Fractional Logistic Map
Atsushi Nagai
A new type of an integrable mapping is presented. This map is equipped with fractional difference and possesses an exact solution, which can be regarded as a discrete analogue of the Mittag-Leffler function.
nlin0206018.tex
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\title{Fractional Logistic Map}
\author{Atsushi Nagai}
\date{}
\begin{document}
\begin{center}
{\bf\Large Fractional Logistic Map}\\
{\large Atsushi Nagai}\footnote{e-mail:a-nagai@sigmath.es.osaka-u.ac.jp}\\
{\it Graduate School of Engineering Science, Osaka University}\\
{\it 1-3 Matikaneyama, Toyonaka 560-8531, Japan}
\end{center}
\begin{abstract}
A new type of an integrable mapping is presented. This map is equipped
with fractional difference and possesses an exact solution, which can
be regarded as a discrete analogue of the Mittag-Leffler function.
\end{abstract}
\section{Introduction}
Logistic map~\cite{Morishita,Hirota5}
\begin{align}\label{eq:logistic}
\frac{u_{n+1} - u_n}{\varepsilon} = a u_n(1-u_{n+1}) \quad (a>0)
\end{align}
is an integrable discretization of the well-known logistic equation.
\begin{align}
\frac{d}{dt}u=a u(1-u) \quad (a>0)
\end{align}
Its origin was a population model in ecology. In addition, it has been
reported~\cite{Sato} that the logistic map is used in many other fields,
for example, agriculture, life sciences and engineering.

Through dependent variable transformation,
\begin{align}
u_n = \frac{1+a\varepsilon}
{\displaystyle a\varepsilon g_n+1+a\varepsilon}
\end{align}
the logistic map~\eqref{eq:logistic} is linearized as 
\begin{align}
& \Delta_{-n} g_{n}
= -a g_{n} \label{eq:linear1},
\end{align}
where $\Delta_{-n}$ is a backward difference operator defined by
\begin{align}
\Delta_{-n}=\varepsilon^{-1}(1-E^{-1}), E^{-1}f_n = f_{n-1}.
\end{align}
Since equation~\eqref{eq:linear1} possesses a solution,
\begin{align}
g_n = g_0(1+a\varepsilon)^{-n}
\end{align}
a solution to the logistic map~\eqref{eq:logistic} is given by
\begin{align}
u_n=\frac{u_0}
{u_0+(1-u_0)(1+a\varepsilon)^{-n}}
\end{align}

The main purpose of this paper is to provide a new type of an integrable
mapping, which can be regarded as an extension of the logistic map and 
is equipped with fractional difference. This map, which we call the 
fractional logistic map here, possesses an exact solution and have 
another parameter $p$ which corresponds to an order of difference.
In section 2, we introduce one definition of fractional difference,
which is a slight modification of Hirota's fractional difference
operator. We also find an eigenfunction of this operator. 
Section 3 is the main consequence of this paper and presents the 
fractional logistic map. We also show its time evolution through 
numerical experiment.

\section{Fractional difference}
We here give a definition of fractional difference operator and 
its eigenfunction. Before going to its definition,
let us introduce fundamental functions $M(a;n)$ defined by
\begin{align}\label{eq:dK0}
M(a;n)=\frac{1}{\Gamma(a)}\varepsilon^{a-1}
\frac{\Gamma(n+a-1)}{\Gamma(n)} 
=\varepsilon^{a-1}\binomial{n+a-2}{n-1}\quad (a>0, n\in{\Bbb Z}),
\end{align}
where $\varepsilon$ is an interval length and 
$\binomial{a}{n} \ (a\in{\Bbb R}, n\in{\Bbb Z})$ stands for
a binomial coefficient defined by
\begin{align*}
\binomial{a}{n}=
\begin{cases}
\displaystyle\frac{a(a-1)\cdots(a-n+1)}{n!}=
\displaystyle\frac{\Gamma(a+1)}{\Gamma(a-n+1)\Gamma(n+1)}
& (n>0)\\
1 & (n=0)\\
0 & (n<0)
\end{cases}
\end{align*}
This function satisfies the following lemma.
\begin{lemma}
The following relations hold.
\begin{align}
&M(a;0)=0 \quad (a>1)\label{eq:dK:-n}\\
&\Delta_{-n}{M}(a+1;n)=
\varepsilon^{-1}({M}(a+1;n)-{M}(a+1;n-1))=
{M}(a;n) \quad (a>0)\label{eq:dK1}
\end{align}
\end{lemma}
{\bf Proof}: Equation~\eqref{eq:dK:-n} is obvious owing to the definition
of binomial coefficient. Equation~\eqref{eq:dK1} is proved by using 
the relation $\Gamma(x+1)=x\Gamma(x)$ as follows.
\begin{align*}
\varepsilon^{-1}&(M(a+1;n)-M(a+1;n-1))
\\
&= \frac{\varepsilon^{a-1}}{\Gamma(a+1)}
\left(\frac{(n+a-1)\Gamma(n+a-1)}{\Gamma(n)}-
\frac{(n-1)\Gamma(n+a-1)}{\Gamma(n)}\right)\\
&= \frac{a \varepsilon^{a-1}}{\Gamma(a+1)}
\frac{\Gamma(n+a-1)}{\Gamma(n)}=M(a;n).
\end{align*}
\hfill$\blacksquare$

Next we go to the definition of fractional difference.
Hirota~\cite{Hirota} took the first $n$ terms of Taylor series of
$\Delta_{-n}^{\alpha}=\varepsilon^{-\alpha}(1-E^{-1})^{\alpha}$ and gave
the following definition.
\begin{defi}\label{def:dfrac1}
Let $\alpha \in {\Bbb R}$. Then
difference operator of order $\alpha$ is defined by
\begin{align}\label{eq:dfrac1}
\Delta_{-n}^\alpha u_n
=
\begin{cases}&\varepsilon^{-\alpha}
\displaystyle\sum_{j=0}^{n-1} \binomial{\alpha}{j}(-1)^j u_{n-j}\quad
\alpha\ne 1,2,\cdots\\
& \varepsilon^{-m}
\displaystyle\sum_{j=0}^{m} \binomial{m}{j}(-1)^j u_{n-j}\quad
\alpha=m\in{\Bbb Z}_{>0}
\end{cases}
\end{align}
\end{defi}
It should be noted that Diaz, Osler~\cite{Diaz} gave another definition
of fractional difference,
\begin{align}
\Delta_{-n}^\alpha u(t)&=\varepsilon^{-\alpha}
\sum_{j=0}^{\infty} \binomial{\alpha}{j}(-1)^j u_{n-j}
\end{align}
We here adopt another difference oparator $\Delta_{*,-n}^{\alpha}$ 
by modifying Hirota's operator.
\begin{defi}\label{def:dfrac2}
Let $\alpha\in{\Bbb R}$ and $m$ be an integer such that $m-1<\alpha\leq m$.
We define difference operator of order $\alpha$, $\Delta_{*,-n}^\alpha$,
by
\begin{align}
\label{eq:dfrac2}
 \Delta_{*,-n}^\alpha u_n \equiv \Delta_{-n}^{\alpha-m}
\Delta_{-n}^m u_n
= \varepsilon^{m-\alpha}\sum_{j=0}^{n-1} \binomial{\alpha-m}{j}
(-1)^j \Delta_{-k}^m u_k|_{k=n-j}
\end{align}
\end{defi}
We define a new function,
\begin{align}
F_a(\lambda,n)= \sum_{j=0}^\infty \lambda^jM(aj+1;n).
=\sum_{j=0}^\infty \lambda^j\varepsilon^{aj}
\frac{\Gamma(n+aj)}{\Gamma(aj+1)\Gamma(n)}
\end{align}
\begin{rem}
Putting $a=1$ in the above definition, we have
\begin{align}
F_1(\lambda,n)&=\sum_{j=0}^\infty \lambda^j\varepsilon^{j}
\frac{\Gamma(n+j)}{\Gamma(j+1)\Gamma(n)}\nonumber\\
&=\sum_{j=0}^\infty (\lambda\varepsilon)^{j}\binomial{n+j-1}{j}
\nonumber\\
&=\sum_{j=0}^\infty (-\lambda\varepsilon)^{j}\binomial{-n}{j}
=(1-\lambda\varepsilon)^{-n}.
\end{align}
\end{rem}
\begin{rem}
In the limit of $\varepsilon\rightarrow 0, n\rightarrow\infty$ with
$t=n\varepsilon$ fixed, the function $F_a(\lambda,n)$ converges to
\begin{align}
F_a(\lambda,n)\rightarrow \sum_{j=0}^\infty \frac{\lambda^j t^{aj}}
{\Gamma(aj+1)} = E_a(\lambda t^a).
\end{align}
$E_a(x)$ is the well-known Mittag-Leffler function and
and its asymptotic behavior is studied in detail in \cite{Sansonne}.
\end{rem}
\begin{theorem}\label{th:dML}
If $a>0$, the function $F_a(\lambda,n)$ is an eigen-function of the 
fractional difference operator \eqref{eq:dfrac2}. That is,
\begin{align}
\Delta_{*,-n}^a F_a(\lambda,n) = 
\lambda F_a(\lambda,n)
\end{align}
\end{theorem}
{\bf Proof of Theorem~\ref{th:dML}}: Let $m$ be an integer such that
$m-1<a\leq m$. Then we have
\begin{align}
\Delta_{*,-n}^a &F_a(\lambda,n)
= \Delta_{*,-n}^a \left(1+
\sum_{j=1}^\infty \lambda^j M(aj+1;n)\right)\nonumber\\
&= \Delta_{*,-n}^a \sum_{j=1}^\infty \lambda^j M(aj+1;n)\nonumber\\
&=\Delta_{-n}^{a-m}\sum_{j=1}^\infty \lambda^j \Delta_{-n}^m 
M(aj+1;n)\nonumber\\
&=\sum_{j=1}^\infty \lambda^j \Delta_{-n}^{a-m} M(aj+1-m;n)
\label{eq:proof1}
\end{align}
Each summand in the above equation is given by
\begin{align}
\Delta_{-n}^{a-m}&M(aj+1-m;n)
=\sum_{k=0}^{n-1}\binomial{a-m}{k}(-1)^k M(aj+1-m;n-k)\nonumber\\
&=\sum_{k=0}^{n-1}\binomial{a-m}{k}(-1)^k 
\binomial{aj-m+n-k-1}{n-k-1}\nonumber\\
&=\sum_{k=0}^{n-1}\binomial{a-m}{k}(-1)^{n-1}
\binomial{-aj+m-1}{n-k-1}\nonumber\\
&=(-1)^{n-1}\binomial{a-aj-1}{n-1}=\binomial{aj-a-1}{n-1}=M(aj-a+1;n).
\label{eq:proof2}
\end{align}
Therefore, substitution of eq.~\eqref{eq:proof2} into 
eq.~\eqref{eq:proof1} gives
\begin{align}
\Delta_{*,-n}^a F_a(\lambda,n)
&=\sum_{j=1}^\infty \lambda^j M(aj-a+1;n)
=\sum_{j=0}^\infty \lambda^{j+1} M(aj+1;n)=\lambda F_a(\lambda,n)
\end{align}
which completes the proof.\hfill$\blacksquare$

\section{Fractional Logistic map}
This section presents a fractionl integrable mapping, which we call
fractional logistic map here. We start with a linear equation
\begin{align}\label{eq:linearp}
& \Delta_{*,-n}^p g_n = -a g_n \quad (0<p\leq 1, a>0)
\end{align}
instead of linear eq.~\eqref{eq:linear1}. The above equation is 
rewritten as
\begin{align}
g_n = (1+a\varepsilon^p)^{-1}\left\{g_{n-1}-
\sum_{j=1}^{n-1} \binomial{p-1}{j}(-1)^j (g_{n-j}-g_{n-j-1})\right\}
\end{align}
Through the similar dependent variable transformation as logistic map,
\begin{align}
u_n = \frac{1+a\varepsilon^p}{a\varepsilon^p g_n +1+a\varepsilon^p}
\end{align}
we finally obtain a fractional logistic map
\begin{align}
\label{eq:flm}
u_n = \frac{1}{1+\displaystyle\frac{1}{1+a\varepsilon^p}\left\{
\frac{1}{u_{n-1}}-
\sum_{j=1}^{n-1} \binomial{p-1}{j}(-1)^j \left(\frac{1}{u_{n-j}}
-\frac{1}{u_{n-j-1}}\right)\right\}}.
\end{align}
Due to the Theorem~\ref{th:dML}, eq.~\eqref{eq:linearp} has a solution,
\begin{align}
g_n = g_0 F_p(-a;n).
\end{align}
Therefore, a solution to the fractional logistic map~\eqref{eq:flm}
is given by
\begin{align}
u_n=\frac{u_0}
{u_0+(1-u_0)F_p(-a;n)}
\end{align}
Putting $p=1$ in eq.~\eqref{eq:flm}, we have
\begin{align}
u_n = \frac{(1+a\varepsilon)u_{n-1}}{1+(1+a\varepsilon)u_{n-1}},
\end{align}
which recovers the logistic map. The following figure illustrates
time evolutions of fractional logistic map with order parameter $p = n/4
(n=1,2,3,4)$. We have put $u_0=0.1$, $a=1.0$
and $\varepsilon=0.1$. 

\begin{figure}[htbp]
\begin{center}
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\put(1236.93,614.00){\usebox{\plotpoint}}
\put(1257.69,614.00){\usebox{\plotpoint}}
\put(1278.44,614.00){\usebox{\plotpoint}}
\put(1299.20,614.00){\usebox{\plotpoint}}
\put(1319.95,614.00){\usebox{\plotpoint}}
\put(1340.71,614.00){\usebox{\plotpoint}}
\put(1361.46,614.00){\usebox{\plotpoint}}
\put(1382.22,614.00){\usebox{\plotpoint}}
\put(1402.97,614.00){\usebox{\plotpoint}}
\put(1423.73,614.00){\usebox{\plotpoint}}
\put(1439,614){\usebox{\plotpoint}}
\end{picture}

\end{center}
\caption{Time evolutions of fractional logistic map}
\end{figure}

Considering the fact that the Mittag-Leffler function has an 
asymptotic behavior~\cite{Sansonne},
\begin{align}
E_a(\lambda t^a) = -\sum_{n=1}^{N-1}\frac{\lambda^{-n}t^{-an}}
{\Gamma(1-an)}+O(t^{-aN}), \quad t\rightarrow\infty, \lambda<0,
\end{align}
we can observe that $u_n$ converges to $1$ at the order 
of $O(1/n^p)$ if $0<p<1$.
The following table illustrates a numerical result in which we 
apply the $\rho$-algorithm~\cite{Brezinski}
\begin{align*}
\rho_{k+1}^n &= \rho_{k-1}^{n+1}+\frac{(k+n)^p-n^p}{
\rho_k^{n+1}-\rho_k^n}\\
& \rho_{-1}^n = 0, \ \rho_0^n = u_n
\end{align*}
to the sequence $\{u_n\}$ in the case $p=1/4$. This table shows
that $u_n$ converges to the value near $1.0$ at the order 
$O(1/n^{1/4})$ as $n$ tends to $+\infty$.

\begin{table}[htbp]
\caption{The $\rho$-algorithm applied to the sequence $\{u_n\}$ 
in the case $p=1/4$}
\begin{center}
\begin{tabular}{|l||l|l|l|l|l|l|}\hline
& $\rho_1^n$ & $\rho_3^n$ & $\rho_5^n$ & $\rho_7^n$ & $\cdots$ &
$\rho_{21}^n$ \\ \hline
1 & 0.1    & 0.19745 & 0.84609 & 0.84288  && 1.00145\\ \hline
2 & 0.14791 & 0.23921 & 0.86006 & 1.30904 && 0.99915\\ \hline
3 & 0.16019 & 0.27802 & 0.99118 & 1.21129 && 1.00131\\ \hline
4 & 0.16764 & 0.31406 & 1.07379&  1.17824 && 1.00139\\ \hline
5 & 0.17313 & 0.34778 & 1.11866 & 1.16134 && 1.00112\\ \hline
6 & 0.17752 & 0.37941 & 1.14064 & 1.15284 && 1.00120\\ \hline
7 & 0.18121 & 0.40908 & 1.14968  & 1.15045 && 1.00120 \\ \hline
8 & 0.18442 & 0.43692 & 1.15161  & 1.15375 && 1.00121 \\ \hline
9 & 0.18726 & 0.46304 & 1.14970  & 1.16376 && 1.00122 \\ \hline
\vdots & \vdots & \vdots & \vdots & \vdots && \vdots \\ \hline
\end{tabular}
\end{center}
\end{table}
\section*{Acknowledgement}
The author would like to express his sincere thanks to Professor 
Yoshinori Kametaka at Osaka University for providing me with 
information of Mittag-Leffler function. He also thanks Professor
Junkichi Satsuma at University of Tokyo for many valuable suggestions.
\begin{thebibliography}{99}
\bibitem{Brezinski} C. Brezinski and M. Redivo Zaglia, {\it 
Extrapolation Methods. Theory and Practice}, North Holland,
Amsterdam, 1991.
\bibitem{Diaz} J. B. Diaz and T. J. Osler, {\it Differences of
Fractional Order}, Math. Comp., {\bf 28}(1974) pp. 185--202.
\bibitem{Hirota5} R. Hirota, {\it Nonlinear Partial Difference
Equations. V. Nonlinear Evolution Equations Reducible to Linear 
Equations}, J. Phys. Soc. Jpn {\bf 46}(1979) pp. 312--319.
\bibitem{Hirota} R. Hirota, Lectures on difference equations,
Science-sha, 2000 [in Japanese].
\bibitem{Morishita} F. Morishita, {\it The fitting of the logistic
equation to the rate of increase of population density}, Researches on
Population Ecology, {\bf 7}(1965) pp. 52--55.
\bibitem{Samko}  S. G. Samko, A. A. Kilbas, O. I. Marichev, {\it Fractional
Integrals and Derivatives}, Gordon and Breach Science Publishers, 1993.
\bibitem{Sansonne} G. Sansonne and J. Gerretsen, {\it Lectures on the 
Theory of Functions of a Complex Variable}, P. Noordhoff-Groningen, 1960.
\bibitem{Sato} D. Satoh and S. Yamada, 
{\it Parameter Estimation of Discrete Logistic Curve Models for Software
Reliability Assessment}, Japan J. Ind. Appl. Math., {\bf 19}(2002) 
pp. 39--54.
\bibitem{Sugimoto} N. Sugimoto, {\it `Generalized' Burgers equations and
fractional calculus}, Nonlinear Wave Motion (ed. A. Jeffrey), Longman 
1989, pp. 162--179.
\end{thebibliography}
\end{document}







nlin0206018.extracted_bib
\begin{thebibliography}{99}
\bibitem{Brezinski} C. Brezinski and M. Redivo Zaglia, {\it 
Extrapolation Methods. Theory and Practice}, North Holland,
Amsterdam, 1991.
\bibitem{Diaz} J. B. Diaz and T. J. Osler, {\it Differences of
Fractional Order}, Math. Comp., {\bf 28}(1974) pp. 185--202.
\bibitem{Hirota5} R. Hirota, {\it Nonlinear Partial Difference
Equations. V. Nonlinear Evolution Equations Reducible to Linear 
Equations}, J. Phys. Soc. Jpn {\bf 46}(1979) pp. 312--319.
\bibitem{Hirota} R. Hirota, Lectures on difference equations,
Science-sha, 2000 [in Japanese].
\bibitem{Morishita} F. Morishita, {\it The fitting of the logistic
equation to the rate of increase of population density}, Researches on
Population Ecology, {\bf 7}(1965) pp. 52--55.
\bibitem{Samko}  S. G. Samko, A. A. Kilbas, O. I. Marichev, {\it Fractional
Integrals and Derivatives}, Gordon and Breach Science Publishers, 1993.
\bibitem{Sansonne} G. Sansonne and J. Gerretsen, {\it Lectures on the 
Theory of Functions of a Complex Variable}, P. Noordhoff-Groningen, 1960.
\bibitem{Sato} D. Satoh and S. Yamada, 
{\it Parameter Estimation of Discrete Logistic Curve Models for Software
Reliability Assessment}, Japan J. Ind. Appl. Math., {\bf 19}(2002) 
pp. 39--54.
\bibitem{Sugimoto} N. Sugimoto, {\it `Generalized' Burgers equations and
fractional calculus}, Nonlinear Wave Motion (ed. A. Jeffrey), Longman 
1989, pp. 162--179.
\end{thebibliography}