# ASYMPTOTIC CYCLES IN FRACTIONAL MAPS
OF ARBITRARY POSITIVE ORDERS

Mark Edelman 1 Avigayil B. Helman 2

###### Abstract

Many natural (biological, physical, etc.) and social systems possess power-law
memory, and their mathematical modeling requires application of discrete and
continuous fractional calculus. Most of these systems are nonlinear and
demonstrate regular and chaotic behavior, which may be very different from the
behavior of memoryless systems. Finding periodic solutions is essential for
understanding regular and chaotic behavior of nonlinear systems. Fractional
systems do not have periodic solutions except fixed points. Instead, they have
asymptotically periodic solutions which, in the case of stable regular
behavior, converge to the periodic sinks (similar to regular dissipative
systems) and, in the case of unstable/chaotic behavior, act as repellers. In
one of his recent papers, the first author derived equations which allow
calculations of asymptotically periodic points for a wide class of discrete
maps with memory. All fractional and fractional difference maps of the orders
$0<\alpha<1$ belong to this class. In this paper we derive the equations that
allow calculations of the coordinates of the asymptotically periodic points
for a wider class of maps which include fractional and fractional difference
maps of the arbitrary positive orders $\alpha>0$. The maps are defined as
convolutions of a generating function $-G_{K}(x)$, which may be the same as in
a corresponding regular map $x_{n+1}=-G_{K}(x_{n})+x_{n}$, with a kernel
$U_{\alpha}(k)$, which defines the type of a map. In the case of fractional
maps, it is $U_{\alpha}(k)=k^{\alpha}$, and it is
$U_{\alpha}(k)=k^{(\alpha)}$, the falling factorial function, in the case of
fractional difference maps. In this paper we define the space of kernel
functions which allow calculations of the periodic points of the corresponding
maps with memory. We also prove that in fractional maps of the orders
$1<\alpha<2$ the total of all physical momenta of period–$l$ points is zero.

MSC 2010: Primary 26A33; Secondary 47H99, 34A99, 37G15, 70K50, 39A70

Key Words and Phrases: fractional maps; periodic points; chaos

##  1 Introduction

In recent years many authors tried to make fractional generalizations of
regular maps. The authors apologize if in the following we are missing some
references to outstanding papers on fractional maps. In what follows, the
authors intentionally omit references to the papers in which fractional maps
are derived as approximate numerical algorithms for solving corresponding
fractional differential equations, fractional maps which contain fractions but
have no memory (this would contradict the spirit of fractional calculus), and
works in which maps are not derived but declared.

Two comprehensive forms of fractional maps, whose introduction has a
reasonable justification, are fractional and fractional difference maps
described in the review articles [16, 17]. Fractional maps were introduced as
the exact solutions of fractional differential equations of kicked systems in
the same way as the universal map was introduced in regular dynamics by
Tarasov and Zaslavsky for systems of the orders $\alpha>1$ [44, 39]. They were
extended for orders $0\leq\alpha\leq 1$ in [6], investigated in [6, 21, 5, 20,
7, 8, 11, 13, 14, 15, 18], and reviewed in [16, 17, 40, 9]. In 2014 Wu,
Baleanu, and Zeng proposed to use solutions of fractional difference equations
to generate fractional difference maps [49, 50, 36] and Edelman introduced the
universal fractional difference families of maps [10, 12]. Fractional
difference maps were investigated in [8, 13, 14, 15, 18, 10, 12, 3, 37, 48,
19]. Applications of fractional and fractional difference maps include
economic modeling [43, 45], secure communication [37], and modeling of
longevity of living species [18].

Fractional and fractional difference maps were first introduced as
generalizations of the regular universal and standard maps. These regular maps
are closely related to physics and Hamiltonian systems and they are area-
preserving. As a result, all integer members of fractional and fractional
difference $\alpha$-families of maps with $\alpha>1$ are area/volume
preserving and behavior of maps of the order $\alpha$ near $\alpha=2$ is of a
special interest. Systems of the order $1<\alpha<2$ behave as systems of the
order two but with dissipation. In biology, fractional order $0<\alpha<2$
equations play a significant role in description of electromagnetic waves
propagating through the biological tissue (for biological applications of
fractional calculus see, e.g., references in review articles [9, 23, 38]).
Equations defining periodic points in fractional and fractional difference
maps of the orders $0<\alpha<1$ were derived in [19]. In this paper we
consider equations that allow calculation of asymptotically periodic points in
fractional and fractional difference maps of any real orders $\alpha>0$.

##  2 Preliminaries

###  2.1 Definition of fractional and fractional difference maps

The definitions (derivations) of fractional and fractional difference families
of maps can be found in [16, 17] and the references therein.

The fractional Caputo universal map of the order $\alpha$ is defined (derived)
as

$\displaystyle
x_{n+1}=\sum^{m-1}_{k=0}\frac{b_{k}}{k!}h^{k}(n+1)^{k}-\frac{h^{\alpha}}{\Gamma(\alpha)}\sum^{n}_{k=0}G_{K}(x_{k})(n-k+1)^{\alpha-1},$
(2.1)

where $\alpha\in\mathbb{R}$, $\alpha\geq 0$, $m=\lceil\alpha\rceil$,
$x_{n}=x(t=nh)$, $t$ is time, $n\in\mathbb{Z}$, $n\geq 0$ (the same as
$n\in\mathbb{N}_{0}$), $b_{k}\in\mathbb{R}$ are constants, and $G_{K}(x)$ is a
function (could be nonlinear) depending on a parameter $K$.

The $h$-difference Caputo universal $\alpha$-family of maps is defined
(derived) as

$\displaystyle
x_{n+1}\\!=\\!\sum^{m-1}_{k=0}\frac{c_{k}}{k!}((n\\!+\\!1)h)^{(k)}_{h}-\frac{h^{\alpha}}{\Gamma(\alpha)}\sum^{n+1-m}_{s=0}\\!(n\\!-\\!s\\!-\\!m\\!+\\!\alpha)^{(\alpha-1)}G_{K}(x_{s+m-1}),$
(2.2)

where $x_{k}=x(kh)$, $\alpha\in\mathbb{R}$, $\alpha\geq 0$,
$m=\lceil\alpha\rceil$, $n\in\mathbb{Z}$, $n\geq m-1$, $c_{k}\in\mathbb{R}$
are constants, and $G_{K}(x)$ is a function with a parameter $K$. In both maps
$h\in\mathbb{R}$ and $h>0$. The definition of the falling factorial
$t^{(\alpha)}$ is

$t^{(\alpha)}=\frac{\Gamma(t+1)}{\Gamma(t+1-\alpha)},\ \ t\neq-1,-2,-3,...$
(2.3)

The falling factorial is asymptotically a power function:

$\lim_{t\rightarrow\infty}\frac{\Gamma(t+1)}{\Gamma(t+1-\alpha)t^{\alpha}}=1,\
\ \ \alpha\in\mathbb{R}.$ (2.4)

The $h$-falling factorial $t^{(\alpha)}_{h}$ is defined as

$t^{(\alpha)}_{h}=h^{\alpha}\frac{\Gamma(\frac{t}{h}+1)}{\Gamma(\frac{t}{h}+1-\alpha)}=h^{\alpha}\Bigl{(}\frac{t}{h}\Bigr{)}^{(\alpha)},\
\ \frac{t}{h}\neq-1,-2,-3,...$ (2.5)

The important particular forms of the Caputo universal map are the Caputo
logistic families of maps with $G_{K}(x)=x-Kx(1-x)$ which converge to the
classical logistic map when $\alpha=1$ and $h=1$ and the Caputo standard
families of maps with $G_{K}(x)=K\sin(x)$, which converge to the classical
standard maps when $\alpha=2$ and $h=1$.

###  2.2 Asymptotically periodic cycles for $0<\alpha<1$

Finding periodic trajectories is essential for understanding maps’ regular and
chaotic dynamics [4].

It is known that fractional systems may not have periodic solutions except
fixed points (see, e.g., [24, 25, 2, 26, 46, 47, 51]) but they may have
asymptotically periodic solutions (see [16, 17]). A formula for calculation of
the asymptotically period two solutions for fractional and fractional
difference maps, without a strict proof, may be found in [14]. Formulae to
find asymptotically periodic points for $0<\alpha<1$ were strictly derived in
[19].

When $0<\alpha<1$, all forms of the universal $\alpha$-families of maps
considered in this paper, Eqs. (2.1) and (2.2), can be written as

$\displaystyle x_{n}=x_{0}-\sum^{n-1}_{k=0}{G}^{0}(x_{k})U_{\alpha}(n-k),$
(2.6)

where ${G}^{0}(x)=h^{\alpha}G_{K}(x)/\Gamma(\alpha)$ and $x_{0}$ is the
initial condition. In fractional maps Eq. (2.1)

$\displaystyle U_{\alpha}(n)=n^{\alpha-1},\ \ \ \ U_{\alpha}(1)=1.$ (2.7)

and in fractional difference maps, Eq. (2.2),

$\displaystyle U_{\alpha}(n)=(n+\alpha-2)^{(\alpha-1)},\ \ \ \ $
$\displaystyle U_{\alpha}(1)=(\alpha-1)^{(\alpha-1)}=\Gamma(\alpha).$ (2.8)

The equations that define asymptotically period $l$ points ($l$-cycles)

$x_{lim,m}=\lim_{N\rightarrow\infty}x_{Nl+m},\ \ \ 0<m<l+1,$ (2.9)

are

$\displaystyle
x_{lim,m+1}-x_{lim,m}=S_{1}{G}^{0}(x_{lim,m})+\sum^{m-1}_{j=1}S_{j+1}{G}^{0}(x_{lim,m-j})$
$\displaystyle+\sum^{l-1}_{j=m}S_{j+1}G^{0}(x_{lim,m-j+l}),\ \ 0<m<l$ (2.10)

and

$\sum^{l}_{j=1}G^{0}(x_{lim,j})=0,$ (2.11)

where

${S}_{1}=-U_{\alpha}(1)+\sum^{\infty}_{k=1}\Bigl{[}U_{\alpha}(lk)-U_{\alpha}(lk+1)\Bigr{]}$
(2.12)

and

$\displaystyle
S_{j+1}=\sum^{\infty}_{k=0}\Bigl{[}U_{\alpha}(lk+j)-U_{\alpha}(lk+j+1)\Bigr{]},\
\ 0<j<l.$ (2.13)

Note that

$\sum^{l}_{j=1}S_{j}=0.$ (2.14)

The results of [19] were obtained for a wide class of kernels $U_{\alpha}(x)$
which belong to the space $\mathbb{D}^{0}(\mathbb{N}_{1})$ of diverging
functions, defined by the following:

######  Definition 2.1.

A function $f(n)$, $n\in\mathbb{N}_{1}$, belongs to the space
$\mathbb{D}^{i}(\mathbb{N}_{1})$ if $|\sum^{\infty}_{k=1}\Delta^{i}f(k)|>N$,
$\forall N$, $N\in\mathbb{N}$, while the series
$\sum^{\infty}_{k=1}\Delta^{i+1}f(k)$ is converging absolutely:

$\displaystyle\mathbb{D}^{i}(\mathbb{N}_{1})\ \ =\ \
\\{f:|\sum^{\infty}_{k=1}\Delta^{i}f(k)|>N,\ \ \forall N,\ \ N\in\mathbb{N},\
\ $ $\displaystyle\sum^{\infty}_{k=1}|\Delta^{i+1}f(k)|=C,\ \
C\in\mathbb{R}_{+}\\}.$ (2.15)

Here $\Delta$ is a forward difference operator defined as

$\Delta f(n)=f(n+1)-f(n).$ (2.16)

We may state the following theorem that was proven in [19]:

######  Theorem 2.1.

If the map Eq. (2.6) with $U_{\alpha}(n)\in\mathbb{D}^{0}(\mathbb{N}_{1})$ has
asymptotically period $l$ points Eq. (2.9) in the domain of the function
$G^{0}(x)$, then they are defined by the system of equations Eq. (2.10) and
Eq.(2.11), with the coefficients defined by Eq. (2.12) and Eq.(2.13).

###### P r o o f..

The proof is given in [19]. $\Box$

##  3 Asymptotically periodic cycles for $1<\alpha<2$

We dedicate a section to the case of fractional order systems with
$1<\alpha<2$ which frequently appear in applications. The fractional standard
and logistic maps of the orders $1<\alpha<2$ were already investigated in [17,
6, 21, 5, 20, 7, 8, 14, 9, 49, 10, 12]. Let’s notice that all phase portraits
in [49] are incorrect. When $1<\alpha<2$, the map equations, Eq. (2.1) and
Eq.(2.2), can be written as

$\displaystyle
x_{n+1}=x_{0}+h(n+1)p_{0}-h\sum^{n}_{k=0}G^{1}(x_{k})U_{\alpha}(n-k+1)+hf_{1}(n).$
(3.1)

Here $G^{1}(x)=h^{\alpha-1}G_{K}(x)/\Gamma(\alpha)$, $x_{0}$ is the initial
coordinate, $U_{\alpha}(n)$ is defined by Eqs. (2.7) and (2.8), $p_{0}$ is the
initial momentum ($b_{1}$ or $c_{1}$ in corresponding formulae), and
$f_{1}(n)=0$ in Eq. (2.1) and
$f_{1}(n)=h^{\alpha-1}G(x_{0})(n-1+\alpha)^{(\alpha-1)}/\Gamma(\alpha)\sim
n^{\alpha-1}$ in Eq. (2.2).

###  3.1 Equations defining periodic points

We define the difference momentum as

$\displaystyle p^{1}_{n+1}=\Delta_{h}x_{n}=\frac{x_{n+1}-x_{n}}{h}.$ (3.2)

Taking into account that $U_{\alpha}(0)=0$ and using $x_{n}$ from Eq. (3.1),
the difference momentum Eq. (3.2) can be written as

$\displaystyle
p^{1}_{n}=p_{0}-\sum^{n-1}_{k=0}G^{1}(x_{k})U^{1}_{\alpha}(n-k)+f_{1}(n-1)-f_{1}(n-2),$
(3.3)

where

$\displaystyle U^{1}_{\alpha}(n)=U_{\alpha}(n)-U_{\alpha}(n-1)$
$\displaystyle=\begin{array}[]{c}\left\\{\begin{array}[]{lll}n^{\alpha-1}-(n-1)^{\alpha-1}\sim
n^{\alpha-2}{\rm and}\ \ {U}^{1}_{\alpha}(1)=1\ \ {\rm in\ \
Eq.~{}(\ref{FrCMapx})};\\\
(n+\alpha-2)^{(\alpha-1)}-(n+\alpha-3)^{(\alpha-1)}\\\
=(\alpha-1)(n+\alpha-3)^{(\alpha-2)}\\\ =(\alpha-1)U_{\alpha-1}(n)\sim
n^{\alpha-2}\\\ {\rm and}\ \ {U}^{1}_{\alpha}(1)=\Gamma(\alpha)\ \ {\rm in\ \
Eq.~{}(\ref{FalFacMap_h})},\end{array}\right.\end{array}$ (3.10)

$f_{1}(n)-f_{1}(n-1)=0$ in Eq. (2.1) and $f_{1}(n)-f_{1}(n-1)\sim
n^{\alpha-2}$ in Eq. (2.2). Note, that the definitions of
${U}^{1}_{\alpha}(1)$ in Eq. (3.10) and $U_{\alpha}(1)$ in Eqs. (2.7), (2.8)
are identical.

Assuming that asymptotically period $l$ points ($l$-cycles) Eq. (2.9) exist,
the corresponding limiting values of difference momentum are defined as

$p^{1}_{lim,m}=\lim_{N\rightarrow\infty}p^{1}_{Nl+m}=\frac{x_{lim,m}-x_{lim,m-1}}{h},\
\ 1<m<l+1$ (3.11)

and

$p^{1}_{lim,1}=\lim_{N\rightarrow\infty}p^{1}_{Nl+1}=\frac{x_{lim,1}-x_{lim,l}}{h}.$
(3.12)

The equations defining asymptotically cyclic points are obtained considering
the limit $N\rightarrow\infty$, in which case the last two terms of Eq. (3.3)
disappear. The asymptotic behavior of ${U}^{1}_{\alpha}(n)$ when $1<\alpha<2$
is the same as the asymptotic behavior of ${U}_{\alpha}(n)$ when $0<\alpha<1$,
which means that $U^{1}_{\alpha}(x)\in\mathbb{D}^{0}(\mathbb{N}_{1})$. This
implies that all steps of the derivation of formulae Eqs. (2.10) - (2.14),
strictly proven in [19], can be repeated here. Subtracting consecutive
limiting values and adding all limiting values of difference momenta produces
the following results:

$\displaystyle
p^{1}_{lim,m+1}-p^{1}_{lim,m}={S}^{1}_{1}{G}^{1}(x_{lim,m})+\sum^{m-1}_{j=1}{S}^{1}_{j+1}{G}^{1}(x_{lim,m-j})$
$\displaystyle+\sum^{l-1}_{j=m}{S}^{1}_{j+1}{G}^{1}(x_{lim,m-j+l}),\ \ 0<m<l$
(3.13)

and

$\sum^{l}_{j=1}{G}^{1}(x_{lim,j})=0,$ (3.14)

where

${S}^{1}_{1}=-{U}^{1}_{\alpha}(1)+\sum^{\infty}_{k=1}\Bigl{[}{U}^{1}_{\alpha}(lk)-{U}^{1}_{\alpha}(lk+1)\Bigr{]},$
(3.15)
$\displaystyle{S}^{1}_{j+1}=\sum^{\infty}_{k=0}\Bigl{[}{U}^{1}_{\alpha}(lk+j)-{U}^{1}_{\alpha}(lk+j+1)\Bigr{]},\
\ 0<j<l,$ (3.16)

and, still,

$\sum^{l}_{j=1}{S}^{1}_{j}=0.$ (3.17)

Using Eqs. (3.11) and (3.12) the system of $l$ equations which defines all $l$
periodic points can be written as (notice that ${G}^{0}(x)=h{G}^{1}(x)$)

$\displaystyle
x_{lim,2}-2x_{lim,1}+x_{lim,l}={S}+\\!\sum^{l-1}_{j=1}{S}^{1}_{j+1}{G}^{0}(x_{lim,2-j}),$
(3.18) $\displaystyle
x_{lim,m+1}-2x_{lim,m}+x_{lim,m-1}={S}^{1}_{1}{G}^{0}(x_{lim,m})$
$\displaystyle+\sum^{m-1}_{j=1}{S}^{1}_{j+1}{G}^{0}(x_{lim,m-j})+\\!\sum^{l-1}_{j=m}{S}^{1}_{j+1}{G}^{0}(x_{lim,m-j+l}),\
1<m<l$ (3.19)

and

$\sum^{l}_{j=1}{G}^{0}(x_{lim,j})=0.$ (3.20)

This is the end of the proof of the following theorem:

######  Theorem 3.1.

If the map Eq. (3.1) with $\Delta
U_{\alpha}(n)\in\mathbb{D}^{0}(\mathbb{N}_{1})$ has asymptotically period $l$
points Eq. (2.9) in the domain of the function $G^{1}(x)$, then they are
defined by the system of equations Eq. (3.18), Eq. (3.19), and Eq.(3.20), with
the coefficients defined by Eq. (3.15) and Eq.(3.16).

###  3.2 Periodic points in the phase space of discrete fractional systems

Let us notice that in fractional difference maps (truly discrete systems),
values of difference momentum coincide with the corresponding values of
physical momentum (see Eqs. (28) and (29) in [10]). In this case, after the
$x$-coordinates of asymptotically periodic points are found, the limiting
values of physical momentum $p$ can be calculated using Eqs. (3.11) and
(3.12). This fact allows us to identify asymptotically periodic points,
$(x_{lim,j},p_{lim,j})$, for $1\leq j\leq l$ in the phase space of a system.

This is not the case in fractional maps, in which a value of physical momentum
$p(t)$ is a continuous solution of a differential equation of a kicked system,
and $p_{n}=p(t_{n})$, where $t_{n}$ is a time instant at which a kick occurs.
In this case, difference momentum is just an instrument to calculate space
coordinates of a cyclic point and a value of physical momentum, which is a
solution of the corresponding differential equation, is given by the formula
(see Eq. (18.196) in [40] and Eq. (38) in [16])

$p_{n+1}=p_{0}-\frac{h^{\alpha-1}}{\Gamma(\alpha-1)}\sum^{n}_{k=0}G_{K}(x_{k})(n-k+1)^{\alpha-2},$
(3.21)

or

$\displaystyle
p_{n}=p_{0}-\sum^{n-1}_{k=0}\tilde{G}^{1}(x_{k})(n-k)^{\tilde{\alpha}-1}=p_{0}-\sum^{n-1}_{k=0}\tilde{G}^{1}(x_{k})U_{\tilde{\alpha}}(n-k),$
(3.22)

where $\tilde{G}^{1}(x)=h^{\tilde{\alpha}}G_{K}(x)/\Gamma(\tilde{\alpha})$ and
$\tilde{\alpha}=\alpha-1$. Comparing Eq. (3.22) to Eq. (2.6), we may derive
equations similar to Eq. (2.10) that define relationships between limiting
values of momentum

$p_{lim,m}=\lim_{N\rightarrow\infty}p_{Nl+m},\ \ \ 0<m<l+1,$ (3.23)

in asymptotically period $l$ cycles:

$\displaystyle
p_{lim,m+1}-p_{lim,m}=S_{1,{\tilde{\alpha}}}\tilde{G}^{1}(x_{lim,m})+\sum^{m-1}_{j=1}S_{j+1,{\tilde{\alpha}}}\tilde{G}^{1}(x_{lim,m-j})$
$\displaystyle+\sum^{l-1}_{j=m}S_{j+1,{\tilde{\alpha}}}\tilde{G}^{1}(x_{lim,m-j+l}),\
\ 0<m<l,$ (3.24)

where $S_{j,{\tilde{\alpha}}}$ is the value of $S_{j}$ calculated for
$\alpha=\tilde{\alpha}$. These equations define only relative values of
momentum $p_{lim,m}$ of $l$-cycles. The equation that closes the system is a
consequence of the following theorem:

######  Theorem 3.2.

In fractional maps of the orders $1<\alpha<2$ the total of all physical
momenta of period–$l$ points is zero:

$\sum^{l}_{j=1}p_{lim,j}=0,\ \ l\in\mathbb{Z},\ \ l>1.$ (3.25)

###### P r o o f..

Let us notice that from the definition Eq. (3.2) follows that in a period $l$
point the total of all difference momenta is zero:

$\sum^{l}_{j=1}p^{1}_{lim,j}=0,\ \ l\in\mathbb{Z},\ \ l>1.$ (3.26)

Using equations Eqs. (3.3) and (3.22) and ignoring $f_{1}(n-1)-f_{1}(n-2)$
terms which disappear in the limit $n\rightarrow\infty$, we may write that

$p_{n}-p^{1}_{n}=\sum^{n}_{m=1}\tilde{G}^{1}(x_{n-m})\Bigl{[}\frac{m^{\alpha-1}-(m-1)^{\alpha-1}}{\alpha-1}-m^{\alpha-2}\Bigr{]}.$
(3.27)

Now let us consider a total of $l$ consecutive differences at large values of
$n$:

$\lim_{N\rightarrow\infty}\sum^{Nl+l}_{n=Nl+1}(p_{n}-p^{1}_{n})\hskip
199.16928pt$
$=\lim_{N\rightarrow\infty}\sum^{Nl+l}_{n=Nl+1}\sum^{n}_{m=1}\tilde{G}^{1}(x_{n-m})\Bigl{[}\frac{m^{\alpha-1}-(m-1)^{\alpha-1}}{\alpha-1}-m^{\alpha-2}\Bigr{]}.$
(3.28)

After changing the order of summation on the right hand side (RHS), this
equality can be written as

$\displaystyle\sum^{l}_{j=1}p_{lim,j}-\sum^{l}_{j=1}p^{1}_{lim,j}=\lim_{N\rightarrow\infty}\sum^{Nl+l}_{m=1}\Bigl{[}\frac{m^{\alpha-1}-(m-1)^{\alpha-1}}{\alpha-1}-m^{\alpha-2}\Bigr{]}$
$\displaystyle\times\sum^{Nl+l}_{n=Nl+1}\tilde{G}^{1}(x_{n-m})+\lim_{N\rightarrow\infty}\sum^{Nl+l}_{m=Nl+1}\Bigl{[}\frac{m^{\alpha-1}-(m-1)^{\alpha-1}}{\alpha-1}-m^{\alpha-2}\Bigr{]}$
$\displaystyle\times\sum^{Nl+l}_{n=m}\tilde{G}^{1}(x_{n-m}).$ (3.29)

The expressions in the square brackets on the RHS of this equation are of the
order $m^{\alpha-3}$. The very last sum of $\tilde{G}^{1}(x_{n-m})$ in the
equation contains not more than $l$ bounded terms and is bounded by some
constant $C_{1}$. Its multiplier in the square brackets is of the order
$m^{\alpha-3}$ and for any $\varepsilon$ there exists $N_{l1}$ such that for
any $N>N_{l1}$ this multiplier will be less than $\varepsilon/(3lC_{1})$.
Then, the whole last term will be less than $\varepsilon/3$. The first term on
the RHS may be split into two terms:

$\displaystyle\lim_{N\rightarrow\infty}\sum^{Nl+l}_{m=1}\Bigl{[}\frac{m^{\alpha-1}-(m-1)^{\alpha-1}}{\alpha-1}-m^{\alpha-2}\Bigr{]}\sum^{Nl+l}_{n=Nl+1}\tilde{G}^{1}(x_{n-m})=$
$\displaystyle\lim_{N_{1}\rightarrow\infty,N-N_{1}\rightarrow\infty}\sum^{N_{1}l+l}_{m=1}\Bigl{[}\frac{2-\alpha}{2}m^{\alpha-3}+O(m^{\alpha-4})\Bigr{]}\sum^{Nl+l}_{n=Nl+1}\tilde{G}^{1}(x_{n-m})$
(3.30)
$\displaystyle+\lim_{N_{1}\rightarrow\infty,N-N_{1}\rightarrow\infty}\sum^{Nl+l}_{m=N_{1}l+l+1}\Bigl{[}\frac{2-\alpha}{2}m^{\alpha-3}+O(m^{\alpha-4})\Bigr{]}\sum^{Nl+l}_{n=Nl+1}\tilde{G}^{1}(x_{n-m}),$

where $N>N_{1}$. The sum of $l$ consecutive values of $\tilde{G}^{1}(x_{n-m})$
is bounded by some constant $C_{2}$ (the limit of this sum is zero, see Eq.
(3.20)). The series of the terms in the square brackets is converging
absolutely. So, the series itself is bounded by some constant $C_{3}$ and for
any $\varepsilon$ there exists $N_{l2}$ such that for any $N_{1}>N_{l2}$ the
total of the terms of this series starting from $N_{1}l+l+1$ will be less than
$\varepsilon/(3C_{2})$. Then, the magnitude of the last term (the third line
of the equation) can be made less than $\varepsilon/3$ by selecting a large
enough value of $N_{1}$. For any $\varepsilon$ we may find a large $N_{l3}$
such that, for any $N>N_{l3}$, $N-N_{1}$ will be large enough to make the sum
of $l$ consecutive values of $\tilde{G}^{1}(x_{n-m})$ less than
$\varepsilon/(3C_{3})$. Then, the first term on the RHS of Eq. (3.2) (the
second line of the equation) will be less than $\varepsilon/3$.

Now, for any $\varepsilon$, if we choose a large $N$ ($N>N_{l1}$ and
$N>N_{l2}$), then the expression on the RHS of Eq. (3.29)) will be less than
$\varepsilon$, which means that the expression on the left is zero and

$\sum^{l}_{j=1}p_{lim,j}=\sum^{l}_{j=1}p^{1}_{lim,j}=0,\ \ l\in\mathbb{Z},\ \
l>1.$ (3.31)

This ends the proof. $\Box$

The results of Theorem 3.1 were previously observed in numerical calculations,
starting from [21] (see also review [9]), but has never been proven before.

As in the case $\alpha<1$, to calculate periodic points, one should first
calculate the sums ${S}^{1}_{j}$.

###  3.3 Calculation of the sums ${S}^{1}_{j}$

Calculation of the sums ${S}^{1}_{j}$, in the case of fractional difference
maps, is quite simple. Let us notice that in this case, the combination of
equations Eqs. (3.15), (3.16), (3.10), (2.12), and (2.13) produces the
following results:

${S}^{1}_{1}(\alpha)=-\Gamma(\alpha)+(2-\alpha)(\alpha-1)\sum^{\infty}_{k=1}\frac{\Gamma(lk+\alpha-2)}{\Gamma(lk+1)}\hskip
28.45274pt$ $\displaystyle=(\alpha-1)S_{1}(\alpha-1)\hskip 142.26378pt$ (3.32)

and

$\displaystyle{S}^{1}_{j+1}(\alpha)=(2-\alpha)(\alpha-1)\sum^{\infty}_{k=0}\frac{\Gamma(lk+j+\alpha-2)}{\Gamma(lk+j+1)}$
$\displaystyle=(\alpha-1)S_{j+1}(\alpha-1),\ \ \ 0<j<l.$ (3.33)

In the last two formulae we explicitly show the dependence of sums on $\alpha$
by adding it as the argument. Because $0<\alpha-1<1$, it is possible to use
numerical formulae and tables derived in [19] to obtain $S_{j}$ and then
${S}^{1}_{j}$.

We will follow the steps described in [19] to calculate the sums in the case
of fractional maps. In this case, Eqs. (3.15) and (3.16) can be written as

$\displaystyle{S}^{1}_{1}=-1+\sum^{N}_{k=1}\Bigl{[}2(lk)^{\alpha-1}-(lk-1)^{\alpha-1}-(lk+1)^{\alpha-1}\Bigr{]}$
$\displaystyle+\sum^{\infty}_{k=N+1}\Bigl{[}2(lk)^{\alpha-1}-(lk-1)^{\alpha-1}-(lk+1)^{\alpha-1}\Bigr{]}$
(3.34)

and

${S}^{1}_{j+1}=\sum^{N}_{k=0}\Bigl{[}2(lk)^{\alpha+j-1}-(lk+j-1)^{\alpha-1}-(lk+j+1)^{\alpha-1}\Bigr{]}$
$+\\!\sum^{\infty}_{k=N+1}\\!\Bigl{[}2(lk)^{\alpha+j-1}\\!-\\!(lk+j-1)^{\alpha-1}\\!-\\!(lk+j+1)^{\alpha-1}\Bigr{]},\
0<j<l,$ (3.35)

where $N$ is a large number (we used $N=20000$ in all calculations). In the
second (the infinite) sums we factor out $(lk)^{\alpha-1}$ and develop the
difference into a Taylor series. After some simple transformations, the
expression to calculate ${S}^{1}_{j}$ can be written as

$\displaystyle{S}^{1}_{1}=-1+\sum^{N}_{k=1}\Bigl{[}2(lk)^{\alpha-1}-(lk-1)^{\alpha-1}-(lk+1)^{\alpha-1}\Bigr{]}$
$\displaystyle+(1-\alpha)(\alpha-2)l^{\alpha-3}\Biggl{[}\zeta_{N}(3-\alpha)+\frac{(\alpha-3)(\alpha-4)}{12l^{2}}\zeta_{N}(5-\alpha)\Biggr{]}$
(3.36) $\displaystyle+\,O(N^{\alpha-6})$

and

$\displaystyle{S}^{1}_{j+1}=\sum^{N}_{k=0}\Bigl{[}2(lk+j)^{\alpha-1}-(lk+j-1)^{\alpha-1}-(lk+j+1)^{\alpha-1}\Bigr{]}$
$\displaystyle+(1-\alpha)(\alpha-2)l^{\alpha-3}\Biggl{\\{}\zeta_{N}(3-\alpha)+\frac{\alpha-3}{l}\Biggl{[}j\zeta_{N}(4-\alpha)$
(3.37)
$\displaystyle+\frac{\alpha-4}{12l}\Biggl{(}(6j^{2}+1)\zeta_{N}(5-\alpha)+\frac{(\alpha-5)j(2j^{2}+1)}{l}\zeta_{N}(6-\alpha)\Biggr{)}\Biggr{]}\Biggr{\\}}$
$\displaystyle+\,O(N^{\alpha-6}),$

where

$\zeta_{N}(m-\alpha)=\zeta(m-\alpha)-\sum^{N}_{k=1}k^{\alpha-m},$ (3.38)

$0<j<l$, and we used a fast method for calculating values of the Riemann
$\zeta$-function. To estimate the accuracy of our computations, we also
calculated the value of $\Sigma S^{1}_{i}$, whose deviation from zero
represents the absolute error. Researchers, who apply and investigate
fractional maps may use the results of the $S^{1}_{j}$ constants calculations
for the period two and three points presented in Tables 1 and 2. It may be
noticed that, in the period two case, the definition of the constant
$S^{1}_{2}$ coincides with the definition of the introduced in [21] constant
$V_{\alpha l}$.

$\alpha$ | $S^{1}_{1}$ | $S^{1}_{2}$ | $\Sigma S^{1}_{i}$
---|---|---|---
1.01 | -.9954781 | .9954781 | 4.44e-16
1.05 | -.9772693 | .9772693 | 2.89e-15
1.1 | -.9542397 | .9542397 | 6.55e-15
1.15 | -.9309180 | .9309180 | 1.30e-14
1.2 | -.9073118 | .9073118 | 1.67e-14
1.25 | -.8834292 | .8834292 | 2.16e-14
1.3 | -.8592791 | .8592791 | 3.72e-14
1.35 | -.8348713 | .8348713 | 3.01e-14
1.4 | -.8102162 | .8102162 | 3.71e-14
1.45 | -.7853250 | .7853250 | 5.47e-14
1.5 | -.7602096 | .7602096 | 5.62e-14
1.55 | -.7348831 | .7348831 | 6.73e-14
1.6 | -.7093590 | .7093590 | 7.38e-14
1.65 | -.6836521 | .6836521 | -7.55e-15
1.7 | -.6577777 | .6577777 | -2.78e-13
1.75 | -.6317523 | .6317523 | -9.10e-13
1.8 | -.6055932 | .6055932 | -2.04e-13
1.85 | -.5793187 | .5793187 | -3.74e-13
1.9 | -.5529480 | .5529480 | -2.76e-12
1.95 | -.5265014 | .5265014 | -1.05e-11
1.99 | -.5053036 | .5053036 | -6.53e-12
Table 1: The values of $S^{1}_{j}$, $1\leq j\leq 2$, for T=2 cycles (fractional maps). $\alpha$ | $S^{1}_{1}$ | $S^{1}_{2}$ | $S^{1}_{3}$ | $\Sigma S^{1}_{i}$
---|---|---|---|---
1.01 | -.998088676 | .994297820 | .003790856 | 5.67E-16
1.05 | -.990205877 | .971261837 | .018944039 | 2.48E-15
1.1 | -.979805615 | .941958610 | .037847005 | 5.56E-15
1.15 | -.968778551 | .912096431 | .056682119 | 1.15E-14
1.2 | -.957104032 | .881682864 | .075421169 | 1.36E-14
1.25 | -.944761480 | .850727005 | .094034474 | 2.07E-14
1.3 | -.931730456 | .819239584 | .112490872 | 2.78E-14
1.35 | -.917990742 | .787233052 | .130757690 | 2.03E-14
1.4 | -.903522418 | .754721680 | .148800738 | 3.69E-14
1.45 | -.888305951 | .721721659 | .166584292 | 9.76E-14
1.5 | -.872322287 | .688251202 | .184071085 | 3.86E-14
1.55 | -.855552955 | .654330642 | .201222312 | 1.79E-13
1.6 | -.837980174 | .619982544 | .217997630 | 2.31E-14
1.65 | -.819586970 | .585231803 | .234355167 | 1.86E-14
1.7 | -.800357297 | .550105757 | .250251540 | -4.96E-13
1.75 | -.780276168 | .514634289 | .265641879 | -8.29E-13
1.8 | -.759329798 | .478849937 | .280479860 | 2.76E-13
1.85 | -.737505746 | .442788001 | .294717744 | 2.04E-12
1.9 | -.714793081 | .406486651 | .308306429 | 6.40E-12
1.95 | -.691182527 | .369987024 | .321195502 | -4.32E-12
1.99 | -.671642567 | .340674171 | .330968396 | 8.96E-12
Table 2: The values of $S^{1}_{j}$, $1\leq j\leq 3$, for period 3 cycles
(fractional maps).

##  4 Asymptotically periodic cycles for $\alpha>2$

Let us consider an arbitrary $m-1<\alpha<m$. The fractional Caputo universal
map Eq. (2.1) can be written as

$x_{n+1}=\sum^{m-1}_{k=0}\frac{b_{k}}{k!}h^{k}(n+1)^{k}-h^{m-1}\sum^{n}_{k=0}G^{m-1}(x_{k})U_{\alpha}(n-k+1),$
(4.1)

where $n\in\mathbb{N}_{0}$, $U_{\alpha}(n)$ is defined by Eq. (2.7), and

$G^{m-1}(x)=G_{K}(x)h^{\alpha-m+1}/\Gamma(\alpha).$

The $h$-difference Caputo universal map Eq. (2.2) can be written as

$\displaystyle
x_{n+1}=x_{0}+\sum^{m-1}_{k=1}\frac{c_{k}}{k!}h^{k}(n+1)n...(n+2-k)$
$\displaystyle-h^{m-1}\sum^{n}_{k=0}G^{m-1}(x_{k})U_{\alpha}(n-k+1)$
$\displaystyle+h^{m-1}\sum^{m-2}_{k=0}G^{m-1}(x_{k})U_{\alpha}(n-k+1),$ (4.2)

where $x_{k}=x(kh)$, $\alpha\in\mathbb{R}$, $\alpha\geq 0$,
$m=\lceil\alpha\rceil$, $n\in\mathbb{N}_{m-1}$, $c_{k}\in\mathbb{R}$, and
$U_{\alpha}(n)$ is defined by Eq. (2.8).

###  4.1 Higher order difference momenta

Let us define order $i>1$ difference momentum as

$\displaystyle
p^{i}_{n+1}=\frac{p^{i-1}_{n+1}-p^{i-1}_{n}}{h}=\Delta^{i}x_{n-i+1},\ \
0<i<m.$ (4.3)

To derive the equations that define periodic points we’ll need to derive the
expression for the limiting ($n\rightarrow\infty$) values of the $m-1$st order
momentum $p^{m-1}$. The last term (the sum) on the RHS of Eq. (4.2) is of the
order $n^{\alpha-1}$ and after the application of the $m-1$st order of the
$h$-difference operator it will be of the order $n^{\alpha-m}$. This term will
tend to zero in the limit $n\rightarrow\infty$ and may be ignored. Now, let
uss consider the $m-1$st order $h$-difference of the first term (which
accounts for the initial conditions) on the RHS of Eq. (4.1) written for
$x_{n}$:

$\sum^{m-1}_{k=0}\frac{b_{k}}{k!}h^{k-m+1}\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(n-i)^{k}.$ (4.4)

First, let us prove the following theorem (identity):

######  Theorem 4.1.

For any $n\in\mathbb{Z}$, $m\in\mathbb{N}_{0}$, and $k\in\mathbb{N}_{0}$ (when
$k=0$, $n\notin[0,m]$) the following identity holds:

$\displaystyle\sum^{m}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m\\\
i\end{array}\right)(n-i)^{k}=\begin{array}[]{c}\left\\{\begin{array}[]{lll}0\
\ {\rm for}\ \ 0\leq k<m\ ,\\\ k!\ \ {\rm for}\ \ k=m\
.\end{array}\right.\end{array}$ (4.10)

###### P r o o f..

We prove the identity by induction. When $m=0$, $m=1$, or $m=2$, for $0\leq
k\leq m$ the identity is easily verifiable. In what follows we use the Pascal
identity

$\left(\begin{array}[]{c}m\\\
i\end{array}\right)=\left(\begin{array}[]{c}m-1\\\
i-1\end{array}\right)+\left(\begin{array}[]{c}m-1\\\ i\end{array}\right).$
(4.11)

Assuming that the identity Eq. (4.10) holds for the total with the positive
upper limit $m-1$, we may write:

$\displaystyle\sum^{m}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m\\\
i\end{array}\right)(n-i)^{k}=n^{k}+\sum^{m}_{i=1}(-1)^{i}\left(\begin{array}[]{c}m\\\
i\end{array}\right)(n-i)^{k}$ (4.16)
$\displaystyle=n^{k}+\sum^{m}_{i=1}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i-1\end{array}\right)(n-i)^{k}+\sum^{m}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(n-i)^{k}-n^{k}$ (4.21)
$\displaystyle=-\\!\sum^{m-1}_{j=0}(-1)^{j}\left(\begin{array}[]{c}m-1\\\
j\end{array}\right)((n\\!-\\!1)\\!-\\!j)^{k}\\!+\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(n\\!-\\!i)^{k},$ (4.26)

where $j=i-1$. If $k\leq m$ then, according to our assumption, the two totals
on the last line in Eq (4.26) are identical and have the opposite signs and
the result is zero. If $k=m$, then

$\displaystyle\sum^{m}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m\\\
i\end{array}\right)(n-i)^{k}=\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)\\{(n-i)^{m}-[(n-i)-1]^{m}\\}$ (4.31)
$\displaystyle=\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)\Biggl{[}m(n-i)^{m-1}-\frac{m(m-1)}{2}(n-i)^{m-2}+\ldots\Biggr{]}$
(4.34) $\displaystyle=m\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(n-i)^{m-1}$ (4.37)
$\displaystyle-\frac{m(m-1)}{2}\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(n-i)^{m-2}+\ldots$ (4.40)
$\displaystyle=m(m-1)!+0+\ldots=m!=k!.$ (4.41)

This ends the proof. $\Box$

As a result of this theorem, we may conclude that the $m-1$st order
$h$-difference Eq. (4.4) is equal to $b_{m-1}$, which is the initial value of
the $m-1$st derivative of $x$.

In the case of the fractional difference map, the the $m-1$st order
$h$-difference of the first term in the expression for $x_{n}$ can be written
like Eq. (4.4), but with the falling factorial instead the power function:

$\sum^{m-1}_{k=0}\frac{c_{k}}{k!}h^{k-m+1}\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(n-i)^{(k)}.$ (4.42)

The proof by induction of the following theorem, is similar to the proof of
Theorem 4.1:

######  Theorem 4.2.

For any $n\in\mathbb{Z}_{+}$ and $m\in\mathbb{Z}_{+}$, $n>m$, and
$k\in\mathbb{Z}_{+}$ the following identity holds:

$\displaystyle\sum^{m}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m\\\
i\end{array}\right)(n-i)^{(k)}=\begin{array}[]{c}\left\\{\begin{array}[]{lll}0\
\ {\rm for}\ \ 0<k<m\ ,\\\ k!\ \ {\rm for}\ \ k=m\
.\end{array}\right.\end{array}$ (4.48)

###### P r o o f..

When $m=2$ and $k$ is 1 or 2 the identity holds. Let us assume that the
theorem holds for the total with the upper limit $m-1$, then:

$\sum^{m}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m\\\
i\end{array}\right)(n-i)^{(k)}=n^{(k)}+\sum^{m}_{i=1}(-1)^{i}\left(\begin{array}[]{c}m\\\
i\end{array}\right)(n-i)^{(k)}$
$=n^{(k)}+\sum^{m}_{i=1}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i-1\end{array}\right)(n-i)^{(k)}+\sum^{m}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(n-i)^{(k)}-n^{(k)}$
$-\sum^{m-1}_{j=0}(-1)^{j}\left(\begin{array}[]{c}m-1\\\
j\end{array}\right)((n\\!-\\!1)\\!-\\!j)^{(k)}+\\!\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(n\\!-\\!i)^{(k)},$ (4.49)

where $j=i-1$. If $k\leq m$ then, according to our assumption, the two totals
on the last line in Eq (4.49) are identical and have the opposite signs and
the result is zero. If $k=m$, then

$\sum^{m}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m\\\
i\end{array}\right)(n-i)^{(k)}=\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)\\{(n-i)^{(m)}\\!-\\![(n-i)\\!-\\!1]^{(m)}\\}$
$=m\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(n-1-i)_{(m-1)}=m(m-1)!=m!=k!.$ (4.50)

This ends the proof. $\Box$

We see that the $m-1$st order $h$-difference Eq. (4.42) is equal to $c_{m-1}$.
Accumulating all the results discussed in this section, in both fractional and
fractional difference cases, the expression for $p^{m-1}_{n}$ can be written
as

$\displaystyle
p^{m-1}_{n}=p_{m-1,0}-\sum^{n-1}_{k=0}G^{m-1}(x_{k})U^{m-1}_{\alpha}(n-k)+F(n),$
(4.51)

where $p_{m-1,0}$ is a constant, $F(n)\rightarrow 0$ when $n\rightarrow\infty$
and the definition of $U^{m-1}_{\alpha}$ is

$\displaystyle
U^{m-1}_{\alpha}(n)=U^{m-2}_{\alpha}(n)-U^{m-2}_{\alpha}(n-1)=\Delta^{m-1}U_{\alpha}(n-m+1).$
(4.52)

In the case of fractional difference maps, the equality

$U_{\alpha}(n)=0,\ \ {\rm when}\ \ n<1$ (4.53)

follows from the definition Eq. (2.8). In fractional maps with $\alpha>1$ this
is true only for $n=0$ and we will postulate this equality for $n<0$.

In fractional maps,

$\displaystyle
U^{m-1}_{\alpha}(n)=\begin{array}[]{c}\left\\{\begin{array}[]{lll}\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(n-i)^{\alpha-1}\ \ {\rm for}\ \ n>m-1,\\\
\sum^{n-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(n-i)^{\alpha-1}\ \ {\rm for}\ \
n<m.\end{array}\right.\end{array}$ (4.61)

In fractional difference maps, assuming that the identity

$U^{m-2}_{\alpha}(n)=(\alpha-1)^{(m-2)}U_{\alpha-m+2}(n),$ (4.62)

which, according to Eq. (3.10), holds for $m=3$, is true, we may prove by
induction that this identity holds for any $m$:

$\displaystyle U^{m-1}_{\alpha}(n)=U^{m-2}_{\alpha}(n)-U^{m-2}_{\alpha}(n-1)$
$\displaystyle=(\alpha-1)^{(m-2)}[U_{\alpha-m+2}(n)-U_{\alpha-m+2}(n-1)]$
$\displaystyle=(\alpha-1)^{(m-2)}U^{1}_{\alpha-m+2}(n)=(\alpha-1)^{(m-1)}U_{\alpha-m+1}(n)$
$\displaystyle=\frac{\Gamma(\alpha)\Gamma(n+\alpha-m)}{\Gamma(\alpha-m+1)\Gamma(n)}.$
(4.63)

For large $n$, $U^{m-1}_{\alpha}(n)\sim n^{\alpha-m}$ in both, fractional and
fractional difference cases.

###  4.2 Equations defining periodic points

Assuming that asymptotically period $l$ points ($l$-cycles) Eq. (2.9) exist,
the corresponding limiting values of the order $i<m=\lceil\alpha\rceil$
difference momentum are defined as

$p^{i}_{lim,k}=\lim_{N\rightarrow\infty}p^{i}_{Nl+k}=\frac{p^{i-1}_{lim,k}-p^{i-1}_{lim,k-1}}{h},\
\ 1<k<l+1$ (4.64)

and

$p^{i}_{lim,1}=\lim_{N\rightarrow\infty}p^{i}_{Nl+1}=\frac{p^{i-1}_{lim,1}-p^{i-1}_{lim,l}}{h},$
(4.65)

which can be written as

$\displaystyle
p^{i}_{lim,k}=\begin{array}[]{c}\left\\{\begin{array}[]{lll}h^{-i}\sum^{i}_{j=0}(-1)^{j}\left(\begin{array}[]{c}i\\\
j\end{array}\right)x_{lim,k-j}\ \ \ \ {\rm for}\ \ i<k<l+1;\\\
h^{-i}\sum^{k-1}_{j=0}(-1)^{j}\left(\begin{array}[]{c}i\\\
j\end{array}\right)x_{lim,k-j}\\\
+h^{-i}\sum^{i}_{j=k}(-1)^{j}\left(\begin{array}[]{c}i\\\
j\end{array}\right)x_{lim,l-j+k}\ \ {\rm for}\ \ 0<k\leq
i.\end{array}\right.\end{array}$ (4.76)

The equations defining asymptotically cyclic points are obtained considering
the limit $N\rightarrow\infty$, in which case the last term of Eq. (4.51)
disappears. If we assume that
$U_{\alpha}(n)\in\mathbb{D}^{m-1}(\mathbb{N}_{1})$ for $m-1<\alpha<m$, then
from the definition of $U^{m-1}_{\alpha}(n)$ Eq. (4.52) (with the extension
Eq. (4.53)) and Definition 2.1 follows that
$U^{m-1}_{\alpha}(x)\in\mathbb{D}^{0}(\mathbb{N}_{1})$. This implies that all
steps of the derivation of formulae Eqs. (2.10) - (2.14), strictly proven in
[19], can be repeated here. Subtracting consecutive limiting values and adding
all limiting values of the $m-1$st difference momenta produces the following
results:

$\displaystyle
p^{m-1}_{lim,k+1}-p^{m-1}_{lim,k}={S}^{m-1}_{1}{G}^{m-1}(x_{lim,k})+\sum^{k-1}_{j=1}{S}^{m-1}_{j+1}{G}^{m-1}(x_{lim,k-j})$
$\displaystyle+\sum^{l-1}_{j=k}{S}^{m-1}_{j+1}{G}^{m-1}(x_{lim,k-j+l}),\ \
0<k<l$ (4.77)

and

$\sum^{l}_{j=1}{G}^{m-1}(x_{lim,j})=0,$ (4.78)

where

${S}^{m-1}_{1}=-{U}^{m-1}_{\alpha}(1)+\sum^{\infty}_{k=1}\Bigl{[}{U}^{m-1}_{\alpha}(lk)-{U}^{m-1}_{\alpha}(lk+1)\Bigr{]},$
(4.79)
$\displaystyle{S}^{m-1}_{j+1}=\sum^{\infty}_{k=0}\Bigl{[}{U}^{m-1}_{\alpha}(lk+j)-{U}^{m-1}_{\alpha}(lk+j+1)\Bigr{]},\
\ 0<j<l,$ (4.80)

and, still,

$\sum^{l}_{j=1}{S}^{m-1}_{j}=0.$ (4.81)

Using Eq. (4.76) the system of $l$ equations which defines all $l$ periodic
points after some transformations and using the Pascal identity can be written
as (notice that ${G}^{0}(x)=h^{m-1}{G}^{m-1}(x)$),

$x_{lim,k+1}+(-1)^{m}x_{lim,k-m+1}-\sum^{m-2}_{j=0}(-1)^{j}\left(\begin{array}[]{c}m\\\
j+1\end{array}\right)x_{lim,k-j}={S}^{m-1}_{1}{G}^{0}(x_{lim,k})$
$+\sum^{k-1}_{j=1}{S}^{m-1}_{j+1}{G}^{0}(x_{lim,k-j})+\sum^{l-1}_{j=k}{S}^{m-1}_{j+1}{G}^{0}(x_{lim,k-j+l}),\
m-1<k<l,$ (4.82)
$x_{lim,m}+(-1)^{m}x_{lim,l}+\sum^{m-1}_{j=1}(-1)^{j}\left(\begin{array}[]{c}m\\\
j\end{array}\right)x_{lim,m-j}={S}^{m-1}_{1}{G}^{0}(x_{lim,m-1})$
$+\sum^{m-2}_{j=1}{S}^{m-1}_{j+1}{G}^{0}(x_{lim,m-1-j})+\\!\sum^{l-1}_{j=m-1}{S}^{m-1}_{j+1}{G}^{0}(x_{lim,m-1-j+l}),\
k=m-1,$ (4.83)
$x_{lim,k+1}+(-1)^{m}x_{lim,l+k-m+1}-\sum^{k-1}_{j=1}(-1)^{j}\left(\begin{array}[]{c}m\\\
j+1\end{array}\right)x_{k-j}$
$-\sum^{m-2}_{j=k}(-1)^{j}\left(\begin{array}[]{c}m\\\
j+1\end{array}\right)x_{lim,l+k-j}={S}^{m-1}_{1}{G}^{0}(x_{lim,k})+\sum^{k-1}_{j=1}{S}^{m-1}_{j+1}{G}^{0}(x_{lim,k-j})$
$+\sum^{l-1}_{j=k}{S}^{m-1}_{j+1}{G}^{0}(x_{lim,k-j+l}),\ 0<k<m-1,$ (4.84)

and

$\sum^{l}_{j=1}{G}^{0}(x_{lim,j})=0.$ (4.85)

This is the end of the proof of the following theorem:

######  Theorem 4.3.

If the map

$x_{n+1}=f(n)-h^{m-1}\sum^{n}_{k=0}G^{m-1}(x_{k})U_{\alpha}(n-k+1),$ (4.86)

where $\lim_{n\rightarrow\infty}\Delta^{m-1}f(n)$ is a constant,
$m=\lceil\alpha\rceil$, $\alpha\in\mathbb{R}$, $\alpha>0$, $h>0$, and
$\Delta^{m-1}U_{\alpha}(n)\in\mathbb{D}^{0}(\mathbb{N}_{1})$ has
asymptotically period $l$ points Eq. (2.9) in the domain of the function
$G^{m-1}(x)$, then they are defined by the system of equations Eqs. (4.82) –
Eq. (4.85), with the coefficients defined by Eq. (4.79) and Eq.(4.80).

Remark. When $m>1$, to use Eq. (4.79) and Eq.(4.80), the domain $U(n)$ is
extended to $n\in\mathbb{N}_{2-m}$ and it is assumed that $U(n)=0$ for
$1-m<n<1$.

Fractional and fractional difference maps belong to the class of maps to which
the theorem applies.

###  4.3 Calculation of the sums $S^{m-1}_{j}$

Calculation of the sums $S^{m-1}_{j}$, in the case of fractional difference
maps, is very simple. In this case, combination of equations Eq. (4.63) and
Eq. (4.80) (which with $j=0$ is equivalent to Eq. (4.79)) produces the
following result:

$\displaystyle
S^{m-1}_{j+1}(\alpha)=(m-\alpha)\sum^{\infty}_{k=0}\frac{\Gamma(\alpha)\Gamma(lk+j+\alpha-m)}{\Gamma(\alpha-m+1)\Gamma(lk+j+1)}=(\alpha-1)S^{m-2}_{j+1}(\alpha-1)$
$\displaystyle=(\alpha-1)^{(m-1)}S_{j+1}(\alpha-m+1),\ \ \ 0\leq j<l,$ (4.87)

where we explicitly show the dependence of the sums on $\alpha$ by adding it
as the argument. Because $0<\alpha-m+1<1$, it is possible to use numerical
formulae and tables derived in [19] to obtain $S_{j}$ and then
${S}^{m-1}_{j}$.

The calculations are more complicated in the case of fractional maps:

$\displaystyle{S}^{m-1}_{1}=-{U}^{m-1}_{\alpha}(1)+\sum^{\infty}_{k=1}\Bigl{[}{U}^{m-1}_{\alpha}(lk)-{U}^{m-1}_{\alpha}(lk+1)\Bigr{]}$
$\displaystyle=-1+\sum^{\lceil\frac{m-1}{l}\rceil-1}_{k=1}\Bigl{\\{}\sum^{lk-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)\Bigl{[}(lk-i)^{\alpha-1}-(lk-i+1)^{\alpha-1}\Bigr{]}$
(4.90) $\displaystyle-(-1)^{lk}\left(\begin{array}[]{c}m-1\\\
lk\end{array}\right)\Bigl{\\}}+\delta_{\lceil\frac{m-1}{l}\rceil[\frac{m-1}{l}]}\Bigl{[}\sum^{lk-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(lk-i)^{\alpha-1}$ (4.95)
$\displaystyle-\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(lk-i+1)^{\alpha-1}\Bigr{]}$ (4.98)
$\displaystyle+\sum^{N}_{k=[\frac{m-1}{l}]+1}\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)\Bigl{[}(lk-i)^{\alpha-1}-(lk-i+1)^{\alpha-1}\Bigr{]}$
(4.101)
$\displaystyle+\sum^{\infty}_{k=N+1}\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)\Bigl{[}(lk-i)^{\alpha-1}-(lk-i+1)^{\alpha-1}\Bigr{]}$
(4.104)

and

$\displaystyle{S}^{m-1}_{j+1}=\sum^{\infty}_{k=0}\Bigl{[}{U}^{m-1}_{\alpha}(lk+j)-{U}^{m-1}_{\alpha}(lk+j+1)\Bigr{]}$
$\displaystyle=\sum^{\lceil\frac{m-j-1}{l}\rceil-1}_{k=0}\Bigl{\\{}\sum^{lk+j-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)\Bigl{[}(lk+j-i)^{\alpha-1}-(lk+j-i+1)^{\alpha-1}\Bigr{]}$
(4.107) $\displaystyle-(-1)^{lk+j}\left(\begin{array}[]{c}m-1\\\
lk+j\end{array}\right)\Bigl{\\}}+\delta_{\lceil\frac{m-j-1}{l}\rceil[\frac{m-j-1}{l}]}\Bigl{[}\sum^{lk+j-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)$ (4.112)
$\displaystyle\times\,(lk+j-i)^{\alpha-1}-\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(lk+j-i+1)^{\alpha-1}\Bigr{]}$ (4.115)
$\displaystyle+\sum^{N}_{k=[\frac{m-j-1}{l}]+1}\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)\Bigl{[}(lk+j-i)^{\alpha-1}-(lk+j-i+1)^{\alpha-1}\Bigr{]}$
(4.118)
$\displaystyle+\\!\\!\sum^{\infty}_{k=N+1}\\!\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m\\!-\\!1\\\
i\end{array}\right)\\!\Bigl{[}(lk+j-i)^{\alpha-1}\\!-\\!(lk+j-i+1)^{\alpha-1}\Bigr{]},\
0<j<l,$ (4.121)

where $\delta_{mn}$ is the Kronecker delta and $N$ is a large number (in all
our calculations we assumed $N=20000$). Both expressions, Eq. (4.104) and Eq.
(4.121), can be written as the totals of two sums $S^{m-1}_{j+1,fin}$ and
$S^{m-1}_{j+1,inf}$ ($0\leq j<l$). $S^{m-1}_{j+1,fin}$ is the sum of a large
but finite number of terms (all lines in the corresponding formulae except the
last one) and can be computed very fast. To calculate the second sum we use
the Taylor expansions, results of Theorem 4.1, and changes in the order of
summation:

$\displaystyle{S}^{m-1}_{j+1,inf}=\\!\sum^{\infty}_{k=N+1}\\!\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)\Bigl{[}(lk+j-i)^{\alpha-1}-(lk+j-i+1)^{\alpha-1}\Bigr{]}$
(4.124)
$\displaystyle=\sum^{\infty}_{k=N+1}\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(lk)^{\alpha-1}\sum^{\infty}_{q=1}\frac{(\alpha-1)^{(q)}}{q!(lk)^{q}}[(j-i)^{q}-(j-i+1)^{q}]$
(4.127)
$\displaystyle=-\sum^{\infty}_{k=N+1}\sum^{\infty}_{q=1}(lk)^{\alpha-1}\frac{(\alpha-1)^{(q)}}{q!(lk)^{q}}\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)\sum^{q-1}_{p=0}\left(\begin{array}[]{c}q\\\
p\end{array}\right)(j-i)^{p}$ (4.132)
$\displaystyle=-\sum^{\infty}_{q=1}\frac{(\alpha-1)^{(q)}}{q!}\sum^{\infty}_{k=N+1}(lk)^{\alpha-1-q}\sum^{q-1}_{p=0}\left(\begin{array}[]{c}q\\\
p\end{array}\right)\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(j-i)^{p}$ (4.137)
$\displaystyle=-\sum^{\infty}_{q=m}l^{\alpha-1-q}\frac{(\alpha-1)^{(q)}}{q!}\sum^{q-1}_{p=m-1}\left(\begin{array}[]{c}q\\\
p\end{array}\right)\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(j-i)^{p}$ (4.142)
$\displaystyle\times\,\sum^{\infty}_{k=N+1}k^{\alpha-1-q}=-\sum^{\infty}_{q=m}l^{\alpha-1-q}\frac{(\alpha-1)^{(q)}}{q!}$
$\displaystyle\times\,\sum^{q-1}_{p=m-1}\left(\begin{array}[]{c}q\\\
p\end{array}\right)\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(j-i)^{p}\zeta_{N}(q+1-\alpha),$ (4.147)

where $0\leq j<l$ and $\zeta_{N}(x)$ is defined by Eq. (3.38). Finally, we may
write

$\displaystyle{S}^{m-1}_{j+1}={S}^{m-1}_{j+1,fin}-l^{\alpha-1-m}(\alpha-1)^{(m)}\zeta_{N}(m+1-\alpha)$
$\displaystyle-\sum^{m+3}_{q=m+1}l^{\alpha-1-q}\frac{(\alpha-1)^{(q)}}{q!}\sum^{q-1}_{p=m-1}\left(\begin{array}[]{c}q\\\
p\end{array}\right)\sum^{m-1}_{i=0}(-1)^{i}\left(\begin{array}[]{c}m-1\\\
i\end{array}\right)(j-i)^{p}$ (4.152)
$\displaystyle\times\,\zeta_{N}(q+1-\alpha)+O(N^{\alpha-m-4}),\ \ \ 0\leq
j<l,$ (4.153)

which allows high accuracy calculations of the sums. It is easy to verify that
in the case $m=2$, this expression coincides with Eq. (3.3) and Eq. (3.3).

##  5 A note on generalized discrete fractional calculus

The notion of the generalized fractional calculus was introduced in [27] and
further developed in multiple papers including [28, 29, 30, 31, 32, 33, 41,
42]. The general fractional integrals and derivatives were defined using
Bernstein functions and Sonine kernels to satisfy the fundamental theorems. In
[42] the author proposed the general fractional (universal) maps as the exact
solutions of equations with a general fractional derivative and periodic
kicks.

Fractional sums and differences and discrete fractional calculus were
introduced at the end of the last century [22, 35]. In [49] the authors
proposed to use solutions of fractional difference equations to generate
fractional difference maps, which kernels are different from the kernels of
fractional maps obtained as solutions of fractional differential equations
with periodic kicks. During the last few years various types of
generalizations of fractional differential and difference operators were
introduced. The authors did not find any references to papers on distributed
and variable fractional order generalizations of fractional difference
operators, but we may expect that solutions of fractional difference equations
of distributed (with the kernels integrated over $m-1<\alpha<m$) and variable
orders ($m-1<\alpha(t)<m$) will be of the form Eq. (4.86) and satisfy the
conditions of Theorem 4.3.

Kernels $U_{\alpha}(n)\in\mathbb{D}^{m-1}(\mathbb{N}_{1})$ were introduced to
allow finding asymptotically periodic solutions of Eq. (4.86). We may call the
map defined by Eq. (4.86), in which
$\lim_{n\rightarrow\infty}\Delta^{m-1}f(n)$ is a constant,
$m=\lceil\alpha\rceil$, $\alpha\in\mathbb{R}$, $\alpha>0$, $h>0$, and
$\Delta^{m-1}U_{\alpha}(n)\in\mathbb{D}^{0}(\mathbb{N}_{1})$, the general
fractional $h$-difference map. Finding and investigating of the operator
$\Delta_{(U),h}^{t}$, such that the map Eq. (4.86) is the solution of the
fractional $h$-difference equation of the form

$\Delta_{(U),h}^{t}x(t)=-G(x(t)),$ (5.1)

and the relationship between the kernels which belong to $\mathbb{D}^{m}$ and
the kernels of the general fractional calculus is an interesting problem, but
it is not the subject of the current paper.

##  6 Conclusion

Let us summarize the main new results obtained in this paper.

The main goal of this paper was to derive the equations which define
asymptotically periodic points in a wide class of maps with memory. The maps
are defined as convolutions with kernels $U_{\alpha}(x)$ which belong to the
space $\mathbb{D}^{m-1}(\mathbb{Z})$ (or
$\Delta^{m-1}U_{\alpha}(x)\in\mathbb{D}^{0}(\mathbb{Z})$) defined by
Definition 2.1. Fractional and fractional difference maps of real positive
orders $\alpha$ belong to this class of maps with $m=\lceil\alpha\rceil$.
Theorem 4.3 formulates the equation and is the main achievement of the paper.
Particular forms of the equations for the cases of $0<\alpha<1$ and
$1<\alpha<2$, which have important in applications, are also formulated.

The coefficients of the derived system of equations Eqs. (4.82)– Eq. (4.85)
are obtained as slowly converging series defined by Eq. (4.79) and Eq.(4.80).
The formulae which allow fast calculations of the coefficients in the cases of
fractional and fractional difference maps are given by Eq. (35) from [19] and
Eq. (4.87) (for fractional difference maps) and by Eq. (4.104), Eq. (4.121),
and Eq. (4.153) (for fractional maps). The versions of these formulae for the
case $0<\alpha<2$ are also obtained and used to calculate the tables of
coefficients: Tables 1–5 from [19] and Tables 1 and 2.

The stable periodic points represent the asymptotic behavior of nonlinear
systems with memory and, in the case of chaotic systems, the periodic points
represent skeletons of chaos [4, 34]. Fractional modifications (addition of
the power-law memory) of conservative regular systems behave similarly to the
corresponding regular systems with dissipation (see, e.g., [52]). In these
cases, periodic points will be dense in the corresponding chaotic attractors.
Our results will be useful to any researcher who works with
fractional/fractional difference maps to calculate the periodic points.

Another result, formulated in Theorem 3.2, is the proof of the conjecture that
in fractional maps of the orders $1<\alpha<2$ the total of all physical
momenta of period–$l$ points is zero. This conjecture was previously used in
investigation of the fractional standard and logistic maps.

One of the fundamental questions is whether there exists a self-similarity in
bifurcation diagrams of nonlinear fractional systems near transition to chaos.
Now, when we have the means to numerically find periodic points and define
bifurcation points, we may try to answer this question. In the following paper
we intend to numerically analyze the bifurcation behavior of the solutions of
the derived equations for the case of the fractional and fractional difference
logistic maps in order to make a conjecture about the existence or non-
existence of the Feigenbaum numbers and their possible values.

## Acknowledgements

The author acknowledges support from Yeshiva University’s 2021–2022 Faculty
Research Fund and expresses his gratitude to the administration of Courant
Institute of Mathematical Sciences at NYU for the opportunity to perform some
of the computations at Courant.

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1 Department of Physics, Stern College at Yeshiva University
245 Lexington Ave, New York, NY 10016, USA
Courant Institute of Mathematical Sciences
New York University
251 Mercer St., New York, NY 10012, USA
Department of Mathematics BCC, CUNY
2155 University Avenue, Bronx, New York 10453, USA
e-mail<EMAIL_ADDRESS>(Corresponding author)
Received: …………….. , Revised: November 13, 2021
2 Stern College at Yeshiva University
245 Lexington Ave, New York, NY 10016, USA
e-mail<EMAIL_ADDRESS>