Periodically Kicked Rotator with Power-Law Memory: Exact Solution and Discrete Maps ^†

Abstract

This article discusses the transformation of a continuous-time model of the fractional system into a discrete-time model of the fractional system. For the continuous-time model, the exact solution of the nonlinear equation with fractional derivatives (FDs) that has the form of the damped rotator type with power non-locality in time is obtained.This equation with two FDs and periodic kicks is solved in the general case for the arbitrary orders of FDs without any approximations. A three-stage method for solving a nonlinear equation with two FDs and deriving discrete maps with memory (DMMs) is proposed. The exact solutions of the nonlinear equation with two FDs are obtained for arbitrary values of the orders of these derivatives. In this article, the orders of two FDs are not related to each other, unlike in previous works. The exact solution of nonlinear equation with two FDs of different orders and periodic kicks are proposed. Using this exact solution, we derive DMMs that describe a kicked damped rotator with power-law non-localities in time. For the discrete-time model, these damped DMMs are described by the exact solution of nonlinear equations with FDs at discrete time points as the functions of all past discrete moments of time. An example of the application, the exact solution and DMMs are proposed for the economic growth model with two-parameter power-law memory and price kicks. It should be emphasized that the manuscript proposes exact analytical solutions to nonlinear equations with FDs, which are derived without any approximations. Therefore, it does not require any numerical proofs, justifications, or numerical validation. The proposed method gives exact analytical solutions, where approximations are not used at all.

1. Introduction

Integro-differential and integral operators, which satisfy some generalizations of the fundamental theorems of calculus, are called fractional integrals (FIs) and fractional derivatives (FDs) and form the fractional calculus (FC) (see fundamental books [1,2,3,4,5], the Handbook reviews [6,7], and numerical methods seen in [8,9,10]). There are many types of FDs and FIs [11,12,13,14]. The history of the FC was first described in the 1868 paper [15] and then in a book [1] and papers [16,17,18], including the history of the application of the FC [19,20,21]. Equations with FDs with respect to time can be used to describe processes and systems with memory and non-locality in time in the various fields of physics [22,23,24,25], including continuum mechanics [26,27,28], physical kinetics [29,30], thermoelasticity, and diffusion [31,32], and in other sciences, including economics [33,34], biology [35], and engineering [36].

In physics, mechanics, and nonlinear dynamics, the discrete maps are usually derived from differential equations of integer order with periodic kicks (for example, see the books about the nonlinear dynamics [37,38,39] and the work on physics [22,40,41,42]). This approach allows us to derive the exact solutions of these nonlinear differential equations without any approximations. Therefore, it is important to have a similar approach to solve nonlinear equations with FDs. Some scientists have tried to generalize this approach to nonlinear equations with FDs to derive the discrete maps with memory (DMMs) and nonlocality in time. Until 2008, nonlinear DMMs were simply postulated in some form (for example, see the papers of Fulinski, Kleczkowski [43], Fick with coauthors [44,45], Giona [46], Gallas [47], and Stanislavsky [48]) and are not exact solutions of any integro-differential equations. Some scientists even proposed a justification that periodic “blows” and kicks knock out the memory of dynamic systems. From this, it was concluded that discrete maps associated with equations with FDs cannot have memory. To solve the problem of finding exact solutions to nonlinear equations with FDs and periodic kicks, deriving the corresponding DMM was proposed by Zaslavsky to various scientists who collaborated with him.The author of this article also offered to solve this problem in 2006. This Zaslavsky problem was successfully solved by the author and then published in 2008 [49]. For the first time, DMMs were obtained from nonlinear equations with FDs without approximations in [49]. Then, this approach was developed in [22,50,51] and then in works [34,52,53,54,55,56,57,58,59,60,61,62,63] in which the following results were derived:

• The DMMs were derived from the equations with the Caputo and Riemann–Liouville FDs in [22,49,50,51], including the generalization of the Henon, dissipative, and Zaslavsky maps with memory [22,52,63].
• The DMMs were also derived from equations with FDs describing economic [34,53], population dynamics [54], and quantum dynamic systems [55].
• For the first time, the DMMs were obtained from the equations with FIs in [56].
• The DMMs were obtained from the equations with the Erdelyi-Kober FDs in [57], the Hadamard-type FDs in [58], and the Hilfer FDs in [59].
• For the first time, the DMMs were derived from the equations with the general FDs and FIs in [60,62].
• The DMMs were obtained from the equations with the distributed-order FDs in [61].
• The first computer simulations of some such DMMs, which are obtained from nonlinear equations with FDs, are proposed in papers [63,64,65].
• New types of the chaotic behavior of systems with nonlocality in time were discovered in these papers [63,64], the 2013 papers [66,67,68], and the 2014 works [69,70].
• Note also the new works of Mendez-Bermudez and Aguilar-Sanchez [71] about tunable subdiffusion in the DDM; Borin [72] about scaling invariance analyses for DDM; Orinaite, Telksniene, Telksnys, and Ragulskis [73] about the changes of the complexity of DMM; Orinaite, Smidtaite, and Ragulskis [74] about Arnold tongues of divergence in the Caputo DMM.

Note some works about the derivation of exact solutions and the DMMs from equations with FDs and FIs without approximations. The importance of this approach to obtaining DMMs is that these maps are derived from the exact solutions of nonlinear equations with FDs and FIs for a very wide class of nonlinearities without any approximations.

For physics, mechanics, and applied sciences, the DMMs are primarily important due to the connection of these maps with fractional differential and integral equations. Therefore, it is important to obtain DMMs from various equations with FDs and FIs without approximations.

Note that the DMMs were considered before the publication of the 2008 article [49], and their maps were proposed without any connection with equations with FDs or any differential equations at all. It should be also noted that, recently, some DMMs are suggested by using the discrete fractional calculus [75,76,77] and discrete general fractional calculus [78,79,80]. Such maps, which are called “fractional difference” maps, were considered by Wu, Baleanu, and Zeng in [81,82,83,84], by Edelman in the 2015 works [85,86,87,88], in 2018 papers [89,90], in the reviews [91,92], and in the paper of Edelman, Helman, and Smidtaite [93,94,95,96,97]. Unfortunately, such fractional discrete maps are not related with the exact solutions of equations with FDs, and such maps were not obtained from equations with FDs without approximations. In addition, there are no well-founded models in physics, biology, or economics that were described by the equation of the discrete fractional calculus. Unfortunately, there are currently no studies of the relationship between discrete FC, described in [75,76,77], and classical FC [1,2,3,4,5,6,7]. However, the description of the new chaotic type of behavior of nonlinear systems with memory and new types of attractors of “fractional difference” maps is important. This gives hope that similar attractors and similar chaotic behavior are realized in discrete maps obtained from equations with FDs without approximations.

One of the interesting systems with memory that is described by two FDs is the periodically kicked damped rotator. This system with memory is a fractional generalization of the system without memory that is described by the second-order differential equation

$$D_t^{2}X\left(t\right)+\lambda D_t^{1}X\left(t\right)=K\mathcal{N}\left[X\left(t\right)\right]\sum _{k=1}^{\infty }\delta \left({\displaystyle \frac{t}{T}}-k\right),$$

(1)

where $D_t^k=d^k/d{t}^k$ are the derivatives of integer order $k>0$, $\lambda$ is a damping constant, T is a kick period, K is an amplitude of these kicks, $\mathcal{N}\left[X\right]$ is a real-valued function, and $\delta \left(t\right)$ is the Dirac delta-function. Equation (1) gives [37], pp. 16–17, the memoryless discrete map

$$X_{n+1}-X_n={\displaystyle \frac{1-e^{-\lambda T}}{\lambda }}\left(P_n+KT\mathcal{N}\left[X_n\right]\right),$$

(2)

$$P_{n+1}=e^{-\lambda T}\left(P_n+KT\mathcal{N}\left[X_n\right]\right).$$

(3)

These equations are proved in Section 1.2 of [37], pp. 16–17. This map, (2), is known as the kicked damped rotator map.

We should note that such universal DMMs were first obtained and described by the authors in works [22,52,63] for the case $\alpha \in (1,2)$ and $\beta =\alpha -1$ in Section 1.3.4 of [52] and Sections 18.11–18.13 in [22]. The first computer simulation of the dissipative standard map with memory ($\mathcal{N}\left[X_n\right]=sin\left(X_n\right)$) is realized by Edelman in [63] for $\alpha =1.9975$.

In this paper, we proposed the generalization of Equation (1) in the form

$$\left(D_t^{\alpha }X\right)\left(t\right)+\lambda \left(D_t^{\beta }X\right)\left(t\right)=K\mathcal{N}\left[X\left(t\right)\right]\sum _{k=1}^{\infty }\delta \left({\displaystyle \frac{t}{T}}-k\right),$$

(4)

where $D_t^{\alpha }$ and $D_t^{\alpha }$ are the FDs of the arbitrary orders $\alpha >\beta >0$ [4]; $\lambda$ is a damping constant; T is a period of the kicks; K is an amplitude of these kicks; $\mathcal{N}\left[X\right]$ is a real-valued function; $\delta \left(t\right)$ is the Dirac delta-function. Equation (4) can be interpreted as the equation of periodically kicked rotator with power-law memory. The structure of the original equation is of significant importance: its left-hand side is linear and has constant coefficients, the right-hand side is generally nonlinear, but due to delta functions, only the function values at discrete time moments are used.

The following results are proposed in this paper.

• For the first time, the exact solution of the equation of the damped rotator with power-law memory is obtained in the general case for the arbitrary orders of two FDs in this paper.
• The exact solutions of the nonlinear Equation (4) with two FDs for the orders $\alpha >\beta >0$ are obtained for arbitrary values of the orders of these FDs. In this article, the orders of two FDs are not related to each other, unlike in previous works [22,52,63], where $\alpha \in (1,2]$ and $\beta =\alpha -1$.
• It should be emphasized that the manuscript proposed exact analytical solutions to nonlinear equations with FDs, which are derived without any approximations. The proposed method gives exact analytical solutions, where approximations are not used at all.
• Using these solutions, we derived the DMMs that describe kicked damped rotator with power-law memory.
• As a simple illustration of the possible directions of the application of the proposed method, the model of economic growth was considered in addition to the well-known model of the fractional damped oscillator with friction, memory, and external kicks.

2. Equation with Two FDs and Periodic Kicks

In this section, the equation of the dynamic systems with periodic kicks and power-law nonlocality in time is proposed as the generalization of the periodically kicked rotator without memory. This equation is an equation with two FDs of the orders $\alpha$ and $\beta$ ($\alpha >\beta >0$), which describe nonlocality in time. The search for an explicit exact solution to this nonlinear equation with two derivatives consists of three stages:

(1) At the first stage, the second fundamental theorem of fractional calculus and the properties of FDs and FIs are used to obtain linear non-homogeneous equation with one FD of the order $\alpha -\beta$.

(2) At the second stage, the exact solution of the non-homogeneous linear equation with one FD is derived using Theorem 4.3 of [4].

(3) At the third stage, using the exact solution, which is written for the discrete moments of time, the difference between these solutions at neighboring time points is found as a function of all past discrete moments of time.

2.1. Fractional Differential Equation with Periodic Kicks

Let us consider the equation with two FDs with the periodic kicks

$$\left(D_{C;0+}^{\alpha }X\right)\left(t\right)+\lambda \left(D_{C;0+}^{\beta }X\right)\left(t\right)=K\mathcal{N}\left[X\left(t\right)\right]\sum _{k=1}^{\infty }\delta \left({\displaystyle \frac{t}{T}}-k\right),$$

(5)

where T is a period, K is an amplitude of these kicks, $\mathcal{N}\left[X\right]$ is a real-valued function, and $D_{C;0+}^{\alpha }$ and $D_{C;0+}^{\beta }$ are the Caputo FDs of the orders $N-1<\alpha <N$, $M-1<\beta <M$, with $N,M\in \mathbb{N}$ and $\alpha >\beta >0$, such that

$$\left(D_{C;0+}^{\alpha }X\right)\left(t\right):=\left(I_{0+}^{N-\alpha }{X}^{\left(N\right)}\right)\left(t\right)={\int }_0^t{h}_{N-\alpha }(t-\tau )X^{\left(N\right)}\left(\tau \right)d\tau ,$$

(6)

$$\left(D_{C;0+}^{\beta }X\right)\left(t\right):=\left(I_{0+}^{M-\beta }{X}^{\left(M\right)}\right)\left(t\right)={\int }_0^t{h}_{M-\beta }(t-\tau )X^{\left(M\right)}\left(\tau \right)d\tau ,$$

(7)

where $h_{\omega }\left(t\right)=t^{\omega -1}/\Gamma \left(\omega \right)$ with $\omega >0$, $\Gamma \left(\omega \right)$ is the gamma function, $X^{\left(N\right)}\left(t\right)=d^{N}X\left(t\right)/d{t}^N$, and $X^{\left(M\right)}\left(t\right)=d^{M}X\left(t\right)/d{t}^M$ with $N,M\in \mathbb{N}$ [4].

The Dirac delta-functions are the generalized functions [98,99]. In order for the left side of Equation (5) to make sense, the function $\mathcal{N}\left[X\right(t\left)\right]$ must be continuous at $t=kT$. In Equation (5) with two FDs, we can use $\mathcal{N}\left[X\right(t\left)\right]$ if $\mathcal{N}\left[X\right]$ is continuous since this situation is analogous to the case of an equation with one FD of the order greater than one.

Let us note well-known terms that are used in the discrete maps [22,37,39,40,41,42].

• If $\mathcal{N}[t,X(t\left)\right]=\mathcal{N}\left[X\right(t\left)\right]$, then the map is called the universal DMM.
• If $\mathcal{N}\left[X\right(t\left)\right]=-X\left(t\right)$, then the map is the Anosov DMM.
• If $-\eta \mathcal{N}\left[X\left(t\right)\right]=(r-1)X\left(t\right)-r{X}^2\left(t\right)$, then the map is the logistic DMM.
• For $\mathcal{N}\left[X\right(t\left)\right]=sin\left(X\right(t\left)\right)$, the map is called the standard or Chirikov–Taylor DMM [41].

Applying the Riemann–Liouville FI $I_{RL;0+}^{\beta }$ of the order $\beta$ to Equation (5) and using the second fundamental theorem for the FDs and FIs, we obtain the equation

$$\left(I_{RL;0+}^{\beta }{D}_{C;0+}^{\alpha }X\right)\left(t\right)+\lambda \left(I_{RL;0+}^{\beta }{D}_{C;0+}^{\beta }X\right)\left(t\right)-K\left(I_{RL;0+}^{\beta }\mathcal{N}\left[X\left(\tau \right)\right]\sum _{k=1}^{\infty }\delta \left({\displaystyle \frac{\tau }{T}}-k\right)\right)\left(t\right)=0,$$

(8)

where

$$\left(I_{RL;0+}^{\beta }f\right)\left(t\right):={\int }_0^t{h}_{\beta }(t-\tau )f\left(\tau \right)d\tau$$

(9)

is the Riemann–Liouville FI of the order $\beta >0$ [1,4].

Equation (8) can be written as the sum of three terms:

$$T_1\left(t\right)=\left(I_{RL;0+}^{\beta }{D}_{C;0+}^{\alpha }X\right)\left(t\right),$$

(10)

$$T_2\left(t\right)=\lambda \left(I_{RL;0+}^{\beta }{D}_{C;0+}^{\beta }X\right)\left(t\right),$$

(11)

$$T_3\left(t\right)=-K\left(I_{RL;0+}^{\beta }\mathcal{N}\left[X\left(\tau \right)\right]\sum _{k=1}^{\infty }\delta \left({\displaystyle \frac{\tau }{T}}-k\right)\right)\left(t\right),$$

(12)

where $M-1<\beta <M$ and $N-1<\alpha <N$, with $N,M\in \mathbb{N}$ and $\alpha >\beta >0$.

Let us consider the transformation of these terms of Equation (8).

2.2. Transformation of First Term of Equation with FDs

Using the definition of the Caputo FD of the order $\alpha \in (N-1,N]$ with $N\in \mathbb{N}$ and the semigroup property of the Riemann–Liouville FIs, we obtain

$$\left(I_{RL;0+}^{\beta }{D}_{C;0+}^{\alpha }X\right)\left(t\right)=\left(I_{RL;0+}^{\beta }{I}_{RL;0+}^{N-\alpha }{D}^{N}X\right)\left(t\right)=\left(I_{RL;0+}^{N-(\alpha -\beta )}{D}^{N}X\right)\left(t\right),$$

(13)

where $\alpha >\beta >0$, $D^{N}X\left(t\right):=d^{N}X\left(t\right)/d{t}^N=X^{\left(N\right)}\left(t\right)$.

Let us consider the following two cases.

(1) If $N-1<\alpha -\beta <N$ and $X\left(t\right)\in A{C}^N[a,b]$ with $(0,(n+1\left)T\right]\subset [a,b]$ and $-\infty <a<b<+\infty$, then we obtain

$$\left(I_{RL;0+}^{N-(\alpha -\beta )}{D}^{N}X\right)\left(t\right)=\left(D^{\alpha -\beta }X\right)\left(t\right)$$

(14)

and

$$\left(I_{RL;0+}^{\beta }{D}_{C;0+}^{\alpha }X\right)\left(t\right)=\left(D_{C;0+}^{\alpha -\beta }X\right)\left(t\right).$$

(15)

In this case, we obtain

$$T_1\left(t\right)=\left(D_{C;0+}^{\alpha -\beta }X\right)\left(t\right).$$

(16)

(2) Let us consider the general case

$$M-1<\beta <M,N-1<\alpha <N,\alpha >\beta >0.$$

(17)

Using that

$$\alpha -\beta =(N-1+\{\alpha \left\}\right)-(M-1+\{\beta \left\}\right)=N-M+\left\{\alpha \right\}-\left\{\beta \right\},$$

(18)

where $\left\{x\right\}=x-\left[x\right]$ and $\left[x\right]$ is the integer value of the number $x\in \mathbb{R}$, we obtain

$$L-1<\alpha -\beta ⩽L,$$

(19)

where

$$L:=\left\{\begin{array}{cc}N-M+1\hfill & \left\{\alpha \right\}>\left\{\beta \right\}\mathrm{if}N⩾M⩾1,\hfill \\ N-M\hfill & \left\{\alpha \right\}<\left\{\beta \right\}\mathrm{if}N>M⩾1,\hfill \\ N-M\hfill & \left\{\alpha \right\}=\left\{\beta \right\}\mathrm{if}N>M,\hfill \end{array}\right.$$

(20)

where $\alpha \in (N-1,N]$, $\beta \in (M-1,M]$, and $N,M\in \mathbb{N}$.

For the case $N-L=0$, we obtain $N-1<\alpha -\beta <N$ and Equation (16).

For the case $1⩽N-L⩽N$, we have the equalities

$$\left(I_{RL;0+}^{\beta }{D}_{C;0+}^{\alpha }X\right)\left(t\right)=\left(I_{RL;0+}^{\beta }{I}_{RL;0+}^{N-\alpha }{D}^{N}X\right)\left(t\right)=$$

$$\left(I_{RL;0+}^{N-(\alpha -\beta )}{D}^{N}X\right)\left(t\right)=\left(I_{RL;0+}^{L-(\alpha -\beta )}{I}^{N-L}{D}^{N-L}{D}^{L}X\right)\left(t\right),$$

(21)

if $X\left(t\right)\in A{C}^N[a,b]$ with $(0,(n+1\left)T\right]\subset [a,b]$ and $-\infty <a<b<+\infty$, $D^{L}X\left(t\right):=X^{\left(L\right)}\left(t\right)$, and $D^{N}X\left(t\right):=X^{\left(N\right)}\left(t\right)$.

Using the second fundamental theorem of the calculus for $(N-L)\in \mathbb{N}$ as

$$\left(I^{N-L}{D}^{N-L}f\right)\left(t\right)=f\left(t\right)-\sum _{m=0}^{N-L-1}{\displaystyle \frac{f^{\left(m\right)}(0+)}{m!}}{t}^m,$$

(22)

where $1⩽N-L⩽N-1$ and $f\left(t\right)=X^{\left(L\right)}\left(t\right)$. Equation (21) gives

$$\left(I_{RL;0+}^{L-(\alpha -\beta )}{I}^{N-L}{D}^{N-L}{D}^{L}X\right)\left(t\right)=\left(I_{RL;0+}^{L-(\alpha -\beta )}{I}^{N-L}{D}^{N-L}{X}^{\left(L\right)}\right)\left(t\right)=$$

$$\left(I_{RL;0+}^{L-(\alpha -\beta )}{X}^{\left(L\right)}\right)\left(t\right)-\sum _{m=0}^{N-L-1}{\displaystyle \frac{X^{(L+m)}(0+)}{m!}}\left(I_{RL;0+}^{L-(\alpha -\beta )}{t}^m\right)\left(t\right)=$$

$$\left(D_{C;0+}^{\alpha -\beta }X\right)\left(t\right)-\sum _{m=0}^{N-L-1}{\displaystyle \frac{X^{(L+m)}(0+)}{\Gamma (m+L-(\alpha -\beta )+1)}}{t}^{m+L-(\alpha -\beta )},$$

(23)

where we use $m!=\Gamma (m+1)$ and Equation 2.1.16 of [4], p. 71, in the form

$$\left(I_{RL;0+}^{L-(\alpha -\beta )}{t}^m\right)\left(t\right)={\displaystyle \frac{\Gamma (m+1)}{\Gamma (m+L-(\alpha -\beta )+1)}}{t}^{m+L-(\alpha -\beta )},$$

(24)

and $\Gamma \left(z\right)$ is the gamma function.

As a result, we obtain

$$T_1\left(t\right)=\left(D_{C;0+}^{\alpha -\beta }X\right)\left(t\right)-\sum _{m=0}^{N-L-1}{\displaystyle \frac{X^{(L+m)}(0+)}{\Gamma (m+L-(\alpha -\beta )+1)}}{t}^{m+L-(\alpha -\beta )}$$

(25)

for the case (17), where $L\in \mathbb{N}$, $1⩽L⩽N-1$, and we obtain

$$T_1\left(t\right)=\left(D_{C;0+}^{\alpha -\beta }X\right)\left(t\right)$$

(26)

for $L=N$, which means $N-1<\alpha -\beta <N$.

2.3. Transformation of Second Term of Equation with FDs

The second fundamental theorem of FC for the Riemann–Liouville FIs integral and the Caputo FDs is described by Lemma 2.22 of [4], p. 96, in the following form: if $X\left(t\right)\in A{C}^M[a,b]$ with $(0,(n+1\left)T\right]\subset [a,b]$ and $-\infty <a<b<+\infty$, and $M-1<\beta <M$, then the equation

$$\left(I_{RL;a+}^{\beta }{D}_{C;a+}^{\beta }X\right)\left(t\right)=X\left(t\right)-\sum _{k=0}^{M-1}{\Sigma }_k^C(a+){(t-a)}^k,$$

(27)

holds for all $t>a$, where

$${\Sigma }_k^C\left(a\right)=\underset{t\to a+}{lim}{\displaystyle \frac{X^{\left(k\right)}\left(t\right)}{k!}},{\Sigma }_0^C\left(a\right)=\underset{t\to a+}{lim}X\left(t\right).$$

(28)

For example, if $0<\beta <1$$(M=1)$, then

$$\left(I_{RL;a+}^{\beta }{D}_{C;a+}^{\beta }X\right)\left(t\right)=X\left(t\right)-X\left(a\right)$$

(29)

holds $X\left(t\right)\in A{C}^1[a,b]$.

As a result, we obtain

$$T_2\left(t\right)=\lambda X\left(t\right)-\lambda \sum _{k=0}^{M-1}{\Sigma }_k^C(0+)t^k$$

(30)

for $\beta \in (M-1,m]$ with $M\in \mathbb{N}$, and

$$T_2\left(t\right)=\lambda X\left(t\right)-\lambda X(0+)$$

(31)

for $\beta \in (0,1]$.

2.4. Transformation of Third Term of Equation with FDs

Using the Riemann–Liouville FI (9), the third term is represented as

$$T_3\left(t\right)=-K\left(I_{RL;0+}^{\beta }\left(\mathcal{N}\left[X\left(\tau \right)\right]\sum _{k=1}^{\infty }\delta \left({\displaystyle \frac{\tau }{T}}-k\right)\right)\right)\left(t\right)=$$

(32)

$$-{\displaystyle \frac{K}{\Gamma \left(\beta \right)}}{\int }_0^t{(t-\tau )}^{\beta -1}\left(\mathcal{N}\left[X\left(\tau \right)\right]\sum _{k=1}^{\infty }\delta \left({\displaystyle \frac{\tau }{T}}-k\right)\right)d\tau .$$

(33)

For $nT<t<(n+1)T$, we obtain

$$T_3\left(t\right)=-{\displaystyle \frac{K}{\Gamma \left(\beta \right)}}\sum _{k=1}^n{\int }_0^t{(t-\tau )}^{\beta -1}\left(\mathcal{N}\left[X\left(\tau \right)\right]\delta \left({\displaystyle \frac{\tau }{T}}-k\right)\right)d\tau .$$

(34)

Using the equation

$${\int }_0^{t}f\left(\tau \right)\delta \left({\displaystyle \frac{\tau }{T}}-k\right)d\tau =Tf\left(kT\right)\theta (t-kT),$$

(35)

where $f\left(\tau \right)$ is continuous function at $\tau =kT$ and $0<kT<t$, the term $T_3\left(t\right)$ for $nT<t<(n+1)T$ takes the form

$$T_3\left(t\right)=-{\displaystyle \frac{KT}{\Gamma \left(\beta \right)}}\sum _{k=1}^n{(t-kT)}^{\beta -1}\mathcal{N}\left[X\left(kT\right)\right]\theta (t-kT),$$

(36)

where $\theta (t-kT)$ is the Heaviside step function.

2.5. Equation with One FD

The integration of Equation (5) with two FDs gives the equation with one FD of the order $\alpha -\beta >0$. Substitution of Equations (26) and (31), or (26), (30), and (36), into equation

$$T_1\left(t\right)+T_2\left(t\right)+T_3\left(t\right)=0$$

(37)

gives the following linear equations with FD.

(1) Let us consider $\beta \in (0,1)$ such that $N-1<\alpha <N$ and $N-1<\alpha -\beta <N$. Substitution of Equations (26), (31), and (36) into Equation (37) gives the linear equation with one FD

$$\left(D^{\alpha -\beta }X\right)\left(t\right)+\lambda (X\left(t\right)-X(0+))={\displaystyle \frac{KT}{\Gamma \left(\beta \right)}}\sum _{k=1}^n{(t-kT)}^{\beta -1}\mathcal{N}\left[X\left(kT\right)\right]\theta (t-kT).$$

(38)

Equation (38) can be represented as the linear equation with FD

$$\left(D_{C;0+}^{\alpha -\beta }X\right)\left(t\right)=-\lambda X\left(t\right)+F_1\left(t\right),$$

(39)

where $t\in (nT,(n+1\left)T\right)$, $n\in \mathbb{N}$, $\lambda \in \mathbb{R}$, and

$$F_1\left(t\right):=\lambda X(0+)-T_3\left(t\right)=$$

$$\lambda X(0+)+{\displaystyle \frac{KT}{\Gamma \left(\beta \right)}}\sum _{k=1}^n{(t-kT)}^{\beta -1}\mathcal{N}\left[X\left(kT\right)\right]\theta (t-kT).$$

(40)

(2) Let us consider the case $M-1<\beta <M$, $N-1<\alpha <N$, $\alpha >\beta >0$. Substitution of Equations (25), (30), and (36) into Equation (37) gives the linear equation with the FD

$$\left(D_{C;0+}^{\alpha -\beta }X\right)\left(t\right)=-\lambda X\left(t\right)+F_2\left(t\right),$$

(41)

where the function $F_2\left(t\right)$ can be written as

$$F_2\left(t\right)=A\left(t\right)+B\left(t\right)-T_3\left(t\right)=$$

$$A\left(t\right)+B\left(t\right)+{\displaystyle \frac{KT}{\Gamma \left(\beta \right)}}\sum _{k=1}^n{(t-kT)}^{\beta -1}\mathcal{N}\left[X\left(kT\right)\right]\theta (t-kT),$$

(42)

where

$$A\left(t\right):=\lambda \sum _{k=0}^{M-1}{\Sigma }_k^C(0+)t^k,$$

(43)

$$B\left(t\right):=\sum _{m=0}^{N-L-1}{\displaystyle \frac{X^{(L+m)}(0+)}{\Gamma (m+L-(\alpha -\beta )+1)}}{t}^{m+L-(\alpha -\beta )},$$

(44)

where L is defined by Equation (20), $\alpha \in (N-1,N]$ and $\beta \in (M-1,M]$.

(3) Let us note a simple particular case of Equations (39) and (41), where $\alpha -\beta$ is a positive integer. If $\left\{\alpha \right\}=\left\{\beta \right\}$, then

$$\alpha -\beta =N-M=L\in \mathbb{N},$$

(45)

and the FD is the integer-order derivative

$$\left(D_{C;0+}^{\alpha -\beta }X\right)\left(t\right)={\displaystyle \frac{d^{L}X\left(t\right)}{d{t}^L}}.$$

(46)

Then, in the case $\left\{\alpha \right\}=\left\{\beta \right\}$, Equations (39) and (41) are differential equations of the integer orders

$${\displaystyle \frac{d^{L}X\left(t\right)}{d{t}^L}}=-\lambda X\left(t\right)+F_1\left(t\right).$$

(47)

$${\displaystyle \frac{d^{L}X\left(t\right)}{d{t}^L}}==-\lambda X\left(t\right)+F_2\left(t\right).$$

(48)

If $\alpha =\beta +1$, then $L=1$, and we obtain a differential equation of the first order.

Remark 1. 

Let us note that Equations (39) and (41) are linear non-homogeneous equation with one FD of the order $\alpha -\beta >0$.

3. Exact Solution of Equation with One FD

3.1. Exact Solution of Equation with One FD

To solve Equations (39) and (41) with FDs, we can use the theorem that was proved in [4], pp. 230–231, as Theorem 4.3. This theorem states that if $F\left(t\right)\in C_{\gamma }[a,b]$ with $(0,(n+1\left)T\right]\subset [a,b]$ and $-\infty <a<b<+\infty$, then the Cauchy problem in the form of the equation

$$\left(D_{C;a+}^{\alpha }X\right)\left(t\right)+\lambda X\left(t\right)=F\left(t\right),$$

(49)

where $\lambda \in \mathbb{R}$ and the initial conditions

$$X^{\left(k\right)}(a+)=b_k\in \mathbb{R},(k=0,1,...,N-1)$$

(50)

has a unique solution $X\left(t\right)\in C_{\gamma }^{\alpha ,N-1}[a,b]$, such that

$$X\left(t\right)=\sum _{k=0}^{N-1}{b}_k{(t-a)}^k{E}_{\alpha ,k+1}[-\lambda {(t-a)}^{\alpha }]+{\int }_a^t{(t-\tau )}^{\alpha -1}{E}_{\alpha ,\alpha }[-\lambda {(t-\tau )}^{\alpha }]F\left(\tau \right)d\tau ,$$

(51)

where $E_{\alpha ,\beta }\left[z\right]$ is the two-parameter Mittag–Leffler function [4,100]. If $\gamma =0$ and $F\left(t\right)\in C[a,b]$, then the solution belongs to the space $X\left(t\right)\in C^{\alpha ,N-1}[a,b]$. The two-parameter Mittag–Leffler function $E_{\alpha ,\beta }\left[z\right]$ is defined [100] by the expression

$$E_{\alpha ,\beta }\left[z\right]=\sum _{k=0}^{\infty }{\displaystyle \frac{z^k}{\Gamma (\alpha k+\beta )}},$$

(52)

where $\alpha >0$, and $\beta$ are arbitrary real or complex numbers. Note that $E_{1,1}\left(z\right)=e^z$.

Here, $C_{\gamma }[a,b]$ is the weighted space [4], p. 4, of functions $f\left(t\right)$ given on finite interval $(a,b]$ such that the function ${(t-a)}^{\gamma }f\left(t\right)\in C[a,b]$. The space $C_{\gamma }^{N-1}[a,b]$ is the weighted space [4], p. 4, of the continuously differentiable functions $f\left(t\right)$ up to order $N-1$ given on finite interval $(a,b]$ such that $f^{\left(N\right)}\left(t\right)\in C_{\gamma }[a,b]$

To solve Equations (39) and (41) for $t\in (nT,(n+1\left)T\right)$, $n\in \mathbb{N}$, we will use Theorem 4.3 of [4], where $\alpha$ should be $\alpha -\beta$ and N should be L, $a=0$. Let $N-1<\alpha <N$, $0\le \gamma <1$, $\gamma ⩽\alpha$, $\lambda \in \mathbb{R}$.

As a result, Equations (39) and (41) have the solutions

$$X\left(t\right)=\sum _{k=0}^{L-1}{b}_k{t}^k{E}_{\alpha -\beta ,k+1}[-\lambda t^{\alpha -\beta }]+{\int }_0^t{(t-\tau )}^{\alpha -\beta -1}{E}_{\alpha -\beta ,\alpha -\beta }[-\lambda {(t-\tau )}^{\alpha -\beta }]F_j\left(\tau \right)d\tau ,$$

(53)

where $j=1$ and $j=2$ for Equations (39) and (41), respectively.

3.2. Calculating Fractional Integrals in Solution

Let us note that the FIs of the functions $A\left(t\right)$ and $B\left(t\right)$ for $t\in \left(nT\right(n+1\left)T\right)$ can be reduced to the integral

$${\int }_0^t{(t-\tau )}^{\alpha -\beta -1}{E}_{\alpha -\beta ,\alpha -\beta }[-\lambda {(t-\tau )}^{\alpha -\beta }]{\tau }^{\mu -1}d\tau .$$

(54)

For integration (54) of the function $A\left(t\right)$, $B\left(t\right)$, we can use Equation (4.4.5) from [100], p. 70, which without tmisprint $\Gamma \left(\alpha \right)$ is

$${\int }_0^t{(t-\tau )}^{\beta -1}{E}_{\alpha ,\beta }\left[\lambda {(t-\tau )}^{\alpha }\right]{\tau }^{\mu -1}d\tau =\Gamma \left(\mu \right)t^{\beta +\mu -1}{E}_{\alpha ,\beta +\mu }\left[\lambda t^{\alpha }\right],$$

(55)

where $\beta >0$ and $\mu >0$. Note that in Equation (55) the parameter $\mu$ can be considered as a positive integer, or as a real positive number.

For our case, in Equation (55), we must use $\alpha -\beta$ instead of $\alpha$ and $\beta$ in the form

$${\int }_0^t{(t-\tau )}^{\alpha -\beta -1}{E}_{\alpha -\beta ,\alpha -\beta }\left[-\lambda {(t-\tau )}^{\alpha -\beta }\right]{\tau }^{\mu -1}d\tau =\Gamma \left(\mu \right)t^{\alpha -\beta +\mu -1}{E}_{\alpha -\beta ,\alpha -\beta +\mu }\left[-\lambda t^{\alpha -\beta }\right]$$

(56)

with $\mu =k+1$ for $A\left(t\right)$ and $\mu =m+L-(\alpha -\beta )+1$ for $B\left(t\right)$.

Then, the integral of the function $A\left(t\right)$ for $t\in (nT,(n+1\left)T\right)$ can be written as

$${\int }_0^t{(t-\tau )}^{\alpha -\beta -1}{E}_{\alpha -\beta ,\alpha -\beta }[-\lambda {(t-\tau )}^{\alpha -\beta }]A\left(\tau \right)d\tau =$$

$$\lambda \sum _{k=0}^{M-1}{\Sigma }_k^C(0+){\int }_0^t{(t-\tau )}^{\alpha -\beta -1}{E}_{\alpha -\beta ,\alpha -\beta }[-\lambda {(t-\tau )}^{\alpha -\beta }]{\tau }^{k}d\tau =$$

$$\lambda \sum _{k=0}^{M-1}{\Sigma }_k^C(0+)\Gamma (k+1)t^{\alpha -\beta +k}{E}_{\alpha -\beta ,\alpha -\beta +k+1}\left[-\lambda t^{\alpha -\beta }\right].$$

(57)

Using Equation (57) with $k=0$, we can obtain the integral of the term $\lambda X(0+)$ in the function $F_1\left(t\right)$ in the form

$${\int }_0^t{(t-\tau )}^{\alpha -\beta -1}{E}_{\alpha -\beta ,\alpha -\beta }[-\lambda {(t-\tau )}^{\alpha -\beta }]\lambda X(0+)d\tau =$$

$$\lambda X(0+){\int }_0^t{(t-\tau )}^{\alpha -\beta -1}{E}_{\alpha -\beta ,\alpha -\beta }[-\lambda {(t-\tau )}^{\alpha -\beta }]{\tau }^{0}d\tau =$$

$$\lambda X(0+)t^{\alpha -\beta }{E}_{\alpha -\beta ,\alpha -\beta +1}\left[-\lambda t^{\alpha -\beta }\right].$$

(58)

The integral of the function $B\left(t\right)$ for $t\in (nT,(n+1\left)T\right)$ can be written as

$${\int }_0^t{(t-\tau )}^{\alpha -1}{E}_{\alpha ,\alpha }[-\lambda {(t-\tau )}^{\alpha }]B\left(\tau \right)d\tau =$$

$$\sum _{m=0}^{N-L-1}{\displaystyle \frac{X^{(L+m)}(0+)}{\Gamma (m+L-(\alpha -\beta )+1)}}{\int }_0^t{(t-\tau )}^{\alpha -1}{E}_{\alpha ,\alpha }[-\lambda {(t-\tau )}^{\alpha }]t^{m+L-(\alpha -\beta )}d\tau =$$

$$\sum _{m=0}^{N-L-1}{X}^{(L+m)}(0+)t^{m+L}{E}_{\alpha -\beta ,m+L+1}\left[-\lambda t^{\alpha -\beta }\right].$$

(59)

For the integration of the function $T_3\left(t\right)$$t\in (nT,(n+1\left)T\right)$, let us define the function $R_{\alpha ,\beta }(t,kT)$ for $t\in (nT,(n+1\left)T\right)$, with $n\in \mathbb{N}$ by the equation

$$R_{\alpha ,\beta }(t,kT):={\int }_0^t{(t-\tau )}^{\alpha -\beta -1}{E}_{\alpha -\beta ,\alpha -\beta }[-\lambda {(t-\tau )}^{\alpha -\beta }]{\displaystyle \frac{{(\tau -kT)}^{\beta -1}}{\Gamma \left(\beta \right)}}\theta (\tau -kT)d\tau =$$

$${\int }_{kT}^t{(t-\tau )}^{\alpha -\beta -1}{E}_{\alpha -\beta ,\alpha -\beta }[-\lambda {(t-\tau )}^{\alpha -\beta }]{\displaystyle \frac{{(\tau -kT)}^{\beta -1}}{\Gamma \left(\beta \right)}}d\tau ,$$

(60)

where we use

$${\int }_0^{t}f\left(\tau \right)\theta (\tau -kT)d\tau ={\int }_{kT}^{t}f\left(\tau \right)d\tau .$$

(61)

Using the variable $s=\tau -kT$ and Equation (55) with $\mu =\beta$ in the form

$$R_{\alpha ,\beta }(t,kT)={\int }_0^{t-kT}{((t-kT)-s)}^{\alpha -\beta -1}{E}_{\alpha -\beta ,\alpha -\beta }\left[\lambda {((t-kT)-s)}^{\alpha -\beta }\right]{\displaystyle \frac{s^{\mu -1}}{\Gamma \left(\mu \right)}}ds=$$

$${(t-kT)}^{\alpha -\beta +\mu -1}{E}_{\alpha -\beta ,\alpha -\beta +\mu }\left[-\lambda {(t-kT)}^{\alpha -\beta }\right]=$$

$${(t-kT)}^{\alpha -1}{E}_{\alpha -\beta ,\alpha }\left[-\lambda {(t-kT)}^{\alpha -\beta }\right],$$

(62)

we obtain that the function $R_{\alpha ,\beta }(t,kT)$ has the form

$$R_{\alpha ,\beta }(t,kT)={(t-kT)}^{\alpha -1}{E}_{\alpha -\beta ,\alpha }\left[-\lambda {(t-kT)}^{\alpha -\beta }\right],$$

(63)

where $\alpha >\beta >0$. Note that function (63) is defined for $t>kT$, where $k\in \mathbb{N}$, ($k=1,2,3,\dots$).

Remark 2. 

We can extend the function (63) to values $t⩽kT$, by

$$R_{\alpha ,\beta }(t,kT)=0\mathit{if}t⩽kT.$$

(64)

This is due to the fact that at $t\in (0,T)$, Equation (5) contains the sum of the Dirac delta function from $k=1$. In other words, there is no kick at $t=0$. Using the fact that function (63) describes the system response to periodic kicks, one can use (64).

Example 1. 

For the case $\left\{\alpha \right\}=\left\{\beta \right\}$, then $\alpha -\beta =N-M=L\in \mathbb{N}$, and function (60) can be written as

$$R_{\alpha ,\beta }(t,kT):={\int }_0^{t-kT}{s}^{L-1}{E}_{L,L}[-\lambda s^L]{\displaystyle \frac{{(t-kT-s)}^{\beta -1}}{\Gamma \left(\beta \right)}}ds=$$

$${(t-kT)}^{\alpha -1}{E}_{L,\alpha }\left[-\lambda {(t-kT)}^L\right],$$

(65)

where $t>kT$ and $k\in \mathbb{N}$. Using the examples of the Mittag–Leffler function $E_{L,L}\left[z\right]$ with $L=1$ and $L=2$ in the form

$$E_{1,1}\left[z\right]=exp\left(z\right),E_{2,2}\left[z\right]={\displaystyle \frac{sinh\left(\sqrt{z}\right)}{\sqrt{z}}},$$

(66)

we can obtain examples of function (60) in the form

$$R_{\alpha ,\beta }(t,kT)=R_{\beta +1,\beta }(t,kT):={\int }_0^{t-kT}exp(-\lambda s){\displaystyle \frac{{(t-kT-s)}^{\beta -1}}{\Gamma \left(\beta \right)}}ds,$$

(67)

which is the Riemann–Liouville FI of the order $\beta >0$ of the function $exp(-\lambda s)$, and

$$R_{\alpha ,\beta }(t,kT)=R_{\beta +2,\beta }(t,kT):={\int }_0^{t-kT}{\displaystyle \frac{sin\left(\sqrt{\lambda s}\right)}{\sqrt{\lambda s}}}{\displaystyle \frac{{(t-kT-s)}^{\beta -1}}{\Gamma \left(\beta \right)}}sds,$$

(68)

where $t>kT$ and $k\in \mathbb{N}$, and we use $sinh\left(ix\right)=isin\left(x\right)$.

Using function (60), the integral of the function $T_3\left(t\right)$ for $t\in (nT,(n+1\left)T\right)$ with $n\in \mathbb{N}$ can be written as

$${\int }_0^t{(t-\tau )}^{\alpha -\beta -1}{E}_{\alpha -\beta ,\alpha -\beta }[-\lambda {(t-\tau )}^{\alpha -\beta }]T_3\left(\tau \right)d\tau =$$

$$-{\displaystyle \frac{KT}{\Gamma \left(\beta \right)}}\sum _{k=1}^n\mathcal{N}\left[X\left(kT\right)\right]{\int }_0^t{(t-\tau )}^{\alpha -\beta -1}{E}_{\alpha -\beta ,\alpha -\beta }[-\lambda {(t-\tau )}^{\alpha -\beta }]{(\tau -kT)}^{\beta -1}\theta (\tau -kT)d\tau =$$

$$-KT\sum _{k=1}^n\mathcal{N}\left[X\left(kT\right)\right]R_{\alpha ,\beta }(t,kT).$$

(69)

As a result, Equations (39) and (41) have the following solutions for $t\in (nT,(n+1\left)T\right)$.

Equation (39), where $L=N$, has the solution

$$X\left(t\right)=\sum _{k=0}^{L-1}{b}_k{t}^k{E}_{\alpha -\beta ,k+1}[-\lambda t^{\alpha -\beta }]+{\int }_0^t{(t-\tau )}^{\alpha -\beta -1}{E}_{\alpha -\beta ,\alpha -\beta }[-\lambda {(t-\tau )}^{\alpha -\beta }]F_1\left(\tau \right)d\tau =$$

$$\sum _{k=0}^{N-1}{b}_k{t}^k{E}_{\alpha -\beta ,k+1}[-\lambda t^{\alpha -\beta }]+\lambda X(0+)t^{\alpha -\beta }{E}_{\alpha -\beta ,\alpha -\beta +1}\left[-\lambda t^{\alpha -\beta }\right]+$$

$$KT\sum _{k=1}^n\mathcal{N}\left[X\left(kT\right)\right]R_{\alpha ,\beta }(t,kT),$$

(70)

where $t\in (nT,(n+1\left)T\right)$ with $n\in \mathbb{N}$.

Equation (41) has the solution

$$X\left(t\right)=\sum _{k=0}^{L-1}{b}_k{t}^k{E}_{\alpha -\beta ,k+1}[-\lambda t^{\alpha -\beta }]+{\int }_0^t{(t-\tau )}^{\alpha -\beta -1}{E}_{\alpha -\beta ,\alpha -\beta }[-\lambda {(t-\tau )}^{\alpha -\beta }]F_2\left(\tau \right)d\tau =$$

$$\sum _{k=0}^{L-1}{b}_k{t}^k{E}_{\alpha -\beta ,k+1}[-\lambda t^{\alpha -\beta }]+\lambda \sum _{k=0}^{M-1}{\Sigma }_k^C(0+)\Gamma (k+1)t^{\alpha -\beta +k}{E}_{\alpha -\beta ,\alpha -\beta +k+1}\left[-\lambda t^{\alpha -\beta }\right]+$$

$$\sum _{m=0}^{N-L-1}{X}^{(L+m)}(0+)t^{m+L}{E}_{\alpha -\beta ,m+L+1}\left[-\lambda t^{\alpha -\beta }\right]+KT\sum _{k=1}^n\mathcal{N}\left[X\left(kT\right)\right]R_{\alpha ,\beta }(t,kT),$$

(71)

where $t\in (nT,(n+1\left)T\right)$ with $n\in \mathbb{N}$.

Note that all terms of the solutions contain the Mittag–Leffler function $E_{\alpha -\beta ,\gamma }[-\lambda {(t-kT)}^{\alpha -\beta }]$ with different values of the parameter $\gamma$ ($\gamma =\alpha$, $\gamma =\alpha -\beta$, $\gamma =\alpha -\beta +1$, $\gamma =k+1$, $\gamma =\alpha -\beta +k+1$, $\gamma =m+L+1$).

Example 2. 

For the case $\left\{\alpha \right\}=\left\{\beta \right\}$, then $\alpha -\beta =N-M=L\in \mathbb{N}$, and solution (70) has the form

$$X\left(t\right)=\sum _{k=0}^{L-1}{X}^{\left(k\right)}(0+)t^k{E}_{L,k+1}[-\lambda t^L]+\lambda X(0+)t^L{E}_{L,L+1}\left[-\lambda t^L\right]+$$

$$KT\sum _{k=1}^n\mathcal{N}\left[X\left(kT\right)\right]R_{\alpha ,\beta }(t,kT).$$

(72)

and solution (71) is

$$X\left(t\right)=\sum _{k=0}^{L-1}{X}^{\left(k\right)}(0+)t^k{E}_{L,k+1}[-\lambda t^L]+\lambda \sum _{k=0}^{M-1}{\Sigma }_k^C(0+)\Gamma (k+1)t^{L+k}{E}_{L,L+k+1}\left[-\lambda t^L\right]+$$

$$\sum _{m=0}^{M-1}{X}^{(L+m)}(0+)t^{m+L}{E}_{L,m+L+1}\left[-\lambda t^L\right]+KT\sum _{k=1}^n\mathcal{N}\left[X\left(kT\right)\right]R_{\alpha ,\beta }(t,kT),$$

(73)

where $t\in (nT,(n+1\left)T\right)$ with $n\in \mathbb{N}$.

For the case $\alpha =\beta +1$, we have $L=1$, and solution (72) has the form

$$X\left(t\right)=X(0+)+KT\sum _{k=1}^n\mathcal{N}\left[X\left(kT\right)\right]R_{\beta }(t-kT),$$

(74)

where $t\in (nT,(n+1\left)T\right)$ with $n\in \mathbb{N}$ and the function

$$R_{\beta }\left(t\right):={\displaystyle \frac{1}{\Gamma \left(\beta \right)}}{\int }_0^t{e}^{-\lambda s}{(t-s)}^{\beta -1}ds=t^{\beta }{E}_{1,\beta +1}[-\lambda t]$$

(75)

with $T>0$. Equation (74) is the Riemann–Liouville FI of the function $e^{-\lambda t}$. Equation (74) for $t\in (nT,(n+1\left)T\right)$ describes the solution for all positive α and β, such that $\alpha =\beta +1$, for example, for $\alpha =1.2$ and $\beta =0.2$ or $\alpha =3.8$ and $\beta =2.8$, and so on.

To simplify the derivation of the discrete map with memory, we will use the function

$$Q_{\alpha -\beta ,L}\left(t\right):=\sum _{k=0}^{L-1}{X}^{\left(k\right)}(0+)t^k{E}_{\alpha -\beta ,k+1}[-\lambda t^{\alpha -\beta }]+$$

$$\lambda \sum _{k=0}^{M-1}{\Sigma }_k^C(0+)\Gamma (k+1)t^{\alpha -\beta +k}{E}_{\alpha -\beta ,\alpha -\beta +k+1}\left[-\lambda t^{\alpha -\beta }\right]+$$

$$\sum _{j=0}^{N-L-1}{X}^{(L+j)}(0+)t^{j+L}{E}_{\alpha -\beta ,j+L+1}\left[-\lambda t^{\alpha -\beta }\right],$$

(76)

if $L\in [1,N-1]$ with $N\in \mathbb{N}$. For $L=N$, the function is defined as

$$Q_{\alpha -\beta ,N}\left(t\right):=\sum _{k=0}^{N-1}{X}^{\left(k\right)}(0+)t^k{E}_{\alpha -\beta ,k+1}[-\lambda t^{\alpha -\beta }]+\lambda X(0+)t^{\alpha -\beta }{E}_{\alpha -\beta ,\alpha -\beta +1}\left[-\lambda t^{\alpha -\beta }\right].$$

(77)

Using Equations (76) and (77), solutions (70) and (71) can be represented as

$$X\left(t\right)=Q_{\alpha -\beta ,L}\left(t\right)+KT\sum _{k=1}^n\mathcal{N}\left[X\left(kT\right)\right]R_{\alpha ,\beta }(t,kT)$$

(78)

for $t\in (nT,(n+1\left)T\right)$ with $n\in \mathbb{N}$ and $L\in [1,N]$ with $N\in \mathbb{N}$. Note the function $R_{\alpha ,\beta }(t,kT)$ is defined for $t\in (0,T)$, and we have

$$X\left(t\right)=Q_{\alpha -\beta ,L}\left(t\right),$$

(79)

since $R_{\alpha ,\beta }(t,kT)=0$ for $t⩽kT$. Note that there are no kicks at $t<T$ since the sum of the delta-function in Equation (5) starts at $k=1$, i.e., st $t=T$. To reflect this fact in one equation, both Equations (79) and (79) can be written by using the Heaviside step function

$$X\left(t\right)=Q_{\alpha -\beta ,L}\left(t\right)+KT\sum _{k=1}^n\mathcal{N}\left[X\left(kT\right)\right]R_{\alpha ,\beta }(t,kT)\theta (t-kT),$$

(80)

where the step function $\theta \left(x\right)=1$ for $x>0$ and $\theta \left(x\right)=0$ for $x⩽0$.

To consider the DMMs, we must use the function

$$P_m\left(t\right):=X^{\left(m\right)}\left(t\right),m=0,\dots ,L.$$

(81)

Using the functions (78) and (81), we obtain the equations

$$P_m\left(t\right)=Q_{\alpha -\beta ,L}^{\left(m\right)}\left(t\right)+KT\sum _{k=1}^n\mathcal{N}\left[X\left(kT\right)\right]R_{\alpha ,\beta }^{\left(m\right)}(t,kT),$$

(82)

where $m=0,\dots ,L$, $Q_{\alpha -\beta ,L}^{\left(m\right)}\left(t\right)$, and $R_{\alpha ,\beta }^{\left(m\right)}(t,kT)$ are integer-order derivatives of the functions $Q_{\alpha -\beta ,L}\left(t\right)$ and $R_{\alpha ,\beta }(t,kT)$.

Remark 3. 

Note that the derivatives $Q_{\alpha -\beta ,L}^{\left(m\right)}\left(t\right)$ of the integer order m can be calculated explicitly.

Using Equation (52), which defines the Mittag–Leffler function, we obtain

$$t^{\beta -1}{E}_{\alpha ,\beta }[-\lambda t^{\alpha }]=\sum _{k=0}^{\infty }{\displaystyle \frac{{(-\lambda )}^k{t}^{\alpha k+\beta -1}}{\Gamma (\alpha k+\beta )}}.$$

(83)

Then, the derivative of the integer order $m\in \mathbb{N}$ has the form

$${\displaystyle \frac{d^m}{d{t}^m}}\left(t^{\beta -1}{E}_{\alpha ,\beta }[-\lambda t^{\alpha }]\right)={\displaystyle \frac{d^m}{d{t}^m}}\left(\sum _{k=0}^{\infty }{\displaystyle \frac{{(-\lambda )}^k{t}^{\alpha k+\beta -1}}{\Gamma (\alpha k+\beta )}}\right)=$$

$$\sum _{k=0}^{\infty }{\displaystyle \frac{{(-\lambda )}^k}{\Gamma (\alpha k+\beta )}}{\displaystyle \frac{d^m{t}^{\alpha k+\beta -1}}{d{t}^m}}=\sum _{k=0}^{\infty }{\displaystyle \frac{{(-\lambda )}^k}{\Gamma (\alpha k+\beta )}}{\displaystyle \frac{\Gamma (\alpha k+\beta )}{\Gamma (\alpha k+\beta -m)}}{t}^{\alpha k+\beta -m-1}=$$

$$\sum _{k=0}^{\infty }{\displaystyle \frac{{(-\lambda )}^k{t}^{\alpha k+\beta -m-1}}{\Gamma (\alpha k+(\beta -m\left)\right)}}=t^{\beta -m-1}\sum _{k=0}^{\infty }{\displaystyle \frac{{(-\lambda t^{\alpha })}^k}{\Gamma (\alpha k+(\beta -m\left)\right)}}=$$

$$t^{\beta -m-1}{E}_{\alpha ,\beta -m}[-\lambda t^{\alpha }].$$

(84)

As a result, we proved the equation

$${\displaystyle \frac{d^m}{d{t}^m}}\left(t^{\beta -1}{E}_{\alpha ,\beta }[-\lambda t^{\alpha }]\right)=t^{\beta -m-1}{E}_{\alpha ,\beta -m}[-\lambda t^{\alpha }]$$

(85)

for $m\in \mathbb{N}$. Using Equation (85), we obtain

$$Q_{\alpha -\beta ,L}^{\left(m\right)}\left(t\right)={\displaystyle \frac{d^m}{d{t}^m}}{Q}_{\alpha -\beta ,L}\left(t\right)=$$

$$\sum _{k=0}^{L-1}{X}^{\left(k\right)}(0+)t^{k-m}{E}_{\alpha -\beta ,k-m+1}[-\lambda t^{\alpha -\beta }]+$$

$$\lambda \sum _{k=0}^{M-1}{\Sigma }_k^C(0+)\Gamma (k+1)t^{\alpha -\beta +k-m}{E}_{\alpha -\beta ,\alpha -\beta +k+1}\left[-\lambda t^{\alpha -\beta }\right]+$$

(86)

$$\sum _{j=0}^{N-L-1}{X}^{(L+j)}(0+)t^{j+L-m}{E}_{\alpha -\beta ,j+L-m+1}\left[-\lambda t^{\alpha -\beta }\right],$$

(87)

where $\alpha -\beta >0$.

Using Equation (85), we also obtain the m-order derivative of the function $R_{\alpha ,\beta }(t,kT)$ in the form

$$R_{\alpha ,\beta }^{\left(m\right)}(t,kT)=R_{\alpha -m,\beta -m}(t,kT)={(t-kT)}^{\alpha -1-m}{E}_{\alpha -\beta ,\alpha -m}\left[-\lambda {(t-kT)}^{\alpha -\beta }\right],$$

(88)

where $t\in (nT,(n+1\left)T\right)$ with $n\in \mathbb{N}$, and $\alpha >\beta >0$. One can use that

$$R_{\alpha ,\beta }^{\left(m\right)}(t,kT)=0$$

(89)

for $t⩽kT$.

The short representation of exact solutions (78) and (82) of the equation with FD allows us to obtain the desired DMMs in the next section.

4. Dissipative Discrete Maps with Memory

Using the exact solutions (78) and (82) of the equation with FDs, we obtain the DMMs a with power-law memory function.

To derive DMMs, we should use discrete moments in time $t=nT$ and $t=(n+1)T$, where $n\in \mathbb{N}$. Using the functions

$$X_j=\underset{ϵ\to 0+}{lim}X(jT-ϵ),$$

(90)

$$P_{m,j}=\underset{ϵ\to 0+}{lim}{P}_m(jT-ϵ),(j=1,\dots ,n+1),$$

(91)

solutions (78) and (82) at the limit $ϵ\to 0+$ can be derived in the following forms.

For $nT<t<(n+1)T$ with $n\in \mathbb{N}$, using the solutions for $t=(n+1)T-ϵ\in (nT,(n+1\left)T\right)$, we obtain

$$P_{m,n+1}=Q_{\alpha -\beta ,L}^{\left(m\right)}\left((n+1)T\right)+KT\sum _{j=1}^n\mathcal{N}\left[X_j\right]R_{\alpha ,\beta }^{\left(m\right)}((n+1)T,jT).$$

(92)

For $(n-1)T<t<nT$ with $n\in \mathbb{N}$,, using $t=nT-ϵ\in \left(\right(n-1)T,nT)$, we obtain

$$P_{m,n}=Q_{\alpha -\beta ,L}^{\left(m\right)}\left(nT\right)+KT\sum _{j=1}^{n-1}\mathcal{N}\left[X_j\right]R_{\alpha ,\beta }^{\left(m\right)}(nT,jT).$$

(93)

Note that

$$P_{m,1}=Q_{\alpha -\beta ,L}^{\left(m\right)}\left(nT\right),$$

(94)

since $R_{\alpha ,\beta }^{\left(m\right)}(t,kT)=0$ for $t⩽kT$.

Subtracting Equation (93) from Equation (92), we obtain

$$P_{m,n+1}-P_{k,n}=Q_{\alpha -\beta ,L}^{\left(m\right)}\left((n+1)T\right)-Q_{\alpha -\beta ,L}^{\left(m\right)}\left(nT\right)+$$

$$KT\mathcal{N}\left[X_n\right]R_{\alpha ,\beta }^{\left(m\right)}((n+1)T,nT)+$$

$$KT\sum _{j=1}^{n-1}\mathcal{N}\left[X_j\right]\left(R_{\alpha ,\beta }^{\left(m\right)}((n+1)T,jT)-R_{\alpha ,\beta }^{\left(m\right)}(nT,jT)\right),$$

(95)

for $n=2,3,\dots$, and for $n=1$, we have

$$P_{m,2}-P_{k,1}=Q_{\alpha -\beta ,L}^{\left(m\right)}\left(2T\right)-Q_{\alpha -\beta ,L}^{\left(m\right)}\left(T\right)+$$

$$KT\mathcal{N}\left[X_1\right]R_{\alpha ,\beta }^{\left(m\right)}(2T,T),$$

(96)

where $m=0,1,\dots ,L$.

For $m=0$, we have $P_{m,n+1}=X_{n+1}$ and $P_{0,n}=X_n$, and the DMMs are

$$X_{n+1}-X_n=Q_{\alpha -\beta ,L}\left((n+1)T\right)-Q_{\alpha -\beta ,L}\left(nT\right)+$$

$$KT\mathcal{N}\left[X_n\right]R_{\alpha ,\beta }((n+1)T,nT)+$$

$$KT\sum _{j=1}^{n-1}\mathcal{N}\left[X_j\right]\left(R_{\alpha ,\beta }((n+1)T,jT)-R_{\alpha ,\beta }(nT,jT)\right)$$

(97)

for $n=2,3,\dots$ and for $n=1$, we have

$$X_2-X_1=Q_{\alpha -\beta ,L}\left(2T\right)-Q_{\alpha -\beta ,L}\left(T\right)+$$

$$KT\mathcal{N}\left[X_1\right]R_{\alpha ,\beta }(2T,T).$$

(98)

These discrete maps are the exact solution of the equations with two FDs at the discrete moments in time.

Example 3. 

For the case $\alpha =\beta +1$, the DMM (97) is described as

$$X_{n+1}-X_n=KT\mathcal{N}\left[X_n\right]R_{\alpha ,\beta }((n+1)T,nT)+KT\sum _{j=1}^{n-1}\mathcal{N}\left[X_j\right]{\mathcal{R}}_{\alpha ,\beta }(n,j),$$

(99)

with the function

$${\mathcal{R}}_{\beta }(n,j):=R_{\beta }\left((n-j+1)T\right)-R_{\beta }\left((n-j)T\right)=$$

$${\left((n-j+1)T\right)}^{\beta }{E}_{1,\beta +1}[-\lambda \left((n-j+1)T\right)]-{\left((n-j)T\right)}^{\beta }{E}_{1,\beta +1}[-\lambda \left((n-j)T\right)],$$

(100)

where $R_{\beta }\left(t\right)$ is defined by Equation (75).

The DMM (100) describes the exact solutions of equations with FDs at discrete moments of time for all possitive α and β, such that $\alpha =\beta +1$, for example, for $\alpha =1.5$ and $\beta =0.5$ or $\alpha =1.8$ and $\beta =0.8$ or $\alpha =8.2$ and $\beta =7.1$, and so on.

Note that such univeral DMMs were first obtained and described by the author in works [22,52,63] for the case $\alpha \in (1,2]$ and $\beta =\alpha -1$ in Section 1.3.4 of [52] and Sections 18.11–18.13 in [22]. The first computer simulation of the dissipative standard map with memory is realized by Edelman in [63] for $\alpha =1.9975$.

5. Economic Model of Growth with Two-Parameter Memory and Price Kicks

As an example of the application of the proposed DMMs, let us consider an economic growth model (EGM) instead of the usual physical model of the rotator.

The following assumptions are used in the economic model of growth.

(1) Let the price P be a function of released product $Y\left(t\right)$, i.e., $P=P\left(Y\right(t\left)\right)$. In the logistic growth model, it is assumed that $P\left(Y\right(t\left)\right)=b-aY\left(t\right)$, where a is the marginal price, and b is the price that is independent of the output.

In economics, we can consider the sudden changes of price in the form of price splashes. Let us assume that the price splashes are periodic with period $\theta >0$, and we will describe them by the Dirac delta function. In this case, we obtain the economic model with the periodic price kicks. This assumption assumes that the price function $P_s\left(Y\left(t\right)\right)$ with the periodic sharp splashes of the price (periodic price kicks) is

$$P_s\left(Y\left(t\right)\right)=P\left(Y\left(t\right)\right)\sum _{k=1}^{\infty }\delta \left({\displaystyle \frac{t}{T}}-k\right),$$

(101)

where $P(Y$) is the continuous function of the output Y.

(2) The amount of net investment is assumed to be the fixed part of the income $P\left(Y\right(t\left)\right)Y\left(t\right)$, such that

$$I\left(t\right)=\mu P_s\left(Y\left(t\right)\right)Y\left(t\right),$$

(102)

where $\mu$ is the norm of net investment ($0<\mu <1$), specifying the share of income, which is spent on the net investment.

(3) In the EGM without memory and lag, it is assumed that the rate of change of the output ($dY\left(t\right)/dt$) is directly proportional to the value of the net investment $I\left(t\right)$ that is described by the accelerator equation

$$I\left(t\right)=v{Y}^{\left(1\right)}\left(t\right),$$

(103)

where $v>0$ is the investment coefficient (the power of the accelerator), and $1/v$ is the marginal productivity of capital (the rate of acceleration).

The standard accelerator equation (103) is generalized in [34] by taking into account the memory effects. The equation of the accelerator with power-law memory [34] can be described as

$$I\left(t\right)=v\left(\alpha \right)\left(D_{C;0+}^{\alpha }Y\right)\left(t\right),$$

(104)

which allows us to take into account the influence of the history of changes in output $Y\left(\tau \right)$ on net investment $I\left(t\right)$.

This allows us a description that takes into account the impact of the history of changes in the dynamics of output $Y\left(\tau \right)$ on the net investment $I\left(t\right)$. As a result, we obtain a growth model in a competitive environment with power-law memory, which is considered in [34,53].

In Section 10.5 of book [34], pp. 210–212, the economic growth model with two parameter memories has been proposed and investigated. This model is based on the accelerator with two memory functions

$$I\left(t\right)=v_1\left(\alpha \right)\left(D_{C;0+}^{\alpha }Y\right)\left(t\right)+v_2\left(\beta \right)\left(D_{C;0+}^{\beta }Y\right)\left(t\right),$$

(105)

where $\alpha >\beta >0$.

The substitution of Equations (101) and (105) into Equation (102) gives

$$\left(D_{C;0+}^{\alpha }Y\right)\left(t\right)+{\displaystyle \frac{v_2\left(\beta \right)}{v_1\left(\alpha \right)}}\left(D_{C;0+}^{\beta }Y\right)\left(t\right)={\displaystyle \frac{\mu }{v_1\left(\alpha \right)}}\mathcal{N}\left[X\left(t\right)\right]\sum _{k=1}^{\infty }\delta \left({\displaystyle \frac{t}{T}}-k\right).$$

(106)

Equation (106) is the equation that describes the EGM growth in a competitive environment with two parameter memory and sharp splashes (price kicks). For price linearity $P\left(Y\right(t\left)\right)=b-aY\left(t\right)$, the nonlinear equation describes the logistic-type EGM with the two parameter memories.

Equation (106) is Equation (5), where $X\left(t\right)=Y\left(t\right)$ and

$$\mathcal{N}\left[X\left(t\right)\right]=Y\left(t\right)P\left(Y\left(t\right)\right),\lambda ={\displaystyle \frac{v_2\left(\beta \right)}{v_1\left(\alpha \right)}},K={\displaystyle \frac{\mu }{v_1\left(\alpha \right)}}.$$

(107)

This fact allows us to write the exact solution of the nonlinear Equation (78) with two FDs as

$$Y\left(t\right)=Q_{\alpha -\beta ,L}\left(t\right)+{\displaystyle \frac{\mu T}{v_1\left(\alpha \right)}}\sum _{k=1}^{n}P\left(Y\left(kT\right)\right)Y\left(kT\right)R_{\alpha ,\beta }(t,kT),$$

(108)

and discrete maps (97) as

$$Y_{n+1}-Y_n=Q_{\alpha -\beta ,L}\left((n+1)T\right)-Q_{\alpha -\beta ,L}\left(nT\right)+$$

$${\displaystyle \frac{\mu T}{v_1\left(\alpha \right)}}P\left(Y_n\right)Y_n{R}_{\alpha ,\beta }((n+1)T,nT)+$$

$${\displaystyle \frac{\mu T}{v_1\left(\alpha \right)}}\sum _{j=1}^{n-1}P\left(Y_j\right)Y_j\left(R_{\alpha ,\beta }((n+1)T,jT)-R_{\alpha ,\beta }(nT,jT)\right),$$

(109)

where $R_{\alpha ,\beta }(t,jT)$ and $Q_{\alpha -\beta ,L}\left(t\right)$ are defined by Equations (60) and (76) with $\lambda =v_2\left(\beta \right)/v_1\left(\alpha \right)$.

Dependence on initial conditions is an important issue. Note that the initial conditions will determine whether the economy will grow or fall. This issue is discussed in detail in the book. In this article, the economic model is simply an example of the application. The dependence of economic dynamics on initial conditions is discussed in detail in the book [34].

6. Conclusions

In this paper, the fractional generalization of periodically kicked damped rotator is proposed. This dynamical system is described by the nonlinear equation with two FDs of the arbitrary positive orders $\alpha$ and $\beta$, where $\alpha >\beta$ and periodic kicks occur. These FDs allow us to describe power-law non-locality in time. The exact solution of the equation with FDs is obtained in the general case for the arbitrary orders of FGDs in this paper. Using the exact solutions, we derived DMMs that describe a kicked damped rotator with power-law non-localities in time. These maps, described as the exact solution of nonlinear equations with FDs, are at the same discrete time points as the function of all past discrete moments of time. Let us emphasize that these nonlinear dissipative DMMs are derived from the equations with two FDs without any approximations.

Let us note the following possible developments, generalizations, and applications of the proposed methods and results.

• One of the most important continuations of the development of the proposed exact solutions and discrete mappings is computer modeling. It can be assumed that the new type of attractors and the new type of chaotic behavior can be demonstrated in the proposed DMMs obtained from nonlinear equations with FDs. This is an important and very interesting direction of research, namely, the search for new types of chaotic behavior and a new type of attractors in dynamic maps with memory, which are exact solutions of equations with FDs. This is especially important due to the fundamental nature of these new types of the chaotic behavior and a new type of attractors. Unfortunately, such research is only developing, and new types of behavior and attractors have been found only for the simplest maps. A computer simulation of the proposed DMMs will allow us to discover and describe new types of chaotic behavior and new types of attractors with memory. However, such computer simulations are open questions at the present time and require new research to make possible great discoveries in the future.
• Another of the most important continuations of the development of the proposed approach to obtaining exact solutions and discrete mappings is the generalization of the approach to nonlinear equations with power memory to a general form of memory. The proposed model and the three-stage method, which is proposed for solving the nonlinear equation with two FDs and deriving DMMs, can be generalized from the power-law type to the wide class of time nonlocalities by using general FDs (for example, see the basic articles by Luchko [101,102,103], subsequent articles by Luchko and co-authors [104,105,106], Ortigueira’s paper [107], and Al-Refai and Fernandez’s papers [108,109]). These generalized DDMs will be generalizations from equations with the one general FD [60,62] to the equations with two general FDs.
• It is very important to generalize the proposed method and to derive the exact solutions of nonlinear equations with FDs from the one-dimensional case to the multidimensional case. Let us emphasize that the first fractional generalization of the proposed method of obtaining exact analytical solutions and DMMs was suggested in the 2010 works [22,52]. In these works, the fractional generalization of the Henon and Zaslavsky maps, which are the two-dimensional dissipative quadratic maps given by the two coupled equations, is proposed. In paper [63], the computer simulation of the fractional Zaslavsky maps is realized. Then, recently in works [110,111,112,113], some multidimensional DMMs are suggested by using the discrete fractional calculus [75,76,77]. Unfortunately, these fractional discrete maps were proposed without any connection with equations with FDs or any differential equations at all. Therefore, these multidimensional DMMs cannot be considered as the exact analytical solutions of nonlinear equations at discrete time points. Let us note that Orinaite, Smidtaite, and Ragulsk in the 2025 paper [74] proposed to derive the multidimensional DMMs as maps of matrices from the exact analytical solutions of nonlinear fractional differential equations with matrices. This OSR approach to the multidimensional maps, which are exact solutions of equation with FDs, is very promising.
• Applications of the proposed method and the exact solutions of nonlinear equations with two FDs can be realized in various studies, for example, in the following areas: (1) in physics and mechanics to describe systems with dissipation (or friction) and memory [55]; (2) in economics and finance to derive various economic and financial models with memory [34,53]; (3) in describing the chemical kinetics and population dynamics [54]; (4) to describe the behavior of engineering systems involving adaptive memory and path losses due to power-law frequency dispersion [114,115,116] since the erasure and loss of information can be interpreted as a fading memory; (5) a very interesting and important application can be found for describing self-organization with memory in complex systems and processes [117].

All these possible developments, generalizations, and applications of the proposed methods and results are open questions at the present time and require new research in the future.