Fractional Maps and Fractional Attractors. Part II: Fractional Difference Caputo \alpha-Families of Maps

# Fractional Maps and Fractional Attractors. Part II: Fractional Difference Caputo α-Families of Maps

## Abstract

In this paper we extend the notion of an -family of maps to discrete systems defined by simple difference equations with the fractional Caputo difference operator. The equations considered are equivalent to maps with falling factorial-law memory which is asymptotically power-law memory. We introduce the fractional difference Universal, Standard, and Logistic -Families of Maps and propose to use them to study general properties of discrete nonlinear systems with asymptotically power-law memory.

1

footnoteinfo]The author acknowledges support from the Joseph Alexander Foundation, Yeshiva University.

yu,ci]Edelman M.

fractinal derivative \sepfractional difference \sepattractor \sepdiscrete map \seppower-law \sepmemory

## 1 Introduction

Systems with memory are common in biology, social sciences, physics, and engineering (see review Edelman (2014c)). The most frequently encountered type of memory in natural and engineering systems is power-law memory. This leads to the possibility of describing them by fractional differential equations, which have power-law kernels. Nonlinear integro-differential fractional equations are difficult to simulate numerically - this is why in Tarasov and Zaslavsky (2008) the authors introduced fractional maps, which are equivalent to fractional differential equations of nonlinear systems experiencing periodic delta function-kicks, and proposed to use them for the investigation of general properties of nonlinear fractional dynamical systems. Bifurcation diagrams in the fractional Logistic Map related to a scheme of numerical integration of fractional differential equations were considered by Stanislavsky (2006).

An adequate description of discrete natural systems with memory can be obtained by using fractional difference equations (see Miller and Ross (1989); Gray and Zhang (1988); Agarwal (2000); Atici and Eloe (2009); Anastassiou (2009); Chen et al. (2011); Wu et al. (2014); Wu and Baleanu (2014)). In Chen et al. (2011); Wu et al. (2014); Wu and Baleanu (2014) the authors demonstrated that in some cases fractional difference equations are equivalent to maps (which we will call fractional difference maps) with falling factorial-law memory, where falling factorial function is defined as

 t(α)=Γ(t+1)Γ(t+1−α). (1)

Falling factorial-law memory is asymptotically power-law memory -

 limt→∞Γ(t+1)Γ(t+1−α)tα=1,   α∈R, (2)

and we may expect that fractional difference maps have properties similar to the properties of fractional maps.

The goal of the present paper is to introduce fractional difference families of maps depending on memory and nonlinearity parameters consistent with the previous research of fractional maps (see Edelman (2014c); Tarasov and Zaslavsky (2008); Edelman and Tarasov (2009); Tarasov (2009a, b, 2011); Edelman (2011); Edelman and Taieb (2013); Edelman (2013a, b)) in order to prepare a background for an investigation of general properties of systems with asymptotically power-law memory.

In the next section (Sec. 2) we will remind the reader how the regular Universal, Standard, and Logistic Maps (see Chirikov (1979.); Lichtenberg and Lieberman (1992); Zaslavsky (2008); May (1976)) are generalized to obtain fractional Caputo -Families of Maps (FM). In Sec. 3 we’ll present some basics on fractional difference/sum operators, which will be used in Sec. 4 to derive the fractional difference Caputo Universal, Standard, and Logistic FMs. In Sec. 5 we’ll present some results on properties of fractional difference Caputo Standard FM.

## 2 Fractional α-Families of Maps

Fractional FM were introduced in Edelman (2013a), further investigated in Edelman (2013b), and reviewed in Edelman (2014c). The Universal FM was obtained by integrating the following equation:

 dαxdtα+GK(x(t−Δ))∞∑k=−∞δ(t−(k+ε))=0, (3)

where , , , , with the initial conditions corresponding to the type of fractional derivative to be used. is a nonlinear function which depends on the nonlinearity parameter . It is called Universal because integration of Eq. (3) in the case and produces the regular Universal Map (see Zaslavsky (2008)). In what follows the author considers Eq. (3) with the left-sided Caputo fractional derivative (see Samko et al. (1993); Kilbas et al. (2006); Podlubny (1999))

 C0Dαtx(t)=0IN−αt DNtx(t) =1Γ(N−α)∫t0DNτx(τ)dτ(t−τ)α−N+1,(N=⌈α⌉), (4)

where , , is a fractional integral, is the gamma function, and the initial conditions are

 (Dktx)(0+)=bk,   k=0,...,N−1. (5)

There are two reasons to restrict the consideration in this paper to the Caputo case (the Riemann-Liouville case won’t be considered): a) as in the case of fractional differential equations, in the case of fractional difference equations it is much easier to define initial conditions for Caputo difference equations than for Riemann-Liouville difference equations; b) the main goal of this work is to compare fractional and fractional difference maps, and the case of Caputo maps serves the purpose. Comparison of the Riemann-Liouville and Caputo Standard Maps was considered in Edelman (2011).

The problem Eqs. (3)–(5) is equivalent to the Volterra integral equation of the second kind () Kilbas et al. (2006)

 x(t)=N−1∑k=0bkk!tk −1Γ(α)∫t0dτGK(x(τ−Δ))(t−τ)1−α∞∑k=−∞δ(τ−(k+ε)). (6)

After the introduction the Caputo Universal FM can be written as (see Tarasov (2011))

 x(s)n+1=N−s−1∑k=0x(k+s)0k!(n+1)k −1Γ(α−s)n∑k=0GK(xk)(n−k+1)α−s−1, (7)

where and .

In the case and with Eq. (7) produces the well–known Standard Map (see Chirikov (1979.)), which on a torus can be written as

 pn+1=pn−Ksin(xn),   (mod 2π), (8)
 xn+1=xn+pn+1,   (mod 2π). (9)

This is why the Caputo Universal FM Eq. (7) with

 GK(x)=Ksin(x) (10)

is called the Caputo Standard FM:

 x(s)n+1=N−s−1∑k=0x(k+s)0k!(n+1)k −KΓ(α−s)n∑k=0sin(xk)(n−k+1)α−s−1, (11)

where .

In the case and Eq. (7) produces the well–known Logistic Map (see May (1976))

 xn+1=Kxn(1−xn). (12)

This is why the Caputo Universal FM Eq. (7) with

 GK(x)=GLK(x)=x−Kx(1−x) (13)

is called the Caputo Logistic FM:

 x(s)n+1=N−s−1∑k=0x(k+s)0k!(n+1)k −1Γ(α−s)n∑k=0x−Kx(1−x)(n−k+1)1+s−α, (14)

where .

The Caputo Standard and Logistic FMs were investigated in detail in Edelman (2014c, 2013a, 2013b) for the case which is important in applications.

• For the Caputo Standard and Logistic FMs are identically zeros: .

• For the Caputo Standard FM is

 xn=x0−KΓ(α)n−1∑k=0sin(xk)(n−k)1−α,  (mod 2π). (15)

and the Caputo Logistic FM is

 xn=x0−1Γ(α)n−1∑k=0x−Kx(1−x)(n−k)1−α. (16)
• For the 1D Standard Map is the Circle Map with zero driving phase

 xn+1=xn−Ksin(xn),    (mod 2π). (17)

and the 1D Logistic FM is the Logistic Map Eq. (12).

• For the Caputo Standard FM is

 pn+1=pn−KΓ(α−1)[n−1∑i=0V2α(n−i+1)sin(xi) +sin(xn)],  (mod 2π), (18) xn+1=xn+p0−KΓ(α)n∑i=0V1α(n−i+1)sin(xi), (mod 2π), (19)

where and the Caputo Logistic FM is

 xn+1=x0+p(n+1)k−1Γ(α)n∑k=0[xk− Kxk(1−xk)](n−k+1)α−1, (20) pn+1=p0−1Γ(α−1)n∑k=0[xk− Kxk(1−xk)](n−k+1)α−2. (21)
• For the Caputo Standard Map is the regular Standard Map as in Eqs. (8) and (9) above. The 2D Logistic Map is

 pn+1=pn+Kxn(1−xn)−xn, (22) xn+1=xn+pn+1. (23)

## 3 Fractional Difference/Sum Operators

In this paper we will adopt the definition of the fractional sum ()/difference () operator introduced in Miller and Ross (1989) as

 aΔ−αtf(t)=1Γ(α)t−α∑s=a(t−s−1)(α−1)f(s). (24)

Here is defined on and on , where , and falling factorial is defined by Eq. (1). As Miller and Ross noticed, their way to introduce the discrete fractional sum operator based on the Green’s function approach is not the only way to do so. In Gray and Zhang (1988) the authors defined the discrete fractional sum operator generalizing the -fold summation formula in a way similar to the way in which the fractional Riemann–Liouville integral is defined in fractional calculus by extending the Cauchy -fold integral formula to the real variables. They mentioned the following theorem but didn’t present a proof. {thm} For

 aΔ−ntf(t)=1(n−1)!t−n∑s=a(t−s−1)(n−1)f(s) =t−n∑s0=as0∑s1=a...sn−2∑sn−1=af(sn−1), (25)

where , are the summation variables. {pf} Indeed, this formula is obviously true for . Let’s assume that Eq. (25) is true for :

 aΔ−(n−1)tf(t)=t−(n−1)∑s=aC(t−s−1,n−2)f(s) =t−(n−1)∑s1=as1∑s2=a...sn−2∑sn−1=af(sn−1), (26)

where is the number of -combinations from a given set of elements. Then, for Eq. (26) gives

 s0∑s1=aC(s0−s1+n−2,n−2)f(s1) =s0∑s1=as1∑s2=a...sn−2∑sn−1=af(sn−1). (27)

Now Eq. (25) can be obtained from

 t−n∑s0=as0∑s1=a...sn−2∑sn−1=af(sn−1) =t−n∑s0=as0∑s1=aC(s0−s1+n−2,n−2)f(s1) =t−n∑s1=af(s1)t−n∑s0=s1C(s0−s1+n−2,n−2) =t−n∑s1=aC(t−s1−1,n−1)f(s1)=aΔ−ntf(t). (28)

Here we used the identity

 t−n∑s0=s1C(s0−s1+n−2,n−2)=C(t−s1−1,n−1), (29)

which is true for and can be proven by induction for any

 t+1−n∑s0=s1C(s0−s1+n−2,n−2)=C(t−s1−1,n−2) +C(t−s1−1,n−1)=C(t−s1,n−1). (30)

This ends the proof.

As we see, two different approaches are consistent with the definition of the fractioanal sum operator given by Miller and Ross (see also Atici and Eloe (2009)). For and Anastassiou (2009) defined the fractional (left) Caputo-like difference operator as

 CaΔαtx(t)=aΔ−(m−α)tΔmx(t) =1Γ(m−α)t−(m−α)∑s=a(t−s−1)(m−α−1)Δmx(s), (31)

where is the -th power of the forward difference operator defined as . The proof (see Miller and Ross (1989) p.146) that in the limit approaches the identity operator can be easily extended to the operator. In this case the definition Eq. (31) can be extended to all real with for . Then, the Anastassiou’s fractional Taylor difference formula Anastassiou (2009)

 x(t)=m−1∑k=0(t−a)(k)k!Δkx(a) +1Γ(α)t−α∑s=a+m−α(t−s−1)(α−1)CaΔαtx(t), (32)

where is defined on , , and for is valid for any real and for integer is identical to the integer discrete Taylor’s formula (see p.28 in Agarwal (2000))

 x(t)=m−1∑k=0(t−a)(k)k!Δkx(a) +1(m−1)!t−m∑s=a(t−s−1)(m−1)Δmx(t). (33)

As it was noticed in Wu et al. (2014) and Wu and Baleanu (2014), Lemma 2.4 from Chen et al. (2011) on the equivalency of the fractional Caputo-like difference and sum equations can be extended to all real and formulated as follows: {thm} The Caputo-like difference equation

 CaΔαtx(t)=f(t+α−1,x(t+α−1) (34)

with the initial conditions

 Δkx(a)=ck,   k=0,1,...,m−1,   m=⌈α⌉ (35)

is equivalent to the fractional sum equation

 x(t)=m−1∑k=0(t−a)(k)k!Δkx(a)+1Γ(α) (36) ×t−α∑s=a+m−α(t−s−1)(α−1)f(s+α−1,x(s+α−1)),

where . Here we should notice that the authors of Wu et al. (2014) and Wu and Baleanu (2014) didn’t consider the Caputo difference operator with integer . As a result, Theorem 3 is not valid for integer values of with their definition .

This theorem in the limiting sense can be extended to all real . Indeed, taking into account that , Eq. (34) for turns into

 x(t)=f(t−1,x(t−1). (37)

For the first sum on the right in Eq. (36) disappears and in the second sum the only remaining term with in the limit turns into .

## 4 Fractional Difference α-Families of Maps

In the following we assume that is a nonlinear function with the nonlinearity parameter and adopt the Miller and Ross proposition to let . Now, with , Theorem 3 can be formulated as {thm} For , the Caputo-like difference equation

 C0Δαtx(t)=−GK(x(t+α−1)), (38)

where , with the initial conditions

 Δkx(0)=ck,   k=0,1,...,m−1,   m=⌈α⌉ (39)

is equivalent to the map with falling factorial-law memory

 xn+1=m−1∑k=0Δkx(0)k!(n+1)(k) −1Γ(α)n+1−m∑s=0(n−s−m+α)(α−1)GK(xs+m−1), (40)

where which we will call the fractional difference Caputo Universal -Family of Maps. The fractional difference Caputo Universal FM is similar to the general form of the Caputo Universal FM Eq. (7). Both of them can be written as

 xn=x0+m−1∑k=1pk(0)k!n(k) −1Γ(α)n−1∑k=MWα(n−k)GK(xk), (41)

where are the initial value of momenta defined as for fractional maps and as for fractional difference maps; for fractional maps and for fractional difference maps; for fractional maps and for the fractional difference maps. for fractional maps and for fractional difference maps. Asymptotically, both expressions for coincide because of Eq. (2).

### 4.1 Fractional Difference Universal αFm

Let’s consider the case . Then the difference Eq. (34) produces

 Δ2xn=−GK(xn+1) (42)

and the equivalent sum equation is

 xn+1=x0+Δx0(n+1)−n−1∑s=0(n−s)GK(xs+1). (43)

After the introduction with the assumption the map equations indeed can be written as the well–known 2D Universal Map

 pn+1=pn−KG(xn), (44)
 xn+1=xn+pn+1, (45)

which for produces the Standard Map Eqs. (8) and (9). In the rest of this paper we’ll call Eq. (40) with the fractional difference Caputo Standard -Family of Maps.

In the case the fractional difference Caputo Universal FM is

 xn+1=xn−GK(xn), (46)

which produces the Logistic Map if . In the rest of this paper we’ll call Eq. (40) with the fractional difference Caputo Logistic -Family of Maps.

### 4.2 α=0 Difference Caputo Standard and Logistic αFMs

• In the case the 0D Standard Map turns into the Sine Map (see, e.g., Lalescu (2010))

 xn+1=−Ksin(xn),   (mod 2π). (47)
• The 0D Logistic Map is

 xn+1=−xn+Kxn(1−xn). (48)

### 4.3 0<α<1 Fractional Difference Caputo Standard and Logistic αFMs

• For the fractional difference Standard Map is

 xn+1=x0 (49) −KΓ(α)n∑s=0Γ(n−s+α)Γ(n−s+1)sin(xs),   (mod 2π),

which after the -shift of the independent variable coincides with the “fractional sine map” proposed in Wu et al. (2014).

• The fractional difference Logistic Map can be writen as

 xn+1=x0 (50) −1Γ(α)n∑s=0Γ(n−s+α)Γ(n−s+1)[xs−Kxs(1−xs)].

The fractional Logistic Map introduced in Wu and Baleanu (2014) does not converge to the Logistic map in the case .

### 4.4 α=1 Difference Caputo Standard and Logistic αFMs

• The difference Caputo Standard FM is identical to the Circle Map with zero driven phase Eq. (17). The map considered in Wu et al. (2014)

 xn+1=xn+Ksin(xn),    (mod 2π) (51)

is obtained from this map by the substitution .

• The Difference Caputo Logistic FM is the regular Logistic Map.

### 4.5 1<α<2 Fractional Difference Caputo Standard and Logistic αFMs

• For the fractional difference Standard Map is

 xn+1=x0+Δx0(n+1)−KΓ(α) (52) ×n−1∑s=0Γ(n−s+α−1)Γ(n−s)sin(xs+1),  (mod 2π).
• The fractional difference Logistic Map is

 xn+1=x0+Δx0(n+1)−1Γ(α) (53) ×n−1∑s=0Γ(n−s+α−1)Γ(n−s)[xs+1−Kxs+1(1−xs+1)].

Let’s introduce ; then these maps can be written as 2D maps with memory:

• The fractional difference Standard Map is

 pn=p1−KΓ(α−1) (54) ×n∑s=2Γ(n−s+α−1)Γ(n−s+1)sin(xs−1),  (mod 2π), xn=xn−1+pn,  (mod 2π),  n≥1, (55)

which in the case is identical to the ”fractional standard map” introduced in Wu et al. (2014) (Eq. (18) with there).

• The fractional difference Logistic Map is

 pn=p1−KΓ(α−1) (56) ×n∑s=2Γ(n−s+α−1)Γ(n−s+1)[xs−1−Kxs−1(1−xs−1)], xn=xn−1+pn,  n≥1. (57)

### 4.6 α=2 Difference Caputo Standard and Logistic αFMs

• The difference Caputo Standard FM is the regular Standard Map Eqs. (8) and (9).

• From Eqs. (44) and (45) the 2D difference Caputo Logistic FM is

 pn+1=pn−xn+Kxn(1−xn), (58)
 xn+1=xn+pn+1, (59)

which is identical to the 2D Logistic Map Eqs. (22) and (23).

## 5 Conclusion

As we saw in Sec. 3, the fractional difference operator is a natural extension of the difference operator. The simplest fractional difference equations (of the Eq. (34) type), where the fractional difference on the left side is equal to a nonlinear function on the right side, are equivalent to maps with falling factorial-law (asymptotically power-law) memory Eq. (36). Systems with power-law memory play an important role in nature (see Edelman (2014c)) and investigation of their general properties is important for understanding behavior of natural systems.

Properties of the fractional difference Caputo Standard FM were investigated in detail in Edelman (2014a) (see also Sec. 3 in Edelman (2014b). Qualitatively, properties of the fractional difference and fractional maps (maps with falling factorial- and power-law memory) are similar. The similarity reveals itself in the dependence of systems’ properties on the memory () and nonlinearity () parameters (bifurcation diagrams, see Figs. 12, and 3), power-law convergence to attractors, non-uniqueness of solutions (intersection of trajectories and overlapping of attractors), and cascade of bifurcations and intermittent cascade of bifurcations type behaviors (see Figs. 4 and 5).

The differences of the properties of the falling factorial-law memory maps from the power-law memory maps are the results of the differences in weights of the recent (with ) values of the maps’ variables at the time instants in the definition of the present values at time and are significant when (especially when ), see Fig. 1.

{ack}

The author acknowledges support from the Joseph Alexander Foundation, Yeshiva University. The author expresses his gratitude to E. Hameiri, H. Weitzner, and G. Ben Arous for the opportunity to complete this work at the Courant Institute and to V. Donnelly for technical help.

1. thanks: [

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