case:
$a<\frac{1}{2}b+\frac{3}{4}b^{2}-\frac{1}{4},a\geq \frac{1}{8}b^{2}-\frac{1}{
8}b^{3}+\frac{1}{64}b^{4}$, $b>0.$
$P_{1}$ is stable in the following cases:
(i) $a<\frac{1}{8}b^{2}-\frac{1}{8}b^{3}+\frac{1}{64}b^{4},b\leq 2$.
(ii) $a<\frac{1}{8}b^{2}-\frac{1}{8}b^{3}+\frac{1}{64}b^{4},a>-\frac{1}{2}b+%
\frac{3}{4},b<2$.
On the other hand, $P_{2}$ is unstable in the following cases:
(i) $a>\frac{1}{8}b^{2}-\frac{1}{8}b^{3}+\frac{1}{64}b^{4},$ $b\geq 2$.
(ii) $a>-\frac{1}{2}b+\frac{3}{4},$ $b<2.$
(iii) $a<\frac{1}{8}b^{2}-\frac{1}{8}b^{3}+\frac{1}{64}b^{4},b>2.$
$P_{2}$ is stable in the following cases:
(i) $a>\frac{1}{8}b^{2}-\frac{1}{8}b^{3}+\frac{1}{64}b^{4},a<-\frac{1}{2}b+%
\frac{3}{4},$ $b<2.$
(ii) $a<\frac{1}{8}b^{2}-\frac{1}{8}b^{3}+\frac{1}{64}b^{4},0\leq b\leq 2.$
(iii) $a<\frac{1}{8}b^{2}-\frac{1}{8}b^{3}+\frac{1}{64}b^{4},a>\frac{1}{2}b+%
\frac{3}{4}b^{2}-\frac{1}{4},-2**\frac{1}{2}b+%
\frac{3}{4}b^{2}-\frac{1}{4},-2****0$, the map (3) exhibits the following dynamical behaviors as shown in Fig. 3: In the interval $01.07$, it does not converge.
However, if we fix parameter $a=1$, and vary $b\geq 0$, the map (3) exhibits the following dynamical behaviors as shown in Fig. 6: For $0****0.675$, the map (3) does not converge. One interesting feature is that this map is not dissipative for all combinations of $a$ and $b$. In fact, there are values for which both Lyapunov exponents are positive, indicating hyperchaos as shown in Figs. 3(b) and 6(b).
Since the map (3) is not everywhere dissipative, its attractor can have a dimension equal to or even greater than 2.0 by virtue of the folding afforded by the quadratic nonlinearity. There are parameters such as $a=0.765$ and $b=0.854$ for which the two Lyapunov exponents are nearly equal and opposite ($0.10710$ and $-0.10744$), implying an attractor with a dimension of 1.9969 by the Kaplan-Yorke conjecture. Furthermore, when both Lyapunov exponents are positive, the dimension in principle could exceed 2.0, and this would be evident by examining the attractor in embeddings higher than 2. Takens' theorem [15] states that an embedding as large as $2D+1$ might be necessary to resolve the overlaps.
As a test of this prediction, the correlation dimension was calculated for various embeddings using the extraplation method of Sprott and Rowlands [16], and the results are plotted in Fig. 9 for the map (3) with $a=1$ and $b=0.675$ where the Lyapunov exponents are 0.171496 and 0.007595. The correlation dimension is approximately constant with a value of about 1.88 for all embeddings greater than 1. Figure 10 shows the regions of the $ab$-plane where the system is dissipative and bounded (in black) and where it is dissipative but area-expanding (in white) as determined from the sign of the numerical average of $\log |2y|$ over the orbit on the attractor.
These results suggest that the Takens' criterion is overly restrictive for the map (3) even though the map is noninvertible for all combinations of $a$ and $b$, and hence there is not a one-to-one reconstruction for the map. On the other hand, it is well known that basin boundaries arise in dissipative dynamical systems when two or more attractors are present. In such situations each attractor has a basin of initial conditions which lead asymptotically to that attractor. The sets that separate different basins are called the basin boundaries. In some cases the basin boundaries can have very complicated fractal structure and hence pose an impediment to predicting long-term behavior. For the map (3) we have calculated the attractors and their basins of attraction on a grid in $ab$-space where the system is chaotic. Three things to note about these plots are: First, there is a very wide variety of possible attractors, only some of which are shown in Figs. 2, 7, and 8. Second, most of the basin boundaries are smooth, but some appear to be fractal, and this is not a result of numerical errors since the structure persists as the number of iterations of each initial condition is increased. Third, it could be mentioned that we looked for coexisting attractors and did not find any for this system except for the trivial case of two stable fixed points shown as region 36 in Fig. 1.
\FRAME{ftbpFU}{308.625pt}{319.1875pt}{0pt}{\Qcb{Variation of the Lyapunov
exponents of map (3) versus the parameter $0**